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'OSMANIA UNIVERSITY LIBRARYCall No. '5f* ? #42 & Accession No. ft^ %Author #e*J /'*"'This book should be returned on or before the date last marked below,

AHISTORYOFGREEK MATHEMATICSVOLUME II

1OFGREEK MATHEMATICSHYSIR THOMAS HÂ*K.C.B., K.C.V.O., K.R.S.SC.J). C'AMH. ; HON. D.SC. OXFORDHONORARY FKLf.OW (KOIIMKHLV I'ELLOw) Ol TRINITY (OLIFOK,*. . , All independent world,Treated out otpure intelligence.'WORDSWORTH.VOLUME HFROMAltlSTAU(1lI(lS TO DIOrilANTUSO X F R i)AT THE CLARENDON PRESS

OXFORD UNIVERSITY PRESSLondon Edinburgh Glasgow CopenhagenNew York Toronto Melbourne (ape TownBombay Caleutta Madras ShanghaiHUMPHREY M1LFORI)Publisher to the University

CONTENTS OF VOLIIXII. ARISTARCHUS OF SAMOS PAGES 1-15XIII. AKCH1MEDKS 16-101. . .Traditions(() Astronomy 17-lX(3) Mechanics 18Summary of main achievements ... . 19- 20Character of treatises 20 22List of works still extant 22-23Traces of lost works 23 25The text of Archimedes 25-27Contents of The Method 27-34On the fyhere and IT 34-50Cylinder.I,Cubic equation arising out of 1 1. 4 43-4(5(ij Archimedes's own 45-46(ii) Dionysodorus's solution 40(lii)DioeWs solution of original problem. 47-49Measurement of a Circle ....... 50 56On Cvnoidt* and Hplietoid* 5664On ^jiirals ......... 64-75On I'htn** KtriangleKratosthenes 104-109Measurement of the Karth 106 1 OSXlV f . CONICSUCTIONS. APOLLONIU8 OF PKRUA .A. HISTORY OF CONICS UP TO APOLLOMUS .. 110 196. 110 126Discoveiy of the ronie sections by Menaeehmus . 110-111Menaeehmus\s probable procedure. . . 111-116Woiks by Aristaeus and Kuclid . . . 116-1174Solid loci 'and 4 solid . . .problems' 117-118Aristaeus's Solid Loci US 119Focus-directrix property known to Kuclid . . 119Proof from Pappus 120121Propositions included in Kuclid's Conies . . 121-12*2Conic sections in Archimedes .... 122-126

179viCONTENTSXIV. CONTINUED.B. APOLLONIUS OF PERGA .... PAGES 126-196The text of the Conies 126-128Apollonius's own account of the Con lets . . . 128-133Extent of claim to originality 182-133Great generality of treatment .... 133Analysis of the Con ics 133-175Book I 133-148Conies obtained in the most general way fromoblique cone 134-138'* *New names, parabola ', ellipse ', hyperbola ' . 138-139Fundamental properties equivalent to Cartesianequations 139-141Transition to new diameter and tangent at itsextremity 141-147First appearance of principal axrs . . .147-148Hook 11 148-15013ookJIl 150 lf>7Book IV 157-158BookV 158 167Normals us maxima and minima 159-163Number of normals from a point .... 163 164Propositions leading immediately to detei munitionof pvolutt' of conic ...... 164-166Construction of normals 166 167Hook VI 167-168Hook Vll 168-174Other works by Apollonius 175-194(

CONTENTSviiUeminus PAGES 222-234Attempt to prove the Parallel-Postulate . . . 227-230On Meteoroloyica of Posidonius 231-232Introduction to the Phaeuomena attributed to deminus 232-234XVI. SOME HANDBOOKS 235-244Cleomedes, De motu circular! 235-238Nicomachus ... 238Theon of Smyrna, Kjrpositio rerutn muthemu tit-arum adteyenduiH Platonem ut Hi it in 238-244XVII. TRIGONOMETRY: HIPPARCIIUS, M EN E LA US, PTO-LEMY 245-297Theodosius 245 216Works by Theodosius ....... 246Contents of the *S)>//'/w' 297The M

viiiCONTENTSXVIII. MENSURATION: HKRON OF ALEXANDRIA. PAGES 298-354Controversies as to Heron's date 298-306Character of works 307-308List of treatises 308-310Geometry(a) Commentary on Euclid's Elements . . . 310 314() The Definitions 314 316Mensuration 316-344The Metrica, Geometrica, Stereomdrica, Geoduexia,Mensure 316 320Contents of the Mt tried 320-344Book I. Measurement of areas 320331(a) Area of scalene triangle .... 320-321Proof of formula A - J{s(8-)(8-b (s-c)\ 321-323(/3)Method of approximating to the square rootof a non-square number .... 323-326(7) Quadrilaterals 326(6) Regular polygons with 3, 4, 5, 6, 7, 8, *J, 10,11, or 12 sides 326-329(0 The circle :',29(f) Segment of a circle 330-331(17) Kllipse,parabolic segment, surface of cylinder,right cone, sphere and segment of sphere . 331Book II. Measurement of volumes . . . 331-335(a)Cone, cylinder, parallelepipedf prism), pyramidand frustum ...... 332(#) Wedge-shaped solid (#o>/n'

CONTENTSixPAGES 357-439() Character of the work ;wide . . .range 357-358List of authors mentioned 358-360(/:$)(y) Translations and editions ..... ......360-361(6) Summary of contents 361-439The Synayoye or Collection ....Book 111. Section (1). On the problem of the twomean proportionals. . . . . .361 362Section (2). The theory of means . . .363-365Section (:5). The * Paradoxes' of Krycinus . . 365-368Section (4). The inscribing of the five regularsolids in a sphere ...... 308-369Book IV. Section (1). Kxtension of theorem ofPythagoras 369 371Section (2). On circles inscribed in the up0rj\ns(' shoemaker's knife') 371 377Sections (*), (4). Methods of squaring the circleand trisecting any angle ..... 377-386(n) The Archimedean spiral .....377 379l#) The conchoid of Nicomedes .... 379(y) The Vita(?9-atri.r ... 379 -382(h) Digression: ;i. spiral on a sphere 382 3S5. . .Triscction (or division in any ratio) of any angle 385-386Section (5). Solution of the ufOcy of Archimedes,On Sj)tfft/tt,Pro 1. S.1by means of conies . . 386-388Book V. Pief.ii-o on the sagacity of Hees . . 389-390-Section (1). Isoperimetry after Zeiiodoius . . 390 39.Section (2). Compaiisoii of volumes of solids havin

x..CONTENTSXIX. CONTINUED.Book VIII (continued)The inclined plane .... PAGES 433 434Construction of a conic through five . .points 434-437Given two conjugate diameters of an ellipse, to findthe axes 437 438Problem of seven hexagons in a circle . . . 438 439Construction of toothed wheels and indented screws 439XX. ALGEBRA: DIOPHANTUS OF ALEXANDRIA .. 410 517Beginnings learnt from Egypt ...... 440*Hau '-calculations 440 441Arithmetical epigrams in the Greek Anthology. . 441 443Indetei urinate equations of first degree .... 443Indeterminate equations of second degree beforo Diophantus443 444Indeterminate equations in Heronian collections . . 444 447Numerical solution of . . .quadratic equations 418Works of Diophantus 44S 450The Arithmetics 449 514The seven lost Books and their . . .place 449 450Relation of ' Porisms 1 to Arithmetica .... 451 452Commentators from Ilypatia downwaids . . . 453Translations and editions ...... 453 455Notation and definitions 455461Sign for unknown =( x) and its . . .origin 450 457Signs for powers of unknown &c 458 459The sign (/1\) for minus and its meaning 459-4GO. . .The methods of Diophantus 462-479I.Diophantus's treatment of . . .equations 462 476(A)Determinate equations(1) Pure determinate equations. . . 462 463(2) Mixed quadratic equations .... 463 465(3) Simultaneousequationsinvolving quadratics 465(4) Cubic equation 465(B) Indeterminate equations(a) Indeterminate equations of the second degree 466-473(1) Single equation 466 468(2) Double equation 468 4731. Double equations of first .degree 469 4722. Double equations of second degree 472-473th) Indeterminate equations of degree higherthan second 473 476(1) Single equations 473-475(2) Double equations 475 476II. Method of limits 476 477III. Method of approximation to limits . , . . 477 479Porisms and propositions in the Theory of Numbers . 479 4H4(a) Theorems on tins composition of numbers as thesum of two squares 481 483(3) On numbers which are the sum of three .squares 483(y) Composition of numbers as the sum of four squares 483-484Conspectus of Arithmetica, with typical solutions . . 484-514The treatise on Polygonal Numbers 514-517

CONTENTSxiXXI. COMMENTATORS AND BYZANTINES . . PAGESAPPENDIX.518 555Sorenus 519 526(a) On Mr Section of a Cylinder 519-522(ft) On tJir Serf inn of a Cone 522-526Theoii of Alexandria 526528Commentary on the Si/nta^'fn ..... 526 527Edition of Euclid's Elements 527528Edition of the Oj/fics of Euclid 528.Ilypatia 52K 529Porphyry. lamblichus ....... 529Proclus 529 537Commentary on Euclid, Hook I . . . . . 530 535(a) Sources of the Commentary .... 530 532(tt) Character of the . . .Commentary .532535Ift/potypotti* of Astronomical . . .Hypothws 535-536Commentary on the Republic ..... 536 537Marinus of Neapolis 537-538Domninus of Larissa ..... 538.Simplicius ......... 538 540Antheming of Trail es ....... 541-543On hurnhn/-niirront . . . . . . .541 543Extracts from Eudemus ...... 539Eutocius ... 540 541The Papyrus of Akhimm . . . . . . .513 545G< ortttcttiff of Heron ' the Younger '. . . . 545Michael IVellux .....lieorgius Pachymcres ....... 546Maximus Planndes . . .Extinction of the square root .....-. 545 546. . . . . 546 549547-549Two problems ........ 549Manuel Moschopoulos 549-550Nii-ol.is Rhabdas 550 554Rule for apiiroximalin^ to square root of a non-squarenumber ......... 553 554loanne* Podiasinuis ........ 554Barlaam . . . . . . . . . .554 555Isaac Ar^yrus ......... 555On Archimedes*:* proof ot the subtungent-property556-561of a spiral ..........INDEX OF GREEK WORDS 563-569ENGLISH INDEX 570 586

XIIAB1STAUCHUS OF SAMOSHISTORIANS of mathematics have, as a rule, given too littleattention to Aristarchus of Samos. Tlie reason is no doubtthat he was an astronomer, and therefore itmight be supposedthat his work would have no sufficient interest for the mathematician.The Greeks knew better; they called him Aristarehus' the mathematician ', to distinguish him from the hostof other Aristarchuses ;he is also included by Vitruviusamong the i'ew great men who possessed an equally profoundknowledge of all branches of science, geometry, astronomy,music, &c.Men of this type- are rare, men such as were, in limes past,'Aristarchus of Samos, Philolaus and Arehytas of Tarentum,Apollonius of Perga, Eratosthenes of Cyreiie, Archimedes andScopinas of Syracuse, who left to posterity many mechanicaland gnomonic* appliances which they invented and explainedon mathematical (lit. 'numerical') principles.' 1That Aristarchus was a very capable geometerisproved byhis extant work On the shcs

2 ARISTARCHUS OF SAMOSstamping the air with impressions like themselves, as it were ',that ' colours in darkness have no colouring ',and that c lightis the colour impinging on a substratum '.Two facts enable us to fix Aristarchus's date approximately.In 281/280B.C. he made an observation of the summersolstice ;and a book of his, presently to be mentioned, waspublished before the date of Archimedes's Psammites or Sandreckoner,a work written before 216 B.C. Aristarchus, therefore,probably lived circa 310-230 B.C., that is, he was olderthan Archimedes by about 25 years.To Aristarchus belongs the high honour of having beenthe first to formulate the Copernican hypothesis, which wasthen abandoned again until it was revived by Copernicushimself. His claim to the title of ' the ancient Copernicus'still, in my opinion, quite unshaken, notwithstanding the ingeniousand elaborate arguments brought forward by Schiaparellito prove that it was Heraclides of Pontus who firstconceived the heliocentric idea. Heraclides is (along with oneEcphantus, a Pythagorean) credited with having been the iirstto hold that the earth revolves about its own axis every 24hours, and he was the first to discover that Mercury and Venusrevolve, like satellites, about the sun. But though this provesthat Heraclides came near, if he did not actually reach, thehypothesis of Tycho Brahe, according to which the earth wasin the centre and the rest of the system, the sun with theplanets revolving round it, revolved round the earth, it doesnot suggest that he moved the earth away from the centre.The contraryis indeed stated by Acitius, who says that 'Heraclidesand Ecphantus make the earth move, not in the sense oftranslation, but by way of turning on an axle, like a wheel,from west to east, about its own centre V None of thechampions of Heraclides have been able to moot this positivestatement. But we have conclusive evidence in favour of theclaim of Aristarchus ; indeed, ancient testimonyis unanimouson the point. Not only does Plutarch tell us that Cleanthesheld that Aristarchus ought to be indicted for the impiety ofputting the Hearth of the Universe in motion ' 2we have the';best possible testimony in the precise statement of a great12Aet. iii. 13. 3, Vors. i 3 , p. 341. 8.Plutarch, De facie in orbe Iwiae, c. 6, pp. 922 F-923 A.is

ARISTARCHUS OF SAMOS 3contemporary, Archimedes. In the Sand-reckoner Archimedeshas this passage.1You [King Celon] are aware that ''universe" is the namethe centre of whichgiven l>y most astronomers to the sphereis the cojitre of the earth, while its radius is equal to thestraight line between the centre of the sun and the centre ofthe earth. This is the common account, as you have heardfrom astronomers. But Aristarchus brought out a book connintingof certain hypotheses, wherein it appears, as a consequenceof the assumptions made, that the universe ismanytimes greater than the " universe "just mentioned. His hypothesesare that the fixed stars and the HUH remain unmoved,that the, earth revolves about the sun in the circumference of acircle, the sun lying in the middle, of the orhit, and that thesituated about the same centre as thesphere of the fixed stars,sun, is so great that the circle in which he supposes the earthto revolve bears such a proportion to the distance of the fixedstars as the centre of the sphere bears to its surface/(The last statement is a variation of a traditional phrase, forwhich there are many parallels (cf. Aristarchus' s Hypothesis 2'that the earth isin the relation of a point and centre to thesphere in which the moon moves'), and is a method of sayingthat the * universe* is infinitely great in relation not merely tothe size of the sun but even to the orbit of the earth in itsrevolution about it the ;assumption was necessary to Aristarchusin order that he might not have to take account ofparallax.)Plutarch, in the passage referred to above, also makes itclear that Aristarchus followed Heraclides in attributing tothe earth the daily rotation about its axis. The bold hypothesisof Aristarchus found few adherents. Seleucus, ofSeleucia on the Tigris, is the only convinced supporter of it ofwhom we hear, and it was speedily abandoned altogether,mainly owing to the groat authority of Hipparchus. Nor'dowe find any trace of the heliocentric hypothesis in Aristarchus'sextant work 0)i the sizes and distances of theHun and Moon. This ispresumably because that work waswritten before the hypothesis was formulated in the bookreferred to by Archimedes. The geometryof the treatiseis, however, unaffected by the difference between the hypotheses.B 2

4 ARISTARCHUS QF SAMOSArchimedes also says that it was Aristarchus who discoveredthat the apparent angular diameter of the sun is aboutl/720th part of the zodiac circle, that is to say, half a degree.We do not know how he arrived at this pretty accurate figure :but, as he is credited with the invention of the lie cr/ca^T/, mayhave used this instrument for the purpose. But here againthe discovery must apparently have been later than the treatiseOn sizes ami distances, for the value of the subtendedangle is there assumed to be 2 (Hypothesis How Aristarchuscame to assume a value so excessive is uncertain.6).Asthe mathematics of his treatise isnot dependent on the actualvalue taken, 2may have been assumed merely by way ofillustration ;or it may have been a guess at the apparentdiameter made before he had thought of attempting to measureit. Aristarchus assumed that the angular diameters ofthe sun and moon at the centre of the earth are equal.The method of the treatise* depends on the, just observation,which is Aristarchus's third ' hypothesis that when ' the moon',appears to us halved, the great circle which divides the 1 ,darkand the bright portions of the moon is in the, direction of oureye' the effect of this (since the moon receives its light from;the sun), is that at the time of the dichotomy the centres ofthe sun, moon and earth form a triangle right-angled at thecentre of the moon. Two other assumptions were necessary:first, an estimate of the size of the angle of the latter triangle,at the centre of the earth at the moment of dichotomy : thisAristarchus assumed (Hypothesis 4)to be, Mess than a quadrantby one-thirtieth of a quadrant', i.e. 87, again an inaccurateestimate, the true value being 89 50'; secondly, an estimateof the breadth of the earth's shadow where the moontraverses it: this he assumed to be 'the breadth of twomoons' (Hypothesis 5).The inaccuracy of the assumptions docs not, however, detractfrom the mathematical interest of the succeeding investigation.Here we find the logical sequence of propositions and the absoluterigour of demonstration characteristic of Greek geometry ;the only remaining drawback would be, the practical difficultyof determining the exact moment when the moon 'appears tous halved'. The form and style, of the book are thoroughlyclassical, as befits the period between Euclid and Archimedes ;

ARISTARCHUS OF SAMOS 5the Greek is even remarkably attractive. The content fromthe mathematical point of view is no less interesting, for wehave here the first specimen extant of pure geometry usedwith a trigonometrical object, in which respect it is a sort offorerunner of Archimedcs's Measurement of a Circle.Aristarchusdoes not actually evaluate the trigonometrical ratioson which the ratios of the sixes and distances to be obtaineddepend ; he finds limits between which they lie, and that bymeans of certain propositions which he assumes without proof,and which therefore must have been generally known tomathematicians of his day. These propositions are the equivalentsof the statements that,(1)if a is what we call the circular measure of an angleand a is less than \ TT, then the ratio sin y/oi decreases, and theratio tan a/cx increase*, as a increases from to TT ;(2) if /3 lie the circular measure of another angle less than\ TT, and a>/3, thensin a 3 tan asin/^

6 ARISTARCHUS OF SAMOS(3)that the diameter of the sun has to the diameter of theearth a ratio greater than 19:3, but less than 43 : 6.The propositions containing these results are Props. 7, 9and 15.Prop. 1 is preliminary, proving that two equal spheres arecomprehended by one cylinder and two unequal spheres by,one cone with its vertex in the direction of the lesser sphere,and the cylinder or cone touches the spheres in circles atright angles to the line of centres. Prop. 2 proves that, ifa sphere be illuminated by another sphere larger than itself,the illuminated portion is greater than a hemisphere. Prop. 3shows that the circle in the moon which divides the dark fromthe bright portionis least when the cone comprehending thesun and the moon has its vertex at our eye. The ' dividingcircle ',as we shall call it for short, which was in Hypothesis 3spoken of as a great circle, isproved in Prop. 4 to be, nota great circle, but a small circle not perceptibly differentfrom a great circle. The proof is typical and is worth givingalong with that of some connected propositions (11 and 12).B isthe centre of the moon, A that of the earth, CD thethe parallelBA meets the circular section of thediameter of the ' dividing circle in the moon EF',diameter in the moon.moon through A and EF in G, and CD in L. Gil, GKare arcs each of which is equal to half the arc CE. ByHypothesis 6 the angle CAD is ' one-fifteenth of a sign'= 2,and the angle BAG = 1.Now, says Aristarchus,1:45[> tan 1: tan 45]>BC:CA,and, a fortiori,BG:BA or BG:BA< 1:45;that is, BG

NowwhenceARISTARCHUS OF SAMOSBIl :11A[= sin HAB :Fsin 1IBA\> LHAE-.LILEA,L11AE< &LURA,Dand (taking the doubles)Z IIAK < ^ Z IIBK.But Z HBK = Z EBG = ^ /* (where U is a right angle) ;therefore Z //î/f < ^ R.But 'a magnitude (arc UK) seen under such an angle isimperceptible to our eye ' ;therefore, a fortiori, the arcs CE, DF are severally imperceptibleto our eye.Q. E. I).An easy deduction from the same figure is Prop. 12, whichshows that the ratio of CD, the diameter of the c dividingcircle', to EF, the diameter of the moon, is < 1 but > f{J .We have Z EBC = Z BAG = 1;Ttherefore(arc AC = f) / (arc AY?),and accordingly (arc CO):(arc fr'A')= 81) : 90.Doubling the arcs, we have(arc G(!D) : (arc EGF) = 89 : 90.But CD : EF > (arc:CGD) (arc EGF)[equivalent to sin a /sin > /3 a//3, where /.CBD = 2 a,and 2 /S= TT];therefore CW : EF [= cos 1] > 89 :1)0,while obviously CD :EF< 1.Prop. 11 finds limits to the ratio EF:BA (the ratio of thediameter of the moon to the distance of its centre fromthe centre of the earth) ;the ratio is < 2 : 45 but > 1 : 30.

8 AKISTARCHUS OF SAMOSThe first part follows from the relation found in Prop. 4,namely BC:BA < 1 :45,forEF=2BC.The second part requires the use of the circle drawn withcentre A and radius AC. This circle is that on which theends of the diameter of the ' dividing circle ' move as the moonmoves in a circle about the earth. If r is the radius ACof this circle, a chord in it equalto r subtends at the centreA an angle of U or 60; and the arc CD subtends at Aan angle of 2.But(arc subtended by CD) : (arc subtended by r)< CD:r,or 2 : GO < CD : r ;that is, CD:CA > 1 :30.And, by similar triangles,GL:GA = OB:A, or GD:GA = 2CB-.HA = FKiBA.Therefore FE:BA > 1 : 30.The proposition which is of the greatest interest 011 thewhole isProp. 7, to the effect that the distance of the xiinfrom the earth is greater titan 18 tidies, but tess than kJOtimes, the distance of the motiii from the earth. This resultrepresents a great improvement on all previous attempts toestimate the relative distances. The first speculation on thesubject was that of Aiiaximander (circa 61 1-545 B.C.),whoseems to have made the distances of the sun and moon fromthe earth to be in the ratio 3 : 2. Eudoxus, according toArchimedes, made the diameter of the sun 9 times that ofthe moon, and Phidias, Archimedes's father, 12 times; and,assuming that the angular diameters of the; two bodies areequal, the ratio of their distances would be the same.Aristarchtis's proof is shortly as follows. A is the centre ofthe sun, 7ithat of the earth, and C that of the moon at themoment of dichotomy, so that the angle ACIi is right. AllEFis a square, and AE isa quadrant of the sun's circular orbit.Join BF, and bisect the angle FJiK by EG, so thatZG'BA'= \R or 22 J.

ARISTARCHUS OF SAMOSI. Now, by Hypothesis 4, Z ABC = 87,so that LUBE LBAC 3;therefore Z GBE : L UBE =%R: -/ R= 15:2,Aso that(1E:11E[ = Ian (,'HE:tnn 1UM] > Z HUE: L HBE> 15 :2.(1)The ratio which lias lo bo proved > 18:1 is AB:B(J orFK-.Kll.Now FG:GE = FB:RE,whence F({* : (!E~ = F1P : BE*and FG :GE = V2 :1= 2:1,> 7:5(this is the approximation to \/2 mentioned by Plato andknown to the Pythagoreans).

10 ARISTARCHUS OF SAMOSTherefore FE :EG> 1 2 : 5 or 36 : 1 5.Compounding this with (1) above, we haveFE 36:2 or 18:1.II. To prove BA < 20 EC.Let BH meet the circle AE in A and draw DK parallelto EB. Circumscribe a circle about the triangle BK D, andlet the chord EL be equal to the radius (/)Now Z SDK = Z DBE = & R>BK = ^ (circumference of circle).Thus (arc BK) : (arc BL) = &:| ,= 1 : 10.And (arc BK :) (arc BL) < BK : rso that arcof the circle.[this is equivalent to a//J < sin a/sin j8,where a < j8 < ^TT]*so thatandButr < lO^Jf,7?D < 20 BK.BD:J3K = AB:BC',therefore AB < 20 ^6 f .Q.E. 1).The remaining results obtained in the treatise can bevisualized by means of the three figures annexed, which havereference to the positions of the sun (centre A), the earth(centre B) and the 1110011 (centre C) during an eclipse. Fig. Ishows the middle position of the moon relatively to the earth'sshadow which is bounded by the cone comprehending the sunand the earth. ON is the arc with centre K along whichmove the extremities of the diameter of the dividing circle inthe moon. Fig. 3 shows the same position of the moon in themiddle of the shadow, but on a larger scale. Fig. 2 showsthe moon at the moment when it has just entered the shadow;and, as the breadth of the earth's shadow is that of two moonsshown touches BN at(Hypothesis 5), the moon in the positionJâiid BL at Z, where L is the middle point of the arc ON.It should be added that, in Fig. 1, Z7Fis the diameter of thecircle in which the sun is touched by the double cone with Bas vertex, which comprehends both the sun and the moon,

ARISTARCHUS OF SAMOS 11while Y, X are the points in which the perpendicular throughA, the centre of the sun, to BA meets the cone enveloping thesun and the earth.NFIG. 1.This being premised, the main results obtained are asfollows :Prop. 13.but0)ON :(diam. of moon) < 2 : 1> 88:45.

12 ARISTARCHUS OF SAMOSBC:CS> 675:1.(2) 0-ZV:(diam. of sun) < 1 :9but > 22 : 225.(3)ON:YZ> 979:10125.Prop. 14 (Fig. 3).Prop. 15.but(Diaiu. of sun) : (diain. of earth) > 19 : 3< 43:6.BVia. 2.Prop. 17. (Diam. of earth); (cliam. of moon) > 108 : 43but 89 : 45. (by Prop. 1 2)

Hence ON :ARISTARCHUS OF SAMOS 13LC = ON 2 : NL*> 89 2 :45 2 ;therefore ON: LP > 7921 : 4050> 88 :45, says Aristarchus.[If '4050 * )(> Developed as a continued fraction, we easily1 11 88 1obtain 1 +^, which is in tact -(2) ON < 2 (diam. of moon).But (diam. of moon) 88:45, from above,and (diam. of moon) : (diam. of sun) > 1 : 20 ; (Prop. 7)therefore, ex ae 88 : 000> 22 : 225.(3) Siii 89 : 1)0 ;(Prop. 12)A Y > 89 : 90.thrrefon*, 22 : 225 ; (Prop. 1 3)therefore, ex a equal i,ON: YZ > 89 x 22 : 90 x 225> 979:10125.

therefore, ex aequ-ali,CM:C8> 15:1.14 ARISTARCHUS OF SAMOSProp. 14 (Fig. 3).The arcs OM, ML, LP, PN are all ;equal therefore so arethe chords. EM, BP are tangents to the circle MQP, so thatCM is perpendicular to BM, while BM is perpendicularTherefore the triangles LOS, GMR are similar.Therefore SO : MR = SL :to ()L.But 80 < 2 MR, since ON < 2MP: (Prop. 13)thereforeSL < 2 RC,and, a fortiori, SR1 :3.CR = BC : CMAgain, MC :> 45: 1 ;(see Prop. 11)And BC:CM > 45 : 1 ;therefore BCiVS > 675 : 1.Prop. 15 (Fig. 1).We have NO :and, a fortiori, YZ : NO(diam. of sun) < 1 :9, (Prop. 13)> 9 : 1 ;therefore, by similar triangles, if Y(), ZN meet in X,AXiXR > 9:1,and convertendo, XA :AR< 9:8.But AB > \$BC, and, a fortiori, > 18 BR,whence AB > \%(AR-AB), or 19 AB > \%AR;that is, AR:AB < 19:18.Therefore, ex aequali,XA:AB 19:3;therefore (diam. of sun) : (diam. of earth) > 19:3.Lastly, since OB :CR> 675 :1,(Prop. 1 4)OBiBR < 675:674.

therefore, ex aequali,AB.BR < 13500: 674ARISTAECHUS OF SAMOS 15But AB-.BC < 20:1;< 6750:337,whence, by inversion anr] componendo,ButRAiAB > 7087:6750. (1)AXiXR = YZ-.NOtherefore, conrertendo,XA :AR> 10125:9146.< 10125:979; (Prop. 13)From this and (1) we have, ex aerjiutli,XA :AI1> 101 25 X 7087: 9146 X 6750> 71755875 : 61735500> 43 :37, a fortiori.[It is difficult not to see in 43 : 37 the expression 1 -h ,6 -f- ftwhich suggests that 43 : 37 was obtained by developing theratio as a continued fraction.]Therefore, conrertendo,XA :XB < 43:6,whence (diaiu. of sun) : (diain. of earth) < 43 : 6. Q. E. D.

XIIIARCHIMEDESTHE siege and capture of Syracuse by Marcellus during thesecond Punic war furnished the occasion for the appearance ofArchimedes as a personage in history;it is witli this historicalevent that most of the detailed stories of him are connected;and the fact that he was killed in the sack of the cityin 212 B.C., when he is supposed to have been 75 years of age,enables us to fix his date at about 287-212 B.C. He was theson of Phidias, the astronomer, and was on intimate termswith, if not related to, King Hicron and his son Gelon. Itappears from a passage of Diodorus that he spent some* timein Egypt, which visit was the occasion of his discovery of theso-called Archimedean screw as a means of pumping water. 1It may be inferred that he studied at Alexandria with thesuccessors of Euclid. It was probably at Alexandria that hemade the acquaintance of COIIOTI of Samos (for whom he hadthe highest regard both as a mathematician and a friend) andof Eratosthenes of Gyrene,.To the former he was in the habitof communicating his discourses before their publication ;while it was to Eratosthenes that he sent The Method, with anintroductory letter which is of the, highest interest, as well as(ifwe may judge by its heading) the famous Cattle-Problem.Traditions.It is natural that history or legend should say more of hismechanical inventions than of his mathematical achievements,which would appeal less to the average mind. His machineswere used with great effect against the Romans in the siegeof Syracuse. Thus he contrived (so we are told) catapults soingeniously constructed as to be equally serviceable at long or1Diodorus, v. 37. 3.

TRADITIONS 17short range, machines for discharging showers of missilesthrough holes made in the walls, and others consisting oflong movable poles projecting beyond the walls which eitherdropped heavy weights on the enemy's ships, or grappled,tkeir prows by means of an iron hand or a beak like that ofa crane, then lifted them into the air and let them fall again. 1Marcelhis issaid to have derided his own engineers with thewords, ' Shall we not make an end of fighting against thisgeometrical Briareus who uses our ships like cups to ladlewater from the sea, drives off our sambuca ignominiouslywitli cudgel-blows, and by the multitude of missiles that hehurls at us all at once outdoes the hundred-handed giants offor the Romans were inmythology?'; but all to no purpose,such abject terror that, 'if they did but see a piece- of ropeor wood projecting abow the wall, they would cry "there itis", declaring that Archimedes was setting some engine inmotion against them, and would turn their backs and runaway Y2 These things, however, were merely the ' diversionsof geometry at play ', :?and Archimedes himself attached noimportance to them. Accordingto Plutarch,'though these inventions had obtained for him the renown ofmore than human sagacity, he yet would not even deign toleave behind him any written work on such subjects, but,regarding as ignoble and sordid the business of mechanics andevery sort of art which is directed to use and profit, he placedhis whole ambition in those speculations the beauty andsubtlety of which is untainted by any admixture of the commonneeds of life/ 4 (a) Astronomy.Archimedes did indeed write one mechanical book, OnSphere-making, which is lost; this described the constructionof a spheiv to imitate the motions of the sun, moon andplanets/' Cicero saw this contrivance and gives a descriptionof it ;he says that itrepresented the periods of the moonand the apparent motion of the sun with such accuracy thatit would even (over a short period) show the eclipses of thesun and moon/1 As Pappus speaks of ' those who understand1Polybius, Hist. viii. 7, 8 ; Livy xxiv. 34 ; Plutarch, Marcellus, cc. 15-17.*lh., c. 17. 3/&., c. 14. 4/&., c. 17.5Carpus in Pappus, viii, p. 1026. 9; Proclus on Eucl. I, p. 41. 16.6Cicero, De rep. i. 21, 22, Tu*c. i. 63, De nat* rfeor. ii. 88.1628.2 C

fi18 ARCHIMEDESthe making of spheres and produce a model of the heavens bymeans of the circular motion of water', it is possible thatArchirnedes's sphere was moved by water. In any case Archimedeswas much occupied with astronomy. Livy calls him'unicus spectator caeli siderumque '.* Hipparchus says, From'these observations it is clear that the differences in the yearsare altogether small, but, as to the solstices, I almost thinkthat Archimedes and I have both erred to the extent of aquarter of a day both in the observation and in the deductiontherefrom'. 2 Archimedes then had evidently considered thelength of the year. Macrobius says he discovered the distancesof the planets,3 and he himself describes in his Sandreckonerthe apparatus by which he measured the apparentangular diameter of the sun.(/3) Mechanic*.Archimedes wrote, as wo shall see, on theoretical mechanics,and it was by theory that he solved the problem To 'move agiven weight by a tjiven force, for it was in reliance ' on theirresistible cogency of his proof 'that he declared to Hieronthat anygiven weight could be moved by any given force;(however small), and boasted that, if he were given a place tostand on, he could move the earth '(na /3o>, KOL KLV> rav ydv,as he said in his Doric dialect). The story, told by Plutarch,is that, 'when Hieron was struck with amazement and askedArchimedes to reduce the problem to practiee and to give anillustration of some great weight moved by a small force, hefixedupon a ship of burden with three masts from the king'sarsenal which had only been drawn up with great labour bymany men, and loading her with many passengers and a fullfreight, himself the while sitting far off, with no great effortbut only holding the end of a compound pulley (rro\v(nra(TTos)quietly in his hand and pulling at it, he drew the ship alongsmoothly and safely as if she were 4moving through the sea.'The story that Archimedes set the Roman ships on firebyan arrangement of burning-glasses or concave mirrors is notfound in any authority earlier than Lucian but it is ;quite13Livy xxiv. 34. 2. 2Ptolemy, Syntaxis, 111. 1, vol. i, p. 194. 23.Maorobius, In Somn. Scip. ii. 3 ;cf. the figures in Tlippolytus, Refnt.,ed. Duncker.p. 66. 52 sq.,4Plutarch, Marcellus, c. 14.

MECHANICS 19likely that he discovered some form of burning-mirror, e.g. aparaboloid of revolution, which would reflectto one point allrays falling on its concave surface in a direction parallel toits axis.Archimedes's own view of the relative importance of hismany discoveries is well shown by his request to his friendsand relatives that they should place upon his tomb a representationof a cylinder circumscribing a sphere, with an inscriptiongiving the ratio which the cylinder bears to the sphere ;infer that he regarded the discovery offrom which we maythis ratio as his greatest achievement.Cicero, when quaestorin Sicily, found the tomb in a neglected state and repairedit 1 ;but it has now disappeared, and no one knows whew, he wasburied.Archimedes's entire preoccupation by his abstract studies isillustrated by a number of stories. We are told that he wouldforget all about his food and such necessities of life, and wouldbo drawing geometrical figures in the ashes of the fire or, whenanointing himself, in the oil on his body. 2 Of the same sortis the tale that, when lie discovered in a bath the solution ofthe question referred to him by Hieron, as to whether a certaincrown supposed to haw been made of gold did not in fact containa certain proportion of silver, he ran naked through thestreet to his home shouting tvpjjKa, tvprjKa* He was killedin the sack of Syracuse by a Roman soldier. The story istold in various forms; the, most picturesqueis that found into a Roman soldierTxet/es, which represents him as sayingwho found him intent on some diagrams which he had drawn'in the dust and came too close, Stand away, fellow, from mydiagram ', whereat the man was so enraged that he killedhim. 4Summaryof main achievements.In geometry Archimedes's work consists in the main oforiginal investigations into the quadrature of curvilinearplane figures and the, quadrature and cubature of curvedsurfaces. These investigations, beginning where Euclid'sBook XII left off, actually (in the words of Chaslea) ' gave1"'4Cicero, TM.W. v. fi4 sq.Tlutarcli, MarctUus, c. 17.Vitrnvius, De architectum, ix. 1. 9, 10.Tzetzes, Chiliad, ii. 35. 135.c 22

20 ARCHIMEDESbirth to the calculus of the infinite conceived and brought toperfection successively by Kepler, Cavalieri, Fermat, Leibnizand Newton'. He performed in fact what is equivalent tointegration in finding the area of a parabolic segment, and ofa spiral, the surface and volume of a sphere and a segment ofa sphere, and the volumes of any segments of the solids ofrevolution of the second degree. In arithmetic he calculatedapproximations to the value of TT, in the course of which calculationhe shows that he could approximate to the value ofsquare roots of large or small non-square numbers he further;invented a system of arithmetical terminology by which hecould express in language any number lip to that which woshould write down with 1 followed by 80,000 million millionIn mechanics he not only worked out the principles ofciphers.the subject but advanced so far as to find the centre of gravityof a segment of a parabola, a semicircle, a cone, a hemisphere,a segment of a sphere, a right segment of a paraboloid anda spheroid of revolution. His mechanics, as we shall see, liasbecome more important in relation to his geometry since thediscovery of the treatise called The Method which was formerlysupposed to be lost. Lastly, he invented the whole science ofhydrostatics, which again he carried so far as to give a mostcomplete investigation of the positions of rest and stability ofa right segment of a paraboloid of revolution floating in afluid with its base either upwards or downwards, but HO thatthe base is either wholly above or wholly below the surface ofthe fluid. This represents a sum of mathematical achievementunsurpassed by any one man in the world's history.Character of treatises.The treatises are, without exception, monuments of mathematicalexposition; the gradual revelation of the plan ofattack, the masterly ordering of the propositions, the sternelimination of everything not immediately relevant to thepurpose, the finish of the whole, are so impressive in theirperfection as to create a feeling akin to awe in the mind ofthe reader. As Plutarch said, 'It is not possible to find ingeometry more difficult and troublesome questions or proofsset out in simpler and clearer propositions'.1There is at the1Plutarch. Marcellus, c. 17.

CHARACTER OF TREATISES 21same time a certain mystery veiling the wayin which hearrived at his results. For it is clear that they were notdiscovered by the steps which lead up to them in the finishedtreatises. If the geometrical treatises stood alone, Archimedesmight seem, as Wallis said, as it were of set purpose'to have covered up the traces of his investigation, as if lie hadgrudged posterity the secret of his method of inquiry, whilelie wished to extort from them assent to his results'. Andindeed (again in the words of Wallis) 'not only Archimedesbut nearlyall the ancients so hid from posterity their methodof Analysis (though it is clear that they had one) that moremodern mathematicians found it easier to invent a newAnalysis than to seek out the old'. A partial exception isnow furnished by The Method of Archimedes, so happily discoveredby Heiberg. In this book Archimedes tells us howh(3 discovered certain theorems in quadrature and cubature,namely by the use of mechanics, weighing elements of afigure against elements oi' another simpler figure the mensurationof which was already known. At the same time he iscareful to insist on the difference between (1) the meanswhich may be sufficient to suggest the truth of theorems,although not furnishing scientific proofs of them, and (2) therigorous demonstrations of them by orthodox geometricalmethods which must follow before they can be finally acceptedus established :'certain things',lie says, 'first became clear to me by amechanical method, although they had to be demonstrated bygeometry afterwards because their investigation by the saidmethod did not furnish an actual demonstration. But it isof course easier, when we have previously acquired, by themethod, some knowledge of the questions, to supply the proofthan it is to find it without any previous knowledge/ 'This',4he adds, is a reason why,in the case of the theorems thatthe, volumes of a cone and a pyramid are one-third of thevolumes of the cylinder and prism respectively having thesame base and equal height, the proofs of which Eudoxus wasthe first to discover, no small share of the 1 credit should begiven to Democritus who was the first to state the fact,though without proof.'Finally, he says that the veryfirst theorem which he foundout by means of mechanics was that of the separate treatise

22 ARCHIMEDESon the Quadrature of the parabola, namely that the area ofanysegment of a section of a right-angled cone a (i.e. parabola] isfour-thirds of that of the triangle which has the same base andThe mechanical proof, however, of this theorem in theheight.Quadrature of the Parabola is different from that in theMethod, and is more complete in that the argumentis clinchedby formally applying the method of exhaustion.List of works stillextant.The extant works of Archimedes in the order in which theyappear in Heiberg's second edition, following the order of themanuscripts so far as the first seven treatises are concerned,are as follows :(5) On the Sphere and Cylinder: two Books.(9) Meamreineiitufa Circle.(7) On Conoids and fyheroidt*.(6) On Mpiral*.(1) On Plane Equilibrium*, Book I.(3) Book II.(10) The Hand- reckoner (Psammites).(2) Quadrature of the Parabola.(8) On Floating Bodlex: two Books.? titovutchion (a fragment).(4) The Method.This, however, was not the order of composition; and,judging (a) by statements in Archimedes's own prefaces tocertain of the treatises and (b) by the use in certain treatisesof results obtained in others, wo can make out an approximatechronological order, which I have indicated in the abovelistby figures in brackets. The treatises On Floating Kodiwwas formerly only known in the Latin translation by Williamof Moerbeke, but the Greek text of ithas now been in greatr")part restored by Heiberg from the. Constantinople manuscriptwhich also contains The Method and the fragment of theOtowutchion.Besides these works we have a collection of propositions(Liber assumptorum) which has reached us through theArabic. Although in the title of the translation by Thabit b.

LIST OF EXTANT WORKS 23Qurra the book is attributed to Archimedes, the propositionscannot be his in their present form, since his name is severaltimes mentioned in them ;but it is quite likely that someof them are of Archimedean origin, notably those about thegeometrical figures called ap/Âoy ('shoemaker's knife') and'(rd\ivw (probably salt-cellar ') respectively and Prop. 8 bearingon the trisection of an angle.There is also the Cattle-Problem in epigrammatic form,which purports by its heading to have been communicated byArchimedes to the mathematicians at Alexandria in a letterto Eratosthenes. Whether the epigrammatic form is due toArchimedes himself or not, there is 110 sufficient reason fordoubting the possibilityproblem by Archimedes.that the substance of it was set as aTraces of lostworks.Of works which are lost we have the following traces.1. Investigations relating to polyltnlra are referred to byPappus who, after alluding to the five regular polyhedra,describes thirteen others discovered by Archimedes which aresemi-regular, being contained by polygons equilateral andequiangular but not all similar. 12. Then; was a book of arithmetical content dedicated toXeuxippus. We learn from Archimedes himself that it dealtwith the nainiiuj of nuuilu'rx (/earoi/o/za*9 rS>v dpidfjL>v)'*andexpounded the system, which we find in the tfa ad-reckoner, ofexpressing numbers higher than those which could be writtenin the ordinary Greek notation, numbers in fact (as we havesaid) up to the enormous tigure represented by 1 followed by80,000 million million ciphers.3. One or more works on mechanics are alluded to containingpropositions not included in the extant treatise On PlaneKquttiLriwfUH. Pappus mentions a, work n Balu nces or Levers(irtpl vy$>v) in which it was pro VIM I (as it also was in Philon'sand Heron's Mechanic*) that 'greater circles overpower lessercircles when they revolve about the same centre V Heron, too,speaks of writings of Archimedes * which bear the title of1Pappus, v, pp. 352 8.2Archimedes, vol. ii, pp. 1216. 18, 236. 17 22 rf. ; p. 220. 4.3Pappus, viii, p. 1068.

24 ARCHIMEDES" works on the lever " f 1.Simplicius refers to problems on Ikecentre of gravity, KevrpoftapiKd, such as the many elegantproblems solved by Archimedes and others, the object of whichis to show how to find the centre of gravity, that is, the pointin a body such that if the body is hung up from it, the bodywill remain at rest in any position. 2 This recalls the assumptionin the Quadrature of the Parabola (6) that, if a body hangsat rest from a point, the centre of gravity of the body and thepoint of suspension are in the same vertical line.Pappus liasa similar remark with reference to a point of support, addingthat the centre of gravityis determined as the intersection oftwo straight lines in the body, through two points of support,which straight lines are vertical when the body is in equilibriumso supported. Pappus also gives the characteristic of the centreof gravity mentioned by Simplicius, observing that this isthe most fundamental principle of the theory of the centre ofgravity, the elementary propositions of which are found inArchimedes's On Equilibriums (Trtpl ii')and Heron'sMechanics. Archimedes himself cites propositions which musthave been proved elsewhere, e.g. that the centre of gravityof a cone divides the axis in the ratio 3:1, the longer segmentbeing that adjacent to the vertex 3 he also ; says that l it isproved in the Equilibriums 'that the centre of gravity of anysegment of a right-angled conoid (i. e. paraboloid of revolution)divides the axis in such *away that the portion towards thevertex is double of the remainder. 4 It is possible that therewas originally a larger work by Archimedes On KrjuilUtriuinHof which the surviving books On Plane Equilibriums formedonly a part in that case ; irepl vyS>v and KtvTpofHapiKd mayonly be alternative titles. Finally, Heron says that Archimedeslaid down a certain procedure in a book bearing thetitle ' Book on Supports*. 64. Theon of Alexandria quotes a proposition from a workof Archimedes called Catoptrica (properties of mirrors) to theeffect that things thrown into water look larger and stilllarger the farther they sink.G Olympiodorus, too, mentionsHeron, Mechanics, i. 32.Simpl. on Arist. De caelo, ii, p. 508 a 30, Brandis ; p. 543. 24, Heib.4Method, Lemma 10.On Floating Bodies,ii. 2.Heron, Mechanics, i. 25.Theon on Ptolemy's Syntaxis, i, p. 29, Halma.

TRACES OF LOST WORKS 25that Archimedes proved the phenomenon of refraction 'bymeans of the ring placed in the vessel (of water)'.* A scholiastto the Pseudo-Euclid's Catoptrica quotes a proof, which heattributes to Archimedes, of the equality of the angles ofincidence and of reflection in a mirror.The textof Archimedes.Heron, Pappus and Theon all cite works of Archimedeswhich no longer survive, a fact which shows that such workswere still extant at Alexandria as late as the third and fourthcenturies A.D. But it is evident that attention came to beconcentrated on two works only, the Measurement of a Circleand On the Sphere and Cylinder. Eutocius (fl.about A.D. 500)only wrote commentaries on these works and on the PlaneEquilibriums, and he does not seem even to have beenacquainted with the (Quadrature of tfie Parahola or the workOn Spirals, although these have survived. Isidorus of Miletusrevised the commentaries of Eutocius on the Measurementof (t (Circle and the two Books On the ti[)ltereand Cylinder,and it would soem to have been in the school of Isidorusthat these treatises were turned from their original Doricinto the ordinary language, with alterations designedthem nurfe intelligible to elementary pupils. But neither into makecollected editionIsidorus's time nor earlier was there,anyof Archimedes's works, so that those which were less readtended to disappear.In the ninth century Leon, who restored the Universityof Constantinople, collected togetherall the works that hecould find at Constantinople, and had the manuscript written(the archetype, lleiberg's A) which, throughits derivatives,was, up to the discovery of the Constantinople manuscript (C)containing Tlie Method, the only source for the Greek text.Leon's manuscript came, in the twelfth century, to theNorman Court at Palermo, and thence passed to the Houseof Hohenstaufen. Then, with all the library of Manfred, itwas given to the Pope by Charles of Anjou after the battleof Benevento in 1266. It was in the Papal Library in theyears 1269 and 1311, but, some time after 1368, passed into1Olympiodoma on Arist. Meteorofayica, ii, p. 94, Ideler ; p. 211.18,Busse.

26 ARCHIMEDESprivate hands. In 1491 it belonged to Georgius Valla, whotranslated from it the portions published in his posthumouswork De expetendis et fugiendis rebus (1501), and intended topublish the whole of Archimedes with Eutocius's commentaries.On Valla's death in 1500 it was bought by AlberiusPius, Prince of Carpi, passing in 1530 to his nephew, RodolphusPius, in whose possessionit remained till 1544. At sometime between 1544 and 1564 itdisappeared, leaving notrace.The greater part of A was translated into Latin in 1269by William of Moerbeke at the Papal Court at Viterbo. Thistranslation, in William's own hand, exists at Rome (Cod.Ottobon. lat. 1850, Heiberg's B), and is one of our primesources, for, although the translation was hastily done qndthe translator sometimes misunderstood the Greek, he followedits wording so closely that his version is, for purposes ofcollation, as good as a Greek manuscript.William used also,for his translation, another manuscript from the same librarywhich contained works not included in A. This manuscriptwas a collection of works on mechanics and optics William;translated from it the two Books On Floating Bodies, and italso contained the Plane Equilibriums and the Quadratureof the Parabola, for which books William used both manuscripts.The four most important extant Greek manuscripts (exceptC, the Constantinople manuscript discovered in 1906) werecopied from A* The earliest is E, the Venice manuscript(Marcianus 305), which was written between the years 1449ai\d 1472. The next is D, the Florence manuscript (Laurent.XXVIII. 4),which was copied in 1491 for Angelo Poliziano,permission having been obtained with some difficulty in consequenceof the jealousy with which Valla guarded his treasure.The other two are G (Paris. 2360) copied from A after it hadpassed to Albertus Pius, and H (Paris. 2361) copied in 1544by Christopherus Auverus for Georges d'Armagnac, Bishopof Rodez. These four manuscripts, with the translation ofWilliam of Moerbeke (B), enable the readings of A to beinferred.A Latin translation was made at the instance of PopeNicholas V about the year 1450 by Jacobus Cremonensis.

THE TEXT OF ARCHIMEDES 27It was made from A, which was therefore accessible to PopeNicholas though it does not seem to have belonged to him.Regiomontanus made a copy of this translation about 1468and revised it with the help of E (the Venice manuscript oftlie Greek text) and a copy of the same translation belongingto Cardinal Bessarion, as well as another 'old copy' whichseems to have been B.The editio prinwps was published at Basel (apad Hervagium)by Thomas Gechauff Venatorius in 1544. The Greektext was based on a Niirnberg MS. (Norirnberg. Cent. V,app. 12) which was copied in the sixteenth century from Abut witli interpolations derived from B; the Latin translationwas Regiomontamis's revision of Jacobus Cremonensis(Norimb. Cent. V, 15).A translation by F. Commandinus published at Venice in1558 contained the Measurement of a Circle, On Spirals, theQuadrature of the Parabola, On do no ids and fyke raids, andthe tia nd-reckoner. This translation was based] on the Baseledition, but Commandinus also consulted E and other Greekmanuscripts.Torelli's edition (Oxford, 1792) also followed the editio{>ri itcep** in the main, but Tore Hi also collated E. The bookwas brought out after Torelli's death by Abram Robertson,who also collated five more manuscripts, including I), Gand II. The collation, however, was not well done, and theedition was not properly corrected when in the press.The second edition of Hei berg's text of all the works ofArchimedes with Eutocius's commentaries, Latin translation,apparatus criticus, &c., is now available (1910-15) and, ofcourse, supersedes the first edition (1880-1) and all others.It naturally includes The Method, the fragment of the tftoma-'chion, and so much of the Greek text of the two Books OnFloatiiKj Bodies as could be restored from the newly discoveredConstantinople manuscript.1Contents of Tlw.Method.Our description of the extant works of Archimedesmay suitably begin with The, Method (thefull title is On1The Works of Archimedes, edited in modern notation by the presentwriter in 1897, was based on Heiberg'sfirst edition, and the Supplement

28 ARCHIMEDESMechanical Theorems, Method (communicated) to Eratosthenes).Premising certain propositions in mechanics mostly takenfrom the Plane Equilibriums, and a lemrna which formsProp. 1 of On Conoids and Spheroids, Archimedes obtains byhis mechanical method the following results. The area of anysegment of a section of a right-angled cone (parabola) is f ofthe triangle with the same base and height (Prop. 1).Theright cylinder circumscribing a sphere or a spheroid of revolutionand with axis equal to the diameter or axis of revolutionof the sphere or spheroid is 1^ times the sphere or spheroidrespectively (Props. 2, 3). Props. 4, 7, 8, 11 find the volume ofany segment cut off, by a plane at right angles to the axis,from any right-angled conoid (paraboloid of revolution),sphere, spheroid, and obtuse-angled conoid (hyperboloid) interms of the cone which has the same base as the segment andequal height. In Props. 5, 6, 9, 10 Archimedes uses his ftiethodto find the centre of gravity of a segment of a paraboloid ofrevolution, a sphere, and a spheroid respectively. Props.12-15 and Prop. 16 are concerned with the cubature of twospecial solid figures. (1) Suppose a prism witji a square baseto have a cylinder inscribed in it, the circular bases of thecylinder being circles inscribed in the squares which arethe bases of the prism, and suppose a plane drawn throughone side of one base of the prism and through that diameter ofthe circle in the opposite base which is parallel to the saidside. This plane cuts off a solid bounded by two planes andby part of the curved .surface of the cylinder (a solid shapedlike a hoof cut off by a plane); and Props. 12-15 prove thatits volume is one-sixth of the volume of the prism. (2) Supposea cylinder inscribed in a cube, so that the circular basesof the cylinder are circles inscribed in two opposite faces ofthe cube, and suppose another cylinder similarly inscribedwith reference to two other opposite faces.The two cylindersenclose a certain' solid which is actually made up of eight'hoofs' like that of Prop. 12. Prop. 16 proves that thevolume of this solid is two-thirds of that of the cube. Archimedesobserves in his preface that a remarkable fact about(1912) containing The Method, on the "original edition of Heiberg (in

THE METHOD 29those solids respectively is that each of them is equal to asolid enclosed by planes, whereas the volume of curvilinearsolids (spheres, spheroids, &c.) is generally only expressible interms of other curvilinear solids (cones and cylinders). Inaccordance with his dictum thatthe results obtained by themechanical method are merely indicated, but not actuallyproved, unless confirmed by the rigorous methods of puregeometry, Archimedes proved the facts about the two lastnamedsolids by the orthodox method of exhaustion asregularly used by him in his other geometrical treatises the;proofs, partly lost, were given in Props. 15 and 16.We will first illustrate the method* by giving the argumentof Froj).1 about the area of a parabolic segment.Lot AB(! be the segment, BD its diameter, (-F the tangentat (\ Lot P 1)0any point on the segment, and lot AKF,MOPNM bo drawn parallel to BI). Join ( B 1 and produceit tomeet MO in N and FA inK, and lot KH bo made equal toK(\Now, by a proposition ' proved in a lemma ' (cf. Quadratureof the Parabola, Prop. 5)= CA :A()= CK : KNAlso, by tho property of the parabola, EB = BD, so thatIt follows that, if HC be regarded as tho bar of a balance,a lino TG equal to PO and placed with its middle point at //balances, about K, the straight line MO placed where it is,i.e. with its middle point at N.Similarly with all linos, as MO, PO, in the triangle CFAand the segment CBA respectively.And there are the same number of these lines. Therefore*

30 ARCHIMEDESthe whole segment of the parabola acting at H balances thetriangle CFA placed where it is.But the centre of gravity of the triangle CFA is at TT,where CW = 2 WK [and the whole triangle may be taken asacting at W],Therefore (segment ABC):&CFA= WK : KH= 1:3,so that (segment ABC) = &CFA Q.E.D.It will be observed that Archimedes takes the segment andthe triangle to be made up of parallel lines indefinitely closetogether. In reality they are made up of indefinitely narrowstrips, but the width (dx, as we might say) being the samefor the elements of the triangle and segment respectively,divides out. And of course the weight of each element inboth is proportional to the area. Archimedes also, withoutmentioning momenta, in effect assumes that the sum of themoments of each particle of a figure, acting where it is, isequal to the moment of- the whole figure appliedat its centre of gravity.We willas one massnow take the case of any segment of a spheroidof revolution, because that will cover several propositions ofArchimedes as particular cases.The ellipsewith axes AA', BB' is a section made by theplane of the paper in a spheroid with axis A A'. It is require* 1to find the volume of any right segment AD(! of the spheroidin terms of the right cone with the same base and height.Let DC be the diameter of the circular base of the segment.Join ABy AB', and produce them to meet the tangent at A' tothe ellipse in K, K', and DC produced in E, F.Conceive a cylinder described with axis AA fand base thecircle on KK f as diameter, and cones described with AG asaxis and bases the circles on EF, DC as diameters.Let N be any point on AG, and let MOPQNQ'P'O'M' bedrawn through N parallel to BB' or DC as shown in thefigure.Produce A'A to H so that HA = A A'.

NowTHE METHODHA : AN = A'A: AN= KA:AQ= MN:NQ31It is now necessary to prove that MN .HNQ = NP 2 + NQ*.MW0'B'GCK A' K'By the property of the ellipse,AN.NA': NP 2 = (|A A')'*:(% BB'fthereforeXQ* : Nl= A X' 2 :AN. NA'whence ^V7 )2 = MQ.QN.Add NQ Zto each side, and we haveTherefore, from above,HA : AN = MN 2 :(NP* + NQ*). (1)But MN*, NP 2 , NQ 2 are to one another as the areas of thecircles with MM', PP', QQ' respectivelyas diameters, and these

32 ARCHIMEDEScircles arc sections made by the plane though iV at rightfangles to AA in the cylinder, the spheroid and the cone AEFrespectively.Therefore, if HA A' be a lever, and the sections of thespheroid and cone be both placed witli their centres of gravityat JET, these sections placed at II balance, about A, the sectionMM' of the cylinder where it is.Treating all the corresponding sections of the segment ofthe spheroid, the cone and the cylinder in the same way,we find that the cylinder with axis AG, where it is, balances,about A the cone AEF and the ysegment ADG together, whenboth are placed with their centres of gravity at //; and,if IF be the centre of gravity of the i.e.cylinder, the middlepoint of AG tHA :AW= (cylinder, axis A G):(cone AEF+ segmt. A DC).fIf we call V the volume* of the cone AEF, and >Ssegment of the spheroid,we havethat of the(cylinder) : (K+ ti)= 3 V.while HA :A W = A A' :A,, : (V + ,S'),G.Therefore AA':%AG = 3V. :(V +andwhenceAgain, letThenV be the volume of the cone A DC.V:V'= RG Z :DG*But DG 2 :AG.GA'= BB' Z :AA' 2 .Therefore F: V = AG 2 : AG. OA'

It* follows thatTHE METHOD 338 = Vr ,%AA'-AG -A >Q_ Y'v.-^-^' + A'GA'Gwhich is the result stated by Archimedes in Prop. 8.The result is the same for the segment of a sphere. Theproof, of course slightly simpler, is given in Prop. 7.In the particular case where the segment is half the sphereor spheroid, the relation becomesand itis 4 V,where>S' = 2 V, (Props. 2, 3)follows that the volume of the whole sphere or spheroidV is the volume of the cone ABB' \i.e. thevolume of the sphere or spheroidiscircumscribing cylinder.In order now to find th

34 ARCHIMEDESTherefore AX:AG = (AA'-%AG):(%AA'-AG)whenceAXiXG = (*AA'-SAG):(2AA'-AG)which is the result obtained by Archimedes in Prop.9 for thesphere and in Prop. 10 for the spheroid.In the case of the hemi-spheroid or hemisphere the ratioXG becomes 5 :3, a result obtained for the hemisphere inAX :Prop. 6.The cases of tho paraboloid of revolution (Props. 4,5) andthe hyperboloid of revolution (Prop. 1 1)follow the same course,and it is unnecessary to reproduce them.For the cases of the two solids dealt with at the ond of thotreatise the reader must bo referred to the propositions themselves.Incidentally, in Prop. 13, Archimedes finds the centreof gravity of the half of a cylinder cut by a piano throughthe axis, or, in other words, the centre of gravity of a semicircle.We will now take the other treatises in tho order in whichthey appear in the editions.On the Sphere and Cylinder, I, II.The main results obtained in Book I are shortly stated ina prefatory letter to Dositheus. Archimedes tells us thatthey are new, and that he is now publishing them for thefirst time, in order that mathematicians may be able to examinethe proofs and judge of their value. The results are(1) that the surface of a sphere is four times that of a greatcircle of the sphere, (2) that the surface of any segment of asphere is equal to a circle the radius of which is equalto thestraight line drawn from the vertex of the segment to a pointon the circumference of the base, (3) that the volume of acylinder circumscribing a sphere and with height equal to thediameter of the sphereisf of the volume of the sphere,(4) that the surface of the circumscribing cylinder includingits bases is also f of the surface of the sphere. It isworthyof note that, while the first and third of these propositionsappear in the book in this order (Props. 33 and 34 respec-

ON THE SPHERE AND CYLINDER, I 35for Archi-tively), this was not the order of their discovery;medes tells us in The Method that'from the theorem that a sphereis four times as great as thecone with a great circle of the sphere as base and wi^h heightequal to the radius of the sphere I conceived the notion thatthe surface of any sphere is four times as great as a greatcircle in it ; for, judging from the fact that any circle is equalto a triangle with base equal to the circumference and heightequal to the radius of the circle, I apprehended that, in likemanner, any sphere is equal to a cone with base equal to thesurface of the sphere and height equal to the radius '.Book I begins with definitions (of 'concave in the sameas applied to curves or broken lines and surfaces, ofdirection '{, 'solid sector* and a 'solid rhombus') followed by fiveAssumptions, all of importance. Of all line* irhlch hare thewniw extremities the xtraijfkt line /x the lea*t, and, if there aretwo curved or bent lines in a plane having the same extremitiesand concave in the same direction, but one iswhollyincluded by, or partly included by and partly common with,the other, then that which isincluded is the lesser of the two.Similarly with plane surfaces and surfaces concave in thesame direction. Lastly, Assumption5 is the famous 'Axiomof Archimedes', which however was, according to Archimedeshimself, used by earlier geometers (Eudoxus in particular), tothe effect that Of unequal magnitudes the (jreatcr exceed*the Jew frysuch a inuynitiule as, when added to itself, can bemade to exceed tiny awiffued magnitude of the same kind]the axiom is of course practically equivalent to Eucl. V, Def. 4,and is closely connected with the theorem of Eucl. X. 1.As, in applying the method of exhaustion, Archimedes usesboth circumscribed and inscribed figures with a view to comprcsshu)them into coalescence with the curvilinear figure tobe measured, he has to begin with propositions showing that,given two unequal magnitudes, then, however near the ratioof the greater to the less is to 1, it is possible to find twostraight lines such that the isgreater to the less in a still lessratio ( > 1),and to circumscribe and inscribe similar polygons toa circle or sector such that the perimeter or the area of thecircumscribed ispolygon to that of the inner in a ratio lessthan the given ratio (Props. 2 6): also, just as Euclid proves

36 ARCHIMEDESthat, if we continuallydouble the number of the sides of theregular polygon inscribed in a circle, segments will ultimately beleft which are together less than any assigned area, Archimedeshas to supplement this (Prop. 6) by proving that, if wo increasethe number of the sides of a circumscribed regular polygonsufficiently, we can make the excess of the area of the polygonover that of the circle less than any given area. Archimedesthen addresses himself to the problems of finding the surface ofany right cone or cylinder, problems finally solved in Props. 1 3(the cylinder) and 14 (the cone). Circumscribing and inscribingregular polygons to the bases of the cone and cylinder, heerects pyramids and prisms respectively on the polygons asbases and circumscribed or inscribed to the cone and cylinderrespectively. In Props.7 and 8 he finds the surface of thepyramids inscribed and circumscribed to the cone, and inProps. 9 and 10 he proves that the surfaces of the inscribedand circumscribed pyramids respectively (excluding the base)are less and greater than the surface of the cone (excludingthe base). Props. 11 and 12 prove the same thing of theprisms inscribed and circumscribed to the cylinder, and finallyProps. 13 and 14 prove, by the method of exhaustion, that thesurface of the cone or cylinder (excluding the bases) is equalto the circle the radius of which is a mean proportionalbetween the 'side' (i.e. generator) of the cone or cylinder andthe radius or diameter of the base (i.e. is equal to TTTS in thecase of the cone and 2nrs in the case of the cylinder, wherer is the radius of the base and s a generator). As Archimedeshere appliesthe method of exhaustion for the first time, wewill illustrate by the case of the cone (Prop. 14).Let A be the base of the cone, C a straight line equal to itsradius, D a line equal to a generator of the cone, E a meanproportional to (7, D, and JB a circle with radius equal to E.

ON THE SPHERE AND CYLINDER, I 37If S is the surface of the cone, we have to prove that $ = B.For, if S is not equal to B, it must be either greaterI.Suppose B < &or less.Circumscribe a regular polygon about B, and inscribe a similarpolygon in it, such that the former lias to the latter a ratio lessthan S: B (Prop. 5).Describe about A a similar polygon andset up from it a pyramid circumscribing the cone.Thou (polygon about 4) (polygon about B):= (polygon about A) -.(surface of pyramid).Therefore(surface of pyramid) = (polygon about B).But (polygon about B) : (polygon in B) < S: JS;therefore (surface of pyramid) : (polygon in B) < ti : B.But this is impossible, since (surface of pyramid) > >S',while(polygon in B) < B',therefore B is not less than &II.Suppose B > S.Circumscribe and inscribe similar regular polygons to Bsuch that the former has to the latter a ratio (polygon in ^4):(surface of pyramid).(The latter inference is clear, because the ratio of C:D isgreater than the ratio of the perpendiculars from the centre ofA and from the vertex of the pyramid respectively on anyside of the polygon in A] in other words, if /? < & < -|TT,sin a > sin /3.)Therefore(surface of pyramid) > (polygon in B).But (polygon about B) : (polygon in B) < B: 8,whence (a fortiori)(polygon about B) : (surface of pyramid) B, while (surfaceof pyramid) < &8,

38 ARCHIMEDESTherefore B is not greater than #.Hence 8, being neither greater nor less than ti, is equal to B.Archimedes next addresses himself to the problem of findingthe surface and volume of a sphere or a segment thereof, buthas to interpolate some propositions about 'solid rhombi'(figures made up of two right cones, unequal or equal, withbases coincident and vertices in opposite directions) the necessityof which will shortly appear.Taking a great circle of the sphere or a segment of it, heinscribes a regular polygonof an even number of sides bisectedFIG. 1.by the diameter AA', and approximates to the surface andvolume of the sphere or segment by making the polygonrevolve about A A' and measuring the surface and volume ofsolid so inscribed (Props. 21-7). He then does the same for thea circumscribed solid (Props. 28-32). Construct the inscribedpolygons as shown in the above figures. Joining BB', CC', ... ,CB', DC' ... we see that BE', CO' ..v are all parallel,and so areAB, CB', DC'....Therefore, by similar triangles,BF:FA = A'B:BA, and= B'F:FK= E'I:IA' in Fig.1(= PM: MN in Fig. 2),

ON THE SPHERE AND CYLINDER, 1 39whence, adding antecedents and consequents, we have(Fig. 1) (BB' + QC' + .. . +EE'): A A' = A'B :BA, (Prop. 21)(Fig. 2) (BB' + CC'+... + %PP'):AM=A'B:BA. (Prop. 22)When we make the polygon revolve about AA ', the surfaceof the inscribed figure so obtained is made up of the surfacesof cones and frusta of cones; Prop. 14 has proved that thesurface of the cone ABB' is what we should write TT . AB .BF,and Prop. 16 has proved that the surface of the frustumBCC'J? is7r.B(!(BF+CG). It follows that, since AB =BC =. . . ,the surface of the inscribed solid isthat is, TT .AB(BB' + CC'+ ... + EE') (Fig. 1), (Prop. 24)or TT . AB (BB' + CC' + ... +%PP') (Fig. 2). (Prop. 35)Hence, from above, the surface of the inscribed solid isA J/, and is therefore less thanthat is, ?r . AP 2 (Prop. 37).IT. A'B. A A' or ?r . A'B .TT . AA' 2 (Prop. 25) or TT . A'A .AM,Similar propositions with regard to surfaces formed by therevolution about AA' of regular circumscribed solids provethat their surfaces aiv greater than tr.AA'* and n.AP 2respectively (Props. 28-30 and Props. 39-40). The case of thesegment is more complicated because the circumscribed polygonwith its sides parallel to AB, J3C ... DP circumscribesthe sector POP'. Consequently,if the segmentis less than asemicircle, as CAC', the base of the circumscribed polygon(cc') is on the side of GY" towards A, and therefore the circumscribedpolygon leaves over a small strip of the inscribed. Thiscomplication is dealt with in Props. 39-40. Having thenarrived at circumscribed and inscribed figures with surfacesgreater and less than n.AA'* and TT. AP* respectively, andhaving proved (Props. 32, 41) that the surfaces of the circumscribedand inscribed figures are to one another in the duplicateratio of their sides, Archimedes proceeds to prove formally, bythe method of exhaustion, that the surfaces of the sphere andsegment are equal to these circles respectively (Props. 33 and42); A IT. A'* is of course equal to four times the great circleof the sphere. The segment is, for convenience, taken to be

40 ARCHIMEDESless than a hemisphere, and Prop. 43 proves that the sameformula applies also to a segment greater than a hemisphere.As regards the volumes different considerations involving'solid rhombi ' come in. For convenience Archimedes takes,in the case of the whole sphere, an inscribed polygon of 4?isides (Fig. 1).It is easily seen that the solid figure formedby its revolution is made up of the following : first, the solidrhombus formed by the revolution of the quadrilateral ABOB'(the volume of this is shown to be equal to the cone with baseequal to the surface of the cone ABB' and height equal to p ythe perpendicular from on AB, Prop. 18); secondly, theextinguisher-shaped figure formed by the revolution of thetriangle BOO about AA' (this figure is equal to the differencebetween two solid rhombi formed by the revolution of TBOB'and TCOC' respectively about AA', where T is the point ofintersection of CB> C B' produced with A'A produced, andthis difference is proved to be equal to a cone with base equalto the surface of the frustum of a cone described by BC in itsrevolution and height equal to p the perpendicular fromBC, Prop. 20) ;and so on ; finally, the figure formed by therevolution of the triangle COD about AA' is the differencebetween a cone and a solid rhombus, which is proved equal toa cone with base equal to the surface of the frustum of a conedescribed by CD in its revolution and height p (Prop. 19).Consequently, by addition, the volume of the whole solid ofrevolution is equal to the cone with base equal to its wholesurface and height p (Prop. 26).But the whole of the surfaceof the solid is less than 4 ?rr 2 and,p< r therefore the volume;of the inscribed solid is less than four times the cone withbase ?rr 2 and height r (Prop. 27).It isthen proved in a similar way that the revolution of4n sides gives a solidthe similar circumscribed polygon ofthe volume of which is greater than four times the same cone(Props. 28-31 Cor.). Lastly, the volumes of the circumscribedand inscribed figures are to one another in the triplicate ratio oftheir sides (Prop. 32) ;onand Archimedes is now in a position toapply the method of exhaustion to prove that the volume ofthe sphereis 4 times the cone with base ?rr 2 and height r(Prop. 34).Dealing with the segment of a sphere, Archimedes takes, for

ON THE SPHERE AND CYLINDER, I 41convenience, a segment less than a hemisphere and, by thesame chain of argument (Props. 38, 40 Corr., 41 and 42), proves(Prop. 44) that the volume of the sector of the sphere boundedby the surface of the segment is equal to a cone with baseequal to the surface of the segment and height equal to theradius, i.e. the cone with base 7T.AP 2 and height r (Fig. 2).It is noteworthy that the proportions obtained in Props. 21,22 (see p.39 above) can be expressed in trigonometrical form.If 471 isthe number of the sides of the polygon inscribed inthe circle, and 2n the number of the sides of the polygoninscribed in the segment, and if the angle AOP is denotedby a, the trigonometrical equivalents of the proportions arerespectively2n 2/i(1) sin-^ + sin ~v '+...+ fiin(2?i 1)- = cot-^-;'2 ti 4/i(2) 2 } sin + sin + ... + sin (n~ !)-> +sin

42 ARCHIMEDES?rr 2 . 2 sin respectively are (when n is indefinitely increased)2 7i/precisely what we should represent by the integralsJoorandjr.Jo,or 27rr 2 (l eosa).Book 11 contains six problems and three theorems. Of thetheorems Prop. 2 completes the investigation of the volume ofany segment of a sphere, Prop. 44 of Book I having onlybrought us to the volume of the corresponding sector. IfABB' be a segment of a sphere cut off by a plane at rightangles to A A', we learnt in I. 44 that the volume of the sectorOBAB' isequal to the cone with base equal to the surfaceof the segment and height equal to the radius, i.e. JTT AB 2 . . r,where r is the radius. The volume of the segmentis thereforeArchimedes wishes to express this as a cone with base thesame as that of the segment. Let AM, the height of the segment,= h.Now AB 2 : BAP = A'A : A'M = 2r :(2r-h).ThereforeThat is, the segment is equalto the cone with the samebase as that of the segment and height Ti(^rh)/(2r h).

ON THE SPHERE AND CYLINDER, II 43This is expressed by Archimedes thus. If 1IM is the heightof the required cone,A'M):A'M, (1)and similarly the cone equal to the segment A'BE' has theheight H'M, whereIf'M : A'M= (OA + :AM) AM.(2)His proof is, of course, not in the above form but purelygeometrical.This proposition leads to the most important proposition inthe Book, Prop. 4, which solves the problem To cut a givensphere by a jjlane in sack a any that the volumes of thesegments are to one another in a glren ratio.If m :Cubic equation arising out of II. 4.n be the given ratio of the cones which are equal tothe segments and the heights of which an; /t, h', we haveand, if we eliminate h' by means of the relation h + hf = 2r,we easily obtain the following cubic equation in A,_.m+ nArchimedes in effect reduces the problem to this equation,which, however, he treats as a particular case of the moregeneral problem corresponding to the equationwhere b is a given length and 2 c*any given area,or x 2 (a x) = be'2 ,where x = 2r h and 3?' = a.Archimedes obtains his cubic equation with one unknownby means of a geometrical elimination of 11, IV from theequation UM = .7Ai/11//'J/, where HiW, H'M have the valuesdetermined by the proportions (1) and (2) above, after whichthe one variable point M remaining corresponds to the oneunknown of the cubic equation. His method is, first, to find

44 ARCHIMEDESvalues for each of the ratios A'H' : H'M and H'H :A'H' whichare alike independent of H, H' and then, secondly, to equatethe ratio compounded of these two to the known value of theratio HH':H'M.(a) We have, from (2),A'H' : H'M= OA :(OA + AM). (3)() From (1) and (2), separando,AH:AM=OA':A'M, (4)A'H':A'M = OA:AM.(5)Equating the values of the ratio A'M :we have OA' : AHwhence HH' : OH' = OH' := Air :OA= OIF: OH,AMgiven by (4), (5),A' 11', (since OA = OA')orElf. A'IF = 011'*,so that HE' : A'H' = OH' 2 : A'H' Z .But, by (5),OA' : A'H' = AM: A'M,and, componendo, OH' : A'H'= A A' : A'M.By substitution in (6),HH' : A'It' = A A' 2 : A'M' 2 .(6)(7)Compounding with (3),we obtainHH' : H'M = (A A' z :A'M*) (OA .:[The algebraical equivalent of this ism + n 4 r 3OA + AM). (8)l, . , , ,which reduces to ----- =4r~-=- 3m3h 2 rh*or 7^--3A 2 r+-r 3 = 0, as above.] JArchimedes expresses the result (8)more simply by producingOA to D so that OA = AD, and then dividing AD at

ON THE SPHERE AND CYLINDER, II 45E %o that AD:DE = HH'iH'M or (m + n):n. We havethen OA = AD and OA + AM = MD, so that (8) reduces toorAD : DE= (A A'* :A'M*) .(AD: MD),MI):I)E= AA'*:A'M*.Now, says Archimedes, D is {riven, since AD OA. Also,yt/): /)/? being a given ratio, DE is given. Hence the problemreduces itself to that of dividing A'D into two parts at^f such thatAID :(a given length) = (a given area) : A'M*.That is,the generalized equationis of the formx 2 (a x) = bo 2 ,as above.(i)Archimedes's own solution of the cubic.Archimedes adds that, 'if the.problem is propounded in thisgeneral form, it requires a Siopta-pos [i.e. it is necessary tobut if the conditions areinvestigate tae limits of possibility],added which exist in the, present case in the[i.e.actualproblem of Prop. 4], it does not require a ôpjoyjoy' (in otherwords, a solution isalways possible). He then promises togive ' at the end ' an analysis and synthesis of both problems[i.e. the Siopia/jLos and the problem itself]. The promisedsolutions do not appear in the treatise as we have it, but'Eutocius gives solutions taken from an old book ' which hemanaged to discover after laborious search, iind which, since itwas partly written in Archimedes's favourite Doric, he withfair reason assumed to contain the missing addendum byArchimedes.In the Archimedean fragment preserved by Eutocius theabove equation, x*(a x)= 6c 2 , i? solved by means of the intersectionof a parabola and a rectangular hyperbola, the equationsof which may be written thus2

46 ARCHIMEDESthe curves touch at the point for which x = fa. If on theother hand be 2 < â 3 ,it isprovedthat there are two realsolutions. In the particular case arising in Prop. 4 it is clearthat the condition for a real solution is satisfied,for theexpression corresponding to be2 is -4r 3 ,and it is only~-necessary J thatm + n4 r 3 ,which is obviously the case.4r 3 should be not greater than ^V 6 " or(ii) Solution of the cubic by Dionysodorus.It is convenient to add here that Eutocius gives, in additionto the solution by Archimedes, two other solutions of ourproblem. One, by Dionysodorus, solves the cubic equation inthe less general form in which it is required for Archimedes'sproposition. This form, obtained from (8) above, by puttingA'M = .r,is4r 2 :^2 = (3 r -,i-):-in + 'iiand the solution is obtained by drawing the parabola andr,ythe rectangular hyperbola which we should represent bythe,equations?'(3r x) = y 2 and-2r~ = try,referred to A'A and the perpendicular to itthrough A as axesof x, y respectively.(We make FA equal to OA, and draw theAHperpendicularof such a length thatFA :AH CE : ED = (m + n):n.)

ON THE SPHERE AND CYLINDER, II 47(iii).Solution of the original problem of II. 4 by Diocles.Diocles proceeded in a different manner, satisfying, bya geometrical construction, not the derivative cubic equation,but the three simultaneous relations which hold in Archimedes'sproposition, namelyHA :h = r :h'll'A': h' = r:hwith the slight generalization that he substitutes for r inthese equations another length a.HRThe problem is, given a straight line A A', a ratio m:n, andanother straight line AK (= a), to divide A A' at a point Mand at the same time to find two points 77, II' on AA'produced such that the above relations (with a in placeof r) hold.The analysis leading to the construction is very ingenious.Place AK (= a) at right angles to AA r ,and parallel to it.and draw A'K' equalSuppose the problem solved, and the points M, //,found.Join KM, produce it, and complete the rectangle KG-EK'.11' all

48 ARCHIMEDESDraw QMN through M parallelmeet KG produced in F.to AK. Produce K'M toBy similar triangles,FA : AM = K'A' :A'M, or FA : h = a :h',whence FA = AH (k, suppose).SimilarlyA'E = A'U' (k', suppose).Again, by similar triangles,(FA + AM):(A'K' + A'M)= AM: A'M= (AK + A M) :(EA' + A'M),or (k + h):(a + h')= (a + h):(k' + h'),i. e. (k + h) (k' + h')= (a + h) (n + h'). (1 )Now, by hypothesis,m :n = (k + h):(k' + h')= (a + h) (a + h'):(k' + h')2[by (1 )]. (2)Measure AH, A'R' on AA' produced both ways equalto a.Draw RP, R'P' at right angles to RR' as shown in the figure.Measure along MN the length MV equal to MA' or k', anddraw PP' through V, A' to meet RP, R'P'.ThenQV=k' + h',whence PV.P'V=2(a + k) (a, + h') ;and, from (2) above,2w :n= 2 (a + h) (a + h'):(k' + h')*= PV.P'V:QV\ (3)Therefore Q is on an ellipse in which PP' is a diameter, andQVis an ordinate to it.Again, oGQNK is equal to dAA'K'K, whenceGQ.QN= AA'. A'K' =(h + h')a = 2ra, (4)and therefore Q is on the rectangular hyperbol^ with KF,KK' as asymptotes and passing through A'.

ON THE SPHERE AND CYLINDER, II 49How this ingenious analysis was suggested it is not possibleto say. It is the equivalent of reducing the four unknownsh> h' y />',k' to two, by putting h = r + x, hf == rx and k' = y,and then reducing the given relations to two equations in x,y,which are coordinates of a point in relation to Ox, Oy as axes,where is the middle point of AA ', and Ox lies along OA',while Oy is perpendicular to it.Our original relations (p. 47) give= A; = -=- = a-7/ ah' r x,ah r + x^mh + kA r + .i: A rx n h ' + k> A* = -,-/= & ------ ' and = r7 r/'We have at once, from the first two equations,whence(am 1 .(;/: -f r) (//+ rt)= 2 m,which is the rectangular hyperbola (-1)above.mwhence we obtain a cubic CM [nation in ./',which mves/ . \9 / \Oiliiv / ,)(r-f xY (r + a s)= - (r ,r)~-, whence --- = ---.u // a ,V 4 /' .'' r + a-f.r-But =r x r + M r xand the equation becomes'/i/(y + r -a:)2= (r + a)2 - ^2,wliich is the ellipse (3) above.1523.2

50 ARCHIMEDESTo return to Archimedes. Book II of our treatise containsfurther problems To find a sphere equal to a given cone or:cylinder (Prop. 1), solved by reduction to the finding of twomean proportionals;to cut a sphere by a plane into twosegments having their surfaces in a given ratio (Prop. 3),which is easy (by means of I. 42, 43) ; given two segments ofspheres, to find a third segment of a sphere similar to oneof the given segments and havingits surface equal to that ofthe other (Prop. 6) ;the same problem with volume substitutedfor surface (Prop. 5),which is again reduced to the findingof two mean proportionals; from a given sphere to cut offa segment having a given ratio to the cone with the samebase and equal height (Prop. 7).The Book concludes withtwo interesting theorems. If a sphere be cut by a plane intotwo segments, the greater of which has its surface equal to fland its volume equal to F, while $', V are the surface andvolume of the lesser, then V: V < 2 AS' : >S" 2 but >&'5:jS"J(Prop. 8) and, of all segments of : spheres which haves theirsurfaces equal, the hemisphere is the greatest in volume(Prop. 9).Measurement of a Circle.The book on the Measurement of a Circle consists of throepropositions only, and is not in its original form, having lost(as the treatise On the Sphere and Cylinder also has) practicallyall trace of the Doric dialect in which Archimedeswrote ;itmay be only a fragment of a larger treatise 4 . Thethree propositions which survive prove (1) that the area ofa circle is equal to that of a right-angled triangle in whichthe perpendicular is equal to the radius, and the base to thecircumference, of the circle, (2) that the area of a circle is tothe square on its diameter as 11 to 14 (tho text of this propositionis, however, unsatisfactory, and it cannot have beenplaced by Archimedes before Prop. 3, on which it depends),(3) that the ratio of the circumference of any circle to Usdiameter (i.e. TT) is < 3^ but > 3^. Prop.1 isproved bythe method of exhaustion in Archimedes's usual form : heapproximates to the area of the circle in both directions(a) by inscribing successive regular polygons with a number of

MEASUREMENT OF A CIRCLE 51sides % continually doubled, beginning from a square, (6) bycircumscribing a similar set of regular polygons beginningfrom a square,itbeing shown that, if the number of thesides of these polygons be continually doubled, more than halfof the portion of the polygon outside the circle will be takenaway each time, so that we shall ultimately arrive at a circumscribedpolygon greater than the circle by a space less thanany assigned area.Prop. 3, containing the arithmetical approximation to TT,the most interesting. The method amounts to calculatingapproximately the perimeter of two regular polygons of 96sides, one of which is circumscribed, and the other inscribed,to the circle : and the calculation starts from a greater anda lesser limit to the value of \/3, whicil Archimedes assumeswithout remark as known, namely265 ^ A/ Q ^ 1 35 1isHow did Archimedes arrive, at those particular approximations?No puzzle has exercised more fascination uponwriters interested in the history of mathematics. De Lagny,Mollwcide, ttu/engeiger, Hauber, Zeuthen, P. Tannery, Heilermann,Hultsch, Hunrath, Wertheim, Bobyniii these are the:names of sonic* of the authors of different conjectures. Thesimplest supposition is certainly that of Hunrath and Hultsch,who suggested that the formula used waswhere a~ is the nearest square number above or below (t~b tas the case may be,. The use of the first part of this formulaby Heron, who made a number of such approximations,isproved by a passage in his Metrica ', where a rule equivalentto this is applied to \/720; the second part of the formula isused by the Arabian Alkarkhi (eleventh century) who drewfrom Greek sources, and one approximation in Heron may beobtained in this way.- Another suggestion (that of Tannery1Heron, Metrica,i. 8.2Sterrom. ii, p. 184. 19, Hultsch; p. 154. 19, Heib. V^4 = 7J=7 1Vinstead of 7 />4.K2

52 ARCHIMEDESand Zeuthen)is that the successive solutions in integers ofthe equationso; a -32/ 2 =l )j5 2 -3y a = -2)may have been found in a similar way to those of theequations &2 2 y= 1 +1 given by Theon of Smyrna afterthe Pythagoreans. The rest of the suggestions amount for themost part to the use of the method of continued fractionsmore or less disguised.Applying the above formula, we easily findorNext, clearing of fractions, we consider 5 as an approximationto \/3 . 3- or \/27, and we havewhenceff > ^ 3 > TT-Clearing of fractions again, and taking 20 as an approximationto \/3 . 15- or >/675, we have28 ra > 15v/3 > 2 *>-5T>which reduces to1351 v.A/o~T50- > V 5-.> T^.265Archimedes first takes the case of the circumscribed polygon.Let CA be the tangent at A to a circular arc with centre 0.Make the angle AOC equal to one-third of a right angle.Bisect the angle AOG by OD, the angle AOD by OK, theangle AOEby OF, and the angle AOF by OG. Produce GAto .4.#, making ^l// equal to AG. The angle GOlf is them^th of a right angle, soequal to the angle FOA which isthat GH is the side of a circumscribed regular polygon with96 sides.Now UA:AC[= V3:l] > 265:153, (l)and OC :CA = 2:1 = 306:153.(2)

MEASUREMENT OF A CIRCLE53And, since 01) bisects the angle CO A,so that(CO + OA): OA = CA :DA,or(CO + 04) :CA = OA:A D.HenceOA :AD> 571 :153, by (1) and (2).And Oir-:AD* = (OA~ + AD 2 ):AD~Therefore, says Archimedes,> (571-+153-): 153-> 349450:23109.OD:DA > 591 1: 153.Next, just as we have found the limit of OD:ADfrom 00: ('A ami the limit of OA :AC?, we find the limitsof OA:AK and OKiAE from the limits of OD:DA andOA :AD,OA-.AG.and so on. This gives ultimately the limit ofDealing with the inscribed polygon, Archimedes gets asimilar series of approximations. ABC being a semicircle, theangle JiAO\a made equal to one-third of a right angle. Then,if the angle KAC is bisected by AD, the angle BAD by AK,the angle BAK by AF, and the angle 7M.F by AG, thestraight line BG is the side of an inscribed polygon with96 sides.

54 ARCHIMEDESNow the triangles ADB, BDd, ACd are similar;therefore AD:DB = BD:Dd = AC : Cd= AB :Bd, since AD bisects Z BAG,But AC:GB < 1351 :780,while BA : BCTherefore AD := 2 : 1 = 1560 : 780.DB< 2911 : 780.Hence AB 2 : BD* < (2911' 2 + 7 80-): 780-< 9082321 :608400,and, says Archimedes,AB:BD < 301 3;|: 780.Next, justas a limit is found for AD:Dli and AB:BDfrom AB:BC and the limit of AC:(JB, so we findAE : EB BE BDand AB : from the limits of AB :and so on, and finally we obtain the limit of AB :limits torand AD :BG.DH,We have therefore in both cases two series of terms a ,a,,

MEASUREMENT OF A CIRCLE 55The series of values found by Archimedes are shown in thefollowing table :a b c n a b c265 306 153 1351 1560 780571 > |V(571~-H532)]153 1 2911 < i/(2911a+ 780 2 )780153 2/5924J-J...780|1823 ([V r {(2334}) a 58+153}]153 393661 TT ...240f> 2339J 1007 (, 3,4 simply stated in their final form withoutthe intermediate step containing the radical exceptin the first* t Here the ratios of a to c are in the first instance reduced to lowerterms.

56 ARCHIMEDEScase of all, where we are told that OD*:AD* > 349450 : 23409and then that OD:DA > 591|:153. At the pointsmarked* and f in the table Archimedes simplifies the ratio a 2: c andr* 3: c before calculating & 2,& 3 respectively, by multiplying eachterm in the first case by T\ and in the second case by -J.He gives no explanation bf the exact figure taken as theapproximation to the square root in each case, or of themethod by which he obtained We it.may, however, be surethat the method amounted to the use of the formula (ce+ 6)2= a 2 2ab + b 1 )much as our method of extracting the squareroot also depends upon it.We have already seen (vol. i, p. 232) that, according toHeron, Archimedes made a still closer approximation to thevalue of 77.On Conoids and Spheroids.The main problems attacked in this treatise are, in Archimedes'smanner, stated in his preface addressed to Dositheus,which also sets out the premisses with regard to the solidfigures in question. These premisses consist of definitions andobvious inferences from them. The figures are (1) the rightangledconoid (paraboloid of revolution), (2) the obtuse-angledconoid (hyperboloid of revolution), and (3) the spheroids(a) the oblong, described by the revolution of an ellipse aboutits 'greater diameter' (major axis), (6) the flat, described by.the revolution of an ellipse about its * lesser diameter ' (minoraxis). Other definitions are those of the vertex and w/.s of thefigures or segments thereof, the vertex of a segment beingthe point of contact of the tangent plane to the solid whichis parallel to the base of the segment. The centre isonlyrecognized in the case of the spheroid; what corresponds tothe centre in the case of the hyperboloidis the 'vertex ofthe enveloping cone' (described by the revolution of whatArchimedes calls the * nearest lines to the section of theobtuse-angled cone', i.e. the asymptotes of the hyperbola),and the line between this point and the vertex of the hyperboloidor segment is called, not the axis or diameter, but (theline) 'adjacent to the axis'. The axis of the segment is inthe case of the paraboloid the line through the vertex of thesegment parallel to the axis of the paraboloid, in the case

ON CONOIDS AND SPHEROIDS 57of the hyperboloid the portion within the solid of the linejoining the vertex of 'the enveloping cone to the vertex ofthe segment and produced, and in the case of the spheroids theline joining the points of contact of the two tangent planesparallel to the base of the segment. Definitions are added ofa segment of a cone ' (the figure cut off towards the vertex byan elliptical, not circular, section of the cone) and a frustum'of a cylinder' (cut off by two parallel elliptical sections).Props. 1 to 18 with a Lemma at the beginning are preliminaryto the main subject of the treatise. The Lemma and Props. 1, 2are general propositions needed afterwards. They includepropositions in summation,2 * ta+ 2a + Sa+... +n,} > n . na > 2 \a + 2 n(na)lastly, Prop. 2 givesseries ax + ,/r, a .2x + (2 ;i:)2,a . 3 x 4- (3 a) 2 ,inequalities of ratios, thus:limits for the sum of n terms of the. . . ,in t ho form of11 {a .nx + ( nx)' 2 \: S1"- 1[ a . rx + (rx) 2 }> (a -f nx) : (\

58 ARCHIMEDESfirst of the three propositions, proved to be to the area ofthe auxiliary circle as the minor axis to the major equilateral;polygons of 4 n sides are inscribed in the circle and comparedwith corresponding polygons inscribed in the ellipse,which aredetermined by the intersections with the ellipse of the doubleordinates passing through the angular points of the polygonsinscribed in the circle, and the method of exhaustion is thenapplied in the usual way.Props. 7, 8 show how, given an ellipsewith centre C and a straight line CO in a plane perpendicular tothat of the ellipse and passing through an axis of it, (1) in thecase where OC is perpendicular to that axis, (2) in the casewhere it is not, we can find an (in general oblique) circularcone with vertexsuch that the given ellipse is a section of it,or, in other words, how we can find the circular sections of thecone with vertex which passes through the circumference ofthe ellipse ; similarly Prop. 9 shows how to find the circularsections of a cylinder with CO as axis and with surface parsingthrough the circumference of an ellipse with centre 6', whereCO is in the plane through an axis of the ellipseand perpendicularto its plane, but is not itself perpendicular to thataxis. Props. 11-18 give simple propertiesof the conoids andspheroids, easily derivable from the properties of the respectiveconies; they explain the nature and relation of the sectionsmade by planes cutting the solids respectively in different ways(planes through the axis, parallel to the axis, through the centreor the vertex of the enveloping cone, perpendicular to the axis,or cutting it obliquely, respectively), with especialthe ellipticalreference tosections of each solid, the similarity of parallelelliptical sections, &c. Then with Prop. 19 the real businessof the treatise begins, namely the investigation of the volumeof segments (right or oblique)spheroids respectively.The method is, in allof the two conoids and thecases, to circumscribe and inscribe tothe segment solid figures made up of cylinders or *frusta ofcylinders ', which can be made to differ as little as we pleasefrom one another, so that the circumscribed and inscribedfigures are, as it were, compressed together and into coincidencewith the segment which is intermediate between them.In each diagram the plane of the paper is a plane throughthe axis of the conoid or spheroid at right angles to the plane

ON CONOIDS AND SPHEROIDS 59of the^section which is the base of the segment, and whichis a circle or an ellipse according as the said base is or is notat right angles to the axis ;the plane of the paper cuts thebase in a diameter of the circle or an axis of the ellipse asthe case may be.The nature of the inscribed and circumscribed figures willbe seen from the above figures showing segments of a paraboloid,a hyperboloid and a spheroid respectively, cut oft*

60 ARCHIMEDESby planes obliquely inclined to the axis. The base of thesegment is an ellipse in which BB' is an axis, and its plane isat right angles to the plane of the paper, which passes throughthe axis of the solid and cuts it in a parabola, a hyperbola,oran ellipse respectively. The axis of the segmentis cut into anumber of equal parts in each case, and planes are drawnthrough each point of section parallelto the base, cutting thesolid in ellipses, similar to the base, in which PPt\ QQ', &c., areaxes.Describing frusta of cylinders with axis AD and passingwe draw thethrough these elliptical sections respectively,circumscribed and inscribed solids consisting of these frusta.It is evident that, beginning from A, the first inscribed frustumis equal to the first circumscribed frustum, the second to thesecond, and so on, but there is one more circumscribed frustumthan inscribed, and the difference between the circumscribedand inscribed solids is equal to the last frustumof which BR'is the base, and ND is the axis. Since ND can be made assmall as we please, the difference between the circumscribedand inscribed solids can be made less than any assigned solidwhatever. Hence we have the requirements for applying themethod of exhaustion.Consider now separately the cases of the paraboloid, thehyperboloid and the spheroid.I. The paraboloid (Props. 20-22).The fruvstum the base of which is the ellipsein which PP' isan axis is proportional to PP'* or PN~, i.e. proportional toAX. Suppose that the axis AD (= c)is divided into n equalparts. Archimedes compares each frustum in the inscribedand circumscribed figure with the frustum of the whole cylinderBF cut off*by the same planes. Thus(first frustum in :BF) (first frustum in inscribed figure)= BD* : PN*Similarly= BD :(second frustum in BF):TN.(second in inscribed figure)= HN:SM,and so on. The last frustum in the cylinder BF has none to

correspondON CONOIDS AND SPHEROIDS 61to it inthe ratio as (BD :the inscribed figure,and we should writezero).Archimedes concludes, by means of a lemma in proportionsforming Prop. 1, that(frustum BF):(inscribed figure)where XO = k, so that BD = TI/J.In like manner,lie concludes that(frustum BF):(circumscribed figure)But, by the Lemma preceding Prop. 1,whence(Frustum BF):(inscr. fig.)> 2 > (frustum BF):(circumscr. fig.).This indicates the desired result, which is then confirmed bythe method of exhaustion, namely that(frustum BF) = 2 (segment of paraboloid),V or, if be the volume of the ' segment of a cone ', with vertexA and base the same as that of the *segment,(volume of segment) = f V.Archimedes, it will be seen, proves in ett'ect that, if k beindefinitely diminished, and n indefinitely increased, while nkremains equal to c tthenlimit of k{k+2k + 3lc+...+(u- l)k\= |

62 ARCHIMEDESII.In the case of the hyperboloid (Props. 25, 26)let the axisAD be divided into n parts, each of length h, and let AA'*=a.Then the ratio of the volume of the frustum of a cylinder onthe ellipse of which any double ordinate QQ' isan axis to thevolume of the corresponding portion of the whole frustum BFtakes a different form ; for, if AM = rh, we have(frustum in BF) := BD* :QM*(frustum on base QQ')= AD.A'D:AM.A'MBy means of this relation Archimedes proves that(frustum BF):(inscribed figure)and(frustum BF) :(circumscribed figure)= n {a . nh + (nh)2} : S^* [ a . rh + (rh) 2 } .But, by Prop. 2,From these relations it is inferred that(frustum BF):(volume of segment) = (a + nh):(%a + ^ nh),or (volume of segment) : (volume of cone ABB')= (AD +30 A): (AD + 20 A);and this is confirmed by the method of exhaustion.The result obtained by Archimedes is equivalent to provingthat, if h be indefinitely diminished while n is indefinitelyincreased but nh remains always equal to 6, thenlimit ofor limit of ti n = & 2 (%a + &),76where

ON CONOIDS AND SPHEROIDS 63so thaiThe limit of this latter expressionis what we should write,6IJoand Archimedes's procedureis the equivalent of this integration.III. In the case of the spheroid (Props. 29, 30) we takea segmentless than half the spheroid.As in the case of the hyperboloid,(frustum in BF) :(frustum on ba vse QQ')= BD 2 :QM 2= AD.A'DiAM.A'M\but, in order to reduce the summation to the same as that inProp. 2, Archimedes expresses AM . A'M in a different formequivalent to the following.Let AD (=b) be divided into n,equal parts of length h,and suppose that A A''= a, CD = \c.Then AD.4'/) =2Ja -Je 2 ,and AM . A'M = a 2 - 2(c + rh)(DM = r/i)Thus in this case we have(frustum BF) :arid(frustum BF) :(inscribed figure)= n (cb + b 2 ) : [n (cb + 6 2 ) -^{c.rk + (rh)2} ](circumscribed figure)= n (cb + b 2 ) : [n (cb + b 2 )- S^"1And, since b = nh, we have, by means of Prop. 2,n(cb + b 2 ) : [n(cb + b 2 )-^n {c . rh + (rh) 2 }]J c . rh + (rh) 2 } ].

64 ARCHIMEDESThe conclusion, confirmed as usual byis thatthe method of exhaustion,(frustum BF):(segment of = spheroid) (c + b): {c + b - (|c + ^b) }= (o + 6):fto + |6),whence (volume of segment) : (volume of cone ABB')= (3CA-AD):(2CA-AD), since GA = JAs a particular case (Props. 27, 28), half the spheroidisdouble of the corresponding cone.Props. 31, 32, concluding the treatise, deduce the similarformula for the volume of the greater segment, namely, in ourfigure,(greater segmt.) :(cone or segmt.of cone with same base and axis)On Spirals.The treativse On Spirals begins with a preface addressed toDositheus in which Archimedes mentions the death of Cononas a grievous loss to mathematics, and then summarizes themain results of the treatises On the Sphere and Cylinder andOn Conoids and Spheroids, observing that the last two propositionsof Book II of the former treatise took the placeof two which, as originally enunciated to Dositheus, werewrong; lastly,he states the main results of the treatiseOn Spirals, premising the definition of a spiral which is asfollows :1If a straight line one extremity of which remains fixed bemade to revolve at a uniform rate in a plane until it returnsto the position from which it started, and if, at the same timeas the straight line is revolving, a point move at a uniformrate along the straight line, starting from the fixed extremity,the pointwill describe a spiral in the plane.'As usual, we have a series of propositions preliminary tothe main subject, first two propositions about uniform motion,

ON SPIRALS 65Prop. 5 states that,then .two simple geometrical propositions, followed by propositions(5-9) which are all of one type.given a circle with centre 0, a tangent to it at A, and c, theFIG. 1.circumference of any circle whatever, it is possibleto drawa straight line OPF meeting the circle in P and the tangentin F such thatFP : OP< (arc AP):Archimedes takes I) a straight line greater than c> 9draws077 parallelto the tangent at A and then says*let PH beplaced equal to 7) verging (vtvovora} towards A '. This is theusual phraseology of the type of problem known as i/eCo-*?where a straight line of given length has to be placed betweentwo lines or curves in such a position that, if produced, itpasses through a given point (this is the meaning of verging).Each of the propositions 5-9 depends on a vtvvis of this kind,c.FIG. 2.which Archimedes assumes as 'possible without showing how'it is effected. Except in the case of Prop. 5, the theoreticalsolution cannot be effected by means of the straight line andcircle; it depends in general on the solution of an equationof the fourth degree, which can be solved by means of tbe

66 ARCHIMEDESpoints of intersection of a certain rectangular hyperbolaand a certain parabola. It is quite possible, however, thatsuch problems were in practice often solved by a mechanicalmethod, namely by placing a ruler, by trial, in the position ofthe required line : for it is only necessary to place the rulerso that it passes through the given point and then turn itround that point as a pivottill the intercept becomes of thegiven length. In Props. 6-9 we have a circle with centre 0,a chord AB less than the diameter in it, OM the perpendicularfrom on AB, BT the tangent at B OT they straight linethrough parallel ;to ABD Eas the case may be, than the ratio BM ::isany ratio less or greater,MO. Props. 6, 7line OFP(Fig. 2) show that it is possible to draw a straightFIG. 3.PBD: Emeeting AB in F and the circle in P such that FP :(OP meeting AB in the case where D\E BM :MO). In Props. 8, 9(Fig. 3) it is proved that it is possible to draw a straight lineOFP meeting AB in F, the circle in P and the tangent at B in

'Than D :E > BM :> OB :ON SPIRALS 67MO, by hypothesis,BT, by similar triangles.Take a straight line P'H' (less than :BT)P'H', and place P'H' between the circle and OTverging towards B'(construction assumed).= OB :Then F'P' : P'B = OP' : P'H'Prop. 8. If D := OB := D:E.P'H'such that DEE isany given ratio < BM: MO, it is requiredto draw OFPG meeting AB in F, the circle in P, and thetangentat B to the circle in G so thatFP : BG = D :E.If OT is parallel to AB and meets the tangent at B in T,BM: MO = OB :BT, by similar triangles,D:E BT.Describe a circle through the three points 0, T, C and let OBproduced meet this circle in K.'Then, since BC > BT, and OK is perpendicular to GT, it ispossible to place QG [between the circle TKG and BC] equal toBK and verging towards'(construction assumed).F- 2

68 ARCHIMEDESLet QGO meet the original circle in P and AB in F.OFPG is the straight line required.ForButwhenceTherefore CG .whenceCG.GT=OG.GQ = OG. BK.OF: OG = BT: GT, by parallels,OF.GT=OG.BT.GT : OF . GT = OG . BK : OGCG:OF=BK:BT= BC:OB= BC:OP.Therefore OP : OF = BC :and hence PF: OP = BG :or PF: BG = OU :CG,BC,.J56 Y = 1) : K.BT,ThenPappus objects to Archimedes's use of the i/C(r*y assumed inProp. 8, in those words :'it seems to be a grave error into which geometersfallwhenever any one discovers the solution of a plane problemby means of conies or linear (higher) curves, or generallysolves itby means of a foreign kind, as is the case e.g. (1) withthe problem in the fifth Book of the Conies of Apolloniusrelating to the parabola, and (2) when Archimedes assumes inhis work on the spiral a i/sCcny of a " solid " character withreference to a circle ;for it is possible without callingin theaid of anything solid to find the proof of the theorem given byArchimedes, that to is, prove that the circumference of thecircle arrived at in the first revolution is equal to the straightline drawn at right angles to the initial line to meet the tangentto the spiral the (i.e. subtangent)/There is, however, this excuse for Archimedes, that he onlyassumes that the problem can be solved and does not assumethe actual solution. Pappus l himself gives a solution of theparticular i/evcris by means of conies. Apollonius wrote twoBooks of i>i5

ON SPIRALS 69Conoicls and Spheroids involving the summation of the series! 2 + 2 2 -f 3 2 -f ... + n 2 .Prop 11 proves another proposition insummation, namely that> (na)2:{na . a 4- -J (na a) 2 }The same propositionis also true if the terms of the seriesare a 2 , (a + 6)2, (a + 2b)2 ... (a + ?il6) a ,and it is assumed inthe more general form in Props. 25, 26.Archimedes now introduces his Definitions, of the spiralitself, the origin, the initial line, the first distance (= theradius vector at the end of one revolution), the second distance(= the equal length added to the radius vector during thesecond complete revolution), and so on ;the first area (the areabounded by the spiral described in the first revolution andthe 'first distance '),the second area (that bounded by the spiraldescribed in the second revolution and the 'second distance '),and so on; the first circle (the circle witli the 'first distance'as radius), the second circle (the circle with radius equal to thesum of the 'first* and 'second distances', or twice the firstdistance), and so on.Props. 12, 14, 15 give the fundamental property of thespiral connecting the length of the radius vector with the anglethrough which the initial line has revolved from its originalposition, and corresponding to the equation in polar coordinatesr = a 0. As Archimedes does not speak of angles greaterthan TT,or 2 TT, he has, in the case of points on any turn afterthe first, to use multiples of the circumferenceof a circle as well as arcs of it. He uses the'first circle* for this purpose. Thus, if P, Qare two points on the first turn,OP :OQ= (arc AKP'):(arc AK (/) ;if P, Q are points on the nth turn of thespiral, and c is the circumference of the first circle,Prop. 13 proves that, if a straight line touches the spiral, it

70 ARCHIMEDEStouches it at one point only. For, if possible, let the tangentat P touch the spiral at another point Q. Then, if we bisectthe angle POQ by OL meeting PQ, in L and the spiralin 11,OP + OQ20R by the property of the spiral.But bythe property of the triangle (assumed, but easily proved)OP + OQ> 20L, so that OL < OR, and some point of PQlies within the spiral. Hence PQ cuts the spiral,contrary to the hypothesis.which isProps. 16, 17 prove that the angle made by the tangentat a point with the radius vector to that pointis obtuse on the''forward sMe, and acute on the backward * ' side, of the radiusvector.Props. 18-20 give the fundamental proposition about Jhetangent, that is to say, they give the length of the subtanyentat any point P (the distance between and the point of intersectionof the tangent with the perpendicular from to OP).Archimedes always deals first with the first turn and thenwith any subsequent turn, and with each complete turn beforeparts or points of any particular turn. Thus he deals withtangents in this order, (1) the tangent at A the end of the firstturn, the (2) tangent at the end of the second and any subsequentturn, (3) the tangent at any intermediate point of thefirst or any subsequent turn. We will take as illustrativethe case of the tangent at any intermediate point P of the firstturn (Prop. 20).If A be the initial line, P any point on the first turn, PTthe tangent at P and OT perpendicular to OP, then it is to beproved that, if ASP be the circle through P with centre 0,meeting PT in $, then(subtangent OT) = (arc ASP).I. If possible, let OT be greater than the arc ASP.Measure off OU such that OU > arc AS'P but < OT.Then the ratio PO:OU is :greaterthan the ratio PO OT,i.e. greater than the ratio of %PS to the perpendicular fromon PS.Therefore (Prop. 7) we can draw a straight line OQF meetingTP produced in F, and the circle in Q, such that

ON SPIRALS71Let OF meet the spiral in Q'.Then we have, alternando, since PO = QO,< (arc PQ):(arc ASP), by hypothesis and a fortiori.Componendo, FO:QO < (arc:4#Q)

72 ARCHIMEDESLet OF'G meet the spiralin R'.Then, sincePO = RO, we have, alter nando,> (arc:PR) (arc ASP), a fortiori,whence F'O : RO < (arc:ASR) (arc 4/SP)< OR': OP,so that jF'O < OJS';wliich is impossible.Therefore OT is not less than the arc ASP. And it wasproved not greater than the same arc. ThereforeAs particular cases (separately proved by Archimedes),ifP be the extremity of the first turn and C Tthe circumferenceof the first circle, the subtangent = q ;if P be the extremityof the second turn and c tttthe circumference of the 'secondcircle', the subtangent = 2r and2 generally,if c n be the;circumference of the nth circle (the circle with the radiusvector to the extremity of the nth turn as radius), the subtangentto the tangent at the extremity of the nth turn = nc n .If P is a point on the nth turn, not the extremity, and thecircle withas centre and OP as radius cuts the initial linein K, while p is the circumference of the circle, the subtangentto the tangent at P = (n l)p + arc KP (measuredcforward').*The remainder of the book (Props. 21-8)is devoted tofinding the areas of portions of the spiraland its severalturns cut off by the initial line or any two radii vectores.We will illustrate by tlie general case (Prop. 26). TakeOBy 0(7, two bounding radii vectores, including an arc BGof the spiral. With centre and radius 00 describe a circle.Divide the angle BOG into any number of equal parts byradii of this circle. The spiral meets these radii in pointsP, Q ... F, Z such that the radii vectores OJB, OP, OQ ... OZ, 001On the whole course of Archimedes's proof of the propertysubtangent, see note in the Appendix.of the

ON SPIRALS 73are in, arithmetical progression. Draw arcs of circle* withradii OB, OP, OQ ...as shown; this produces a figure circumscribedto the spiral and consisting of the sum of small sectorsof circles, and an inscribed figure of the same kind. As thefirst sector in the circumscribed figure is equal to the secondsector in the inscribed, it is easily seen that the areas of thethe differencecircumscribed and inscribed figures differ bybetween the sectors OzG and OBp''; therefore, by increasingthe number of divisions of the angle BOG, we can make thedifference between the areas of the circumscribed and inscribedfigures as small as we please ;we have, therefore, theelements necessary for the application of the method ofexhaustion.If there are n radii OB, OP ... 00, there are (n 1) parts ofthe angle BOG. Since the angles of tall the small sectors areequal, the sectors are as the square on their radii.Thus (whole sector 0//C 1 ) : (circumscribed figure)= (TI- l)OC*:(OP 2 + OQ 2 + ... + OC 2 ),and (whole sector Ob'C) : (inscribed figure)

74 ARCHIMEDESAnd OB, OP, OQ,. . .OZ, OG is an arithmetical progressionof n terms ; therefore (cf. Prop.1 1 and Cor.),-( 1)OC 2 :(OP 2 + OQ 2 + . . . + OC 2 )Compressing the circumscribed and inscribed figures togetherin the usual way, Archimedes proves by exhaustion that(sector OVC) : (area of spiral OBC)If 05 = 6, OC=c, and (c-b)= (n-l)h, Archimedes'sresult is the equivalent of saying that, when h diminishes and11 increases indefinitely, while c b remains constant,limit of h {1that is,with our notation,!'JbIn particular, the area included by the first turn and theinitial line is bounded by the radii vectores andthe area, therefore, is to the circle with radius 2?ra as ^|to (2?ra)2,that is to say, it is^ of the circle or ^This is separately proved in Prop. 24 by means of Prop. 10and Corr. 1, 2.The area of the ring added while the radius vector describesthe second turn is the area bounded by the radii vectores 2 no,and 4?ra, and is to the circle with radius 4?ra in the ratiothe ratio is 7 : 12 (Prop. 25).If jRjbe the area of the first turn of the spiral bounded bythe initial line, JK 2the area of the ring added by the secondcomplete turn, R. 6that of the ring added by the third turn,and so on, then (Prop. 27)Also ^=

Lastly, ifON SPIRALS 75E be the portion of the sector b'OC bounded byb'B, the arc b'zC of the circle and the arc BG of the spiral, andF the portion cut oft 1between the arc BG of the spiral, theradius OG and the arc intercepted between OB and OG ofthe circle with centreand radius OJ5, it isproved thatE:F= {05 + (0(7- 05)}:(05+ 4(0(7-05)} (Prop. 28).On Plane Equilibriums, I, II.In this treatise we have the fundamental principles ofmechanics established by the methods of geometry in itsstrictest sense. There were doubtless earlier treatises onmechanics, but itmay be assumed that none of them hadbeen worked out with such geometrical rigour. Archimedesbegins with seven Postulates including the following principles.Equal weights at equal distances balance ; if unequalweights operate at equal distances, the larger weighs downthe smaller. If when equal weights are in equilibrium somethingbe added to, or subtracted from, one of them, equilibriumis not maintained but the weight which is increased or is notdiminished prevails.When equal and similar plane figurescoincide if applied to one another, their centres of gravitysimilarly coincide; and in figures which are unequal butsimilar the centres of gravity will be 'similarly situated'.In any figure the contour of which is concave in one and thesame direction the centre of gravity must be within the figure.Simple propositions (15) follow, deduced by reductio adabsurdum', these load to the fundamental theorem, provedfirst for commensurable and then by reductio ad abswrdumfor incommensurable magnitudes, that Two magnitudes,whether commensurable or incommensurable, balance at. distancesreciprocally proportional to the magnitudes (Props.6, 7). Prop. 8 shows how to find the centre of gravity ofa part of a magnitude when the centres of gravity of theother part and of the whole magnitude are given. Archimedesthen addresses himself to the main problems of Book I, namelyto find the centres of gravity of (l)a parallelogram (Props.9, 10), (2) a triangle (Props. 13, 14), and (3) a paralleltrapezium(Prop. 15), and here we have an illustration of theextraordinary rigour which he requires in his geometrical

76 ARCHIMEDESproofs. We do not find him here assuming, as in The Method,that, if all the lines that can be drawn in a figure parallel to(and including) one side have their middle points in a straightline, the centre of gravity must lie somewhere on that straightline ;he is not content to regard the figure as made up of aninfinity of such parallel lines; pure geometry realizes thatthe parallelogramis made up of elementary parallelograms,indefinitely narrow if you please, but still parallelograms, andthe triangle of elementary trapezia, not straight lines, sothat to assume directly that the centre of gravity lies on thestraight line bisecting the parallelograms would really bea petitio principii. Accordingly the result, no doubt discoveredin the informal way, is clinched by a proof by reductioad absurdum in each case. In the case of the parallelogramABCJJ (Prop. 9),if the centre of gravityis not on the straightline EF bisecting two opposite sides, let it be at H. DrawHK parallel to AD. Then it is possible by bisecting AE, ED,then bisecting the halves, and so on, ultimately to reacha length less than KIL Let this be done, and through thepoints of division of AD draw parallels to AB or DO makinga number of equal and similar parallelograms as in the figure.The centre of gravity of each of these parallelograms issimilarly situated with regard to it. Hence we have a numberof equal magnitudes with their centres of gravity at equaldistances along a straight line. Therefore the centre ofgravity of the whole is on the line joining the centres of gravityof the two middle parallelograms (Prop. 5, Cor. 2).But thisis impossible, because // is outside those parallelograms.Therefore the centre of gravity cannot but lie on EF.Similarly the centre of gravity lies on the straight linebisecting the other opposite sides AB, CD; therefore it lies atthe intersection of this line with EF, i.e. at the point ofintersection of the diagonals.

ON PLANE EQUILIBRIUMS, I 77Tho proof in the case of the triangleis similar (Prop. 13).Let AD be .the median through A. The centre of gravitymust lie on AD.For, if not, let it be at //, and draw 111 parallelto BC.Then, if we, bisect DC, then bisect the halves, and so on,we shall arrive at a length DE less than IH. Divide BC intolengths equal to DE, draw parallels to DA through the pointsof division, and complete the small parallelogramsas shown inthe figure.The centres of gravity of the whole parallelograms 8N, Tl\lie on AD (Prop. 9) ;therefore the centre of gravity of theligure formed by them all lies on AD\ let it bo (>. Join OH,and produceit to meet in F the parallel through C to AD.n be the number of parts intoNow it is easy to see that, ifwhich DC, AC are divided respectively,(sum of small teAMR, MLti ... ARN, NUP ...):(AttC)whence~ 1 : it,;(sum of small As) : (sum of parallelograms) = 1 : (?i I).Therefore the centre of gravity of the figure made up of allthe small trianglesis at a point X on OH produced such thatBut VII : 110 < CE : EDor (n - 1) : 1 ;therefore XH > VII.It follows that the centre of gravity of all the smalltriangles taken together lies at X notwithstanding that allthe triangles lie on one side of the parallel to A D drawnthrough X : which is impossible.

78 ARCHIMEDESHence the centre of gravity of the whole triangle cannotbut lie on AD.It lies, similarly, on either of the other two medians; sothat it is at the intersection of any two medians (Prop. 14).Archimedes gives alternative proofs of a direct character,both for the parallelogram and the triangle, depending on thepostulate that the centres of gravity of similar figures are*similarly situated' in regard to them (Prop.parallelogram, Props. 11, 12 and part 2 of Prop.10 for the13 for thetriangle).The geometry of Prop. 15 deducing the centre of gravity ofa trapeziumis also interesting.It is proved that, if AD, BOare the parallel sides (AD being the smaller), and EF is thestraight line joining their middle points, the centre of gravityis at a point(? on EF such thatGE:GF = (2BC + AD):(2AD + BC).Book II of the treatise is entirely devoted to finding thocentres of gravity of a parabolic segment (Props. 1-8) andof a portion of it cut off by a parallel to the base (Props. 9, 10).Prop. 1 (really a particular case of I. 6, 7) proves that, if P, P'be the areas of two parabolic segments and 7J,E their centresof gravity, the centre of gravity of both taken together isat a pointon DE such thatP:P'=CE:CD.

ON PLANE EQUILIBRIUMS, I, II 79This is % merely preliminary. Then begins the real argument,the course of which is characteristic and deserves to be set out.Archimedes uses a series of figures inscribed to the segment,as he says,'in the recognized manner' (yi/o>pf/bia>?).The ruleis as follows.Inscribe in the segment the triangle ABB' witlithe same base and height; the vertex A is then the pointof contact of the tangent parallel to BB'. Do the same withthe remaining segments cut off by AB, AB', then with thesegments remaining, and so on. If BRQlÂP'Q'R'B' is sucha figure, the diameters through Q, Q', P, P', 11, R' bisect thestraight lines ABy AB', AQ, AQ\ QB y $]? respectively,BB and/ is divided by the diameters into parts which are allequal. It is easy to prove also that PP', QQ', RR' are allparallel to BB' yand that AL:LM:MN:NO = 1:3:5:7, thesame relation holding if the number of sides of the polygonis increased; i.e. the segments of AO arc always in the ratioof the successive, odd numbers (Lemmas to Pi-op. 2).Thecentre of gravity of the inscribed figure lies on AO (Prop. 2).If there be two parabolic segments, and two figures inscribedin them 'in the recognized manner' with an equal ndmber ofsides, the centres of gravity divide the respective axes in thesame proportion, for the ratio depends on the same ratio of oddnumbers 1:3:5:7... (Prop. 3).The centre of gravity of theparabolic segment itself lies on the diameter AO (this is provedin Prop. 4 by reductio ad absurdum in exactly the same wayas for the triangle in I. 13). It is next proved (Prop. 5) thatthe centre of gravity of the segmentis nearer to the vertex Athan the centre of gravity of the inscribed figure is; but thatit is possible to inscribe in the segment in the recognizedmanner a figure such that the distance between the centres ofgravity of the segment and of the inscribed figureany assigned length, for we have only to increase the numberof sides sufficiently (Prop. 6). Incidentally, it is observed inProp. 4 that, if in any segment the triangle with the samebase and equal heightis inscribed, the triangle is greater thanis less thanhalf the segment, whence it follows that, each time we increasethe number of sides in the inscribed figure, we take awaymore than half of the segments remaining over ;and in Prop. 5that corresponding segments on opposite sides of the axis, e. g.Q'R'B' have their axes equal- and therefore are equal inQRBy

80 ARCHIMEDESarea. Lastly (Prop. 7), if there be two parabolic segments,their centres of gravity divide their diameters in the sameratio (Archimedes enunciates this of similar segments only,but it is true of any two segments and is required of any twosegments in Prop. 8). Prop. 8 now finds the centre of gravityof any segment by using the last proposition.It is thegeometrical equivalent of the solution of a simple equation inthe ratio (m, say) of A G to AO, where G is the centre ofthe sum of the two seg-gravity of the segment.Since the segment = f (&ABB') tments AQB, AQ'B' = %(&ABB').Further, if QD, Q'D' are the diameters of these segments,QD, Q'D' are equal, and, since the centresof gravity //, H' of the segments divideQD, Q'D' proportionally, HH' is parallelto QQ', and the centre of gravity of thetwo segments togetheris at K, the pointwhere HH' meets AO.Now AO = 4^17 (Lemma 3 to Prop.2), and QD = %AO-AV=AV. ButH divides QDdivides AO (Prop. 7);in the same ratio as (7thereforeVK = QH = m. QD = m.AV.Taking moments about A of the segment, the triangle A KB'and the sum of the small segments, we have (dividing out byAV and A ABB')orand in = -|.That is,= 9,or A (f : (]() = 3:2.The final proposition (10) finds the centre of gravityof theportion of a parabola cut oft* between two parallel chords PP' 9BB'. If PP' is the shorter of the chords and the diameterbisecting PP', BB' meets them in N, respectively, ArchimedesNO proves that, if be divided into five equal parts ofwhich LM is the middle one (L being nearer to J\T than M is),

ON PLANE EQUILIBRIUMS, II 81the centre of gravity G of the portion of the parabola betweenPP' and BB' divides LM in such a way thatLG : GM = BO* .(2PN+ BO) PN* : .(2 BO + PN).The geometrical proofis somewhat difficult, and uses a veryremarkable Lemma which forms Prop.9. If a, 6, c, d, x, y arestraight lines satisfying the conditionsa b c ,d _ xa d ~~~ (&, 2a + 4b + 6v + 3dand5a+106--+ 10c + 5then mustx + y = fa.The proof is entirely geometrical, but amounts of course tothe elimination of three quantities 6, c, (/ from the above fourequations.IThe Sand-reckoner (Psammites orArenarius).have already described in a previous chapter the remarkablesystem, explained in this treatise and in a lost work,'Apxai, Principles, addressed to Zeuxippus, for expressing verylarge numbers which were beyond the range of the ordinaryGreek arithmetical notation. Archimedes showed that hissystem would enable any number to be expressed upto thatwhich in our notation would require 80,000 million millionciphers and then proceeded to prove that this system morethan sufficed to express the number of grains of sand whichit would take to fill the universe, on a reasonable view (as itseemed to him) of the size to be attributed to the universe.Interesting as the book is for the course of the argument bywhich Archimedes establishes this, it is, in addition, a documentof the first importance historically. It is here that welearn that Aristarchus put forward the Copernican theory ofthe universe, with the *ûn in the centre and the planetsincluding the earth revolving round it, and that Aristarchusfurther discovered the angular diameter of the sun to be ôthof the circle of the zodiac or half a degree.in order to calculate a safe figure1533.2 ftSince Archimedes,(not too small) for the size

82 ARCHIMEDESof the universe, has to make certain assumptions as to thesizes and distances of the sun and moon and their relationto the size of the universe, he takes the opportunity ofquoting earlier views. Some have tried, he says, to provethat the perimeter of the earth is about 300,000 stades; inorder to be quite safe he will take it to be about ten timesthis, or 3,000,000 stades, and not greater. The diameter ofthe earth, like most earlier astronomers, he takes to bethan that of the sun.than that of the moon but lessgreaterEudoxus, he says, declared the diameter of the sun to be ninetimes that of the moon, Phidias, his own father, twelve times,while Aristarchus tried to prove that it is greater than 1 8 butless than 20 times the diameter of the moori; he will again beon the safe side and take it to be 30 times, but not mom Theposition is rather more difficult as regards the ratio of thedistance of the sun to the size of the universe. Here he seizesupon a dictum of Aristarchus that the sphere of the fixedstars is so great that the circle in which he supposes the earthto revolve (round the sun)'bears such a proportion to thedistance of the fixed stars as the centre of the sphere bears toits surface '.If this is taken in a strictly mathematical sense,it means that the sphere of the fixed stars is infinite in size,which would not suit Archimcdes's purpose to ; get anothermeaning out of it he presses the point that Aristarchus'swords cannot be taken quite literally because the centre, beingwithout magnitude, cannot be in any ratio to any other magnitude;hence he suggests that a reasonable interpretation ofthe statement would be to suppose that,sphere with radius equal to the distance between the centreof the sun and the centre of the earth, thenif we conceive a(diam. of earth) Cdiam. of said :sphere)= (diam. of said sphere):(diam. of sphere of fixed stars).This is, of course, an arbitrary interpretation ;Aristarchuspresumably meant no such thing, but merely that the size ofthe earth is negligible in comparison with that of the sphereof the fixed stars. However, the solution of Archimedes'sproblem demands some assumption of the kind, and, in makingthis assumption, he was no doubt awai;c that he was takinga liberty with Aristarchus for the sake of giving his hypothesisan air of authority.

THE SAND-RECKONER 83Arahimedes has, lastly, to compare the diameter of the sunwith the circumference of the circle described by its centre.Aristarchus had made the apparent diameter of the sun y^thof the said circumference ;Archimedes will prove that thesaid circumference cannot contain as many as 1,000 sun'sdiameters, or that the diameter of the sun is greater than theside of a regular chiliagon inscribed in the circle. First hemade an experiment of his own to determine the apparentdiameter of the sun. With a small cylinder or disc in a planeat right angles to a long straight stick and movcablc along it,he observed the sun at the moment when it cleared thehorizon in rising, moving the disc till it just covered and justfailed to cover the sun as he looked along the straight stick.He thus found the angular diameter to lie between Tf^JB and^ J Li, where R is a right angle. But as, under his assumptions,the size of the earth is not negligible in comparison withthe sun's circle, he had to allow for parallax and find limitsfor the angle subtended by the sun at the centre of the earth.This hr does by a geometrical argument very much in themanner of Aristarchus.Let the circles with centres 0, C represent sections of the sunand earth respectively, E the position of the observer observingG2

84 ARCHIMEDESthe sun when it has just cleared the horizon. Draw from Etwo tangents EP, EQ to the circle with centre 0, and fromC let CF, GG be drawn touching the same circle. With centreC and radius CO describe a circle :this will represent the pathof the centre of the sun round the earth. Let this circle meetthe tangents from C in A, B, and join AB meeting CO in M.Archimedes's observation has shown that^R> Z.PEQ >jfaR;and he proceeds to prove that AB is less than the side of aregular polygon of 656 sides inscribed in the circle AOB,but greater than the side of an inscribed regular polygon of1,000 sides, in other words, thatThe first relation is obvious, for, since CO > EO,Next, the perimeter ofL PEQ > Z FCG.any polygoninscribed in the circleAOB is less than ^ CO-2 2 - (i.e. T times the diameter) ;Therefore AB < ^ -\ 4 -CO or T | CO,AB < T^ CO.and, a fortiori,Now, the triangles CAM, COF being equal in allrespects,AM= OF, so that AB = 20F= (diameter of sun) > C//+ OK,since the diameter of the sun is greater than that of the earth ;thereforeCH+OK < yfoCO, and HK > -f^CO.And CO > CF, while HK < EQ, so that EQ > ftWe can now compare the angles OCF, OEQ ;L OCF rtan OCF}II\C\\* ^/ f\ ~Hi~\ I i. f\ I/V1 IDoubling the angles, we haveEQ> CF> -loci a fortiori.

THE SAND-RECKONER 85Hence AB is greater than the side of a regular polygon of812 sides, and a fortiori greater than the side of a regularpolygon of 1,000 sides, inscribed in the circle AOB.The perimeter of the chiliagon, as of any regular polygonwith more sides than six, inscribed in the circle AOB is greaterthan 3 times the diameter of the sun's orbit, but is less than1,000 times the diameter of the sun, and a fortiori less than30,000 times the diameter of the earth;therefore(diameter of sun's orbit) < 10,000 (diam. of earth)< 10,000,000,000 stades.But (diam. of earth) : (diam. of sun's orbit)= (diam. of sim's orbit):(diam. of universe) ;therefore the universe, or the sphere of the fixed stars, is lessthan 10,0003times the sphere in which the sun's orbit is agreat circle.Archimedes takes a quantity of sand not greater thana poppy-seed and assumes that it contains not more than 1 0,000grains ;the diameter of a poppy-seed he takes to be not lessthan 4*Q-thof a finger-breadth ;thus a sphere of diameter1finger-breadth is not greater than 64,000 poppy-seeds andtherefore contains not more than 640,000,000 grains of sand('6 units of second order + 40,000,000 units of first order')and a fortiori not more than 1,000,000,000 ('10 units ofsecond order of numbers '). Gradually increasing the diameterof the sphere by multiplyingit each time by 100 (making thesphere 1,000,000 times larger each time) and substituting for10,000 finger-breadths a stadium (< 10,000 finger-breadths),lie finds the number of grains of sand in a sphere of diameter10,000,000,000 stadia to bo less than '1,000 units of seventhorder of numbers' or 10 fll ,and the number in a sphere 10,000 3times this size to be less than ' 10,000,000 units of the eighthorder of numbers' or 10 c:i .The Quadrature of the Parabola.In the preface, addressed to Dositheus after the death ofCoiion, Archimedes claims originality for the solution of theproblem of finding the area of a segment of a parabola cut offby any chord, which he says he first discovered by means ofmechanics and then confirmed by means of geometry, usingthe lemma that, if there are two unequal areas (or magnitudes

86 ARCHIMEDESgenerally), then however small the excess of the greater overthe lesser, it can by being continually added to itself be madeto exceed the greater in other;words, he confirmed the solutionmeans ofby the method of exhaustion. One solution bymechanics is, as we have seen, given in The Method; thepresent treatise contains a solution by means of mechanicsconfirmed by the method of exhaustion (Props. 1-17), andthen gives an entirely independent solution by means of puregeometry, also confirmed by exhahstion (Props. 18-24).I. The mechanical solution depends upon two properties ofthe parabola proved in Props. 4, 5. If Qq be the base, and Pthe vertex, of a parabolic segment, P isthe point of contactof the tangent parallel to Qq, the diameter PV through Pbisects Qq in and, if VP V9 produced meets the tangent at Qin T, then TP = PV. These properties, along with the fundamentalproperty that QV' 2 varies as Archimedes uses toPV9prove that, if EO be any parallel to TV meeting QT, QP(produced, if necessary), the curve, and Qq in E, t\ 11,respectively, thenQV: VO = OF:FR,and :QO Oq= ER :RO. (Props. 4, 5.)Now suppose a parabolic segment Ql^q so placed in relationto a horizontal straight line QA through Q that the diameterbisecting Qq is at right angles to QA, i.e. vertical, and let thetangent at Q meet the diameter qO through q in E. ProduceQO to A, making OA equal to OQ.Divide Qq into any number of equal parts at O Ol , 2 . . . n ,and through these points draw parallels to OE,i. e. verticallines meeting OQ in H 19 H. 2 , ..., EQ in E 19 E^ ..., and the

THE QUADRATURE OF THE PARABOLA 87curve in R ly R^ ... Join QR . lF, Q/? 2 meeting O El lin t\, and so on.,and produceit to meet OE inO HI H 2 H 3Now Archimedes has proved in a series of propositions(6-13) that, if a trapezium such as 1E 1 EÔ% is suspendedfrom //^Yând an area P suspended at A balances 1 /? 1 A? 2 2so suspended, itwill take a greater area than P suspended atA to balance the same trapezium suspended from J/ 2anda less area than P to balance the same trapezium suspendedfrom //,.A similar proposition holds with regard to a trianglewhere it is and suspended at Q andsuch as E n lllt Q suspendedll n respectively.Suppose (Props. 14, 15) the triangle QqE suspended whereit is from OQ, and suppose that the trapezium EO^ suspendedwhere it is, is balanced by an area l\ suspended at A, thetrapezium A T ,(A,, suspended where it is, is balanced by 7suspended at A, and so on, and finally the triangle M n O n Q,suspended where it is, is balanced by P n+l suspended at A;then Pl+ J^ + . so that. . -f jfji+1 at A balances the whole triangle,.sincethe whole triangle may be regarded as suspended fromthe point on OQ vertically above its centre of gravity.Now AO:OJI l= QO:OJll= #,0^0^, by Prop. 5,= (trapezium EO^):(trapezium

88 ARCHIMEDESthat is, it takes the trapezium F0 lsuspendedat A to balancethe trapezium E0 l suspended at H r And P lbalances E0 lwhere it is.Therefore(FOJ > P rSimilarly (^1^2) > P 'ind so n -Again AO'.OH^ = E, O l 0^:= (trapezium EÔ^:9(trapezium K^0^that is, (R>i0 2 )at A will balance (#a 2 ) suspended at H 19while P 2at A balances (Efl^) suspended where it iswhence F 2> U/^.Therefore (^02) > ^2 > (#1^)*(^2^3) > ^ > ^2â> an(l so on ;and finally,AA T w O n Q > 7^ 41 >By addition,therefore, a fortiori,That is to say, we have an inscribed figure consisting oftrapezia and a triangle which is less, and a circumscribedfigure composed in the same way which is greater, thanIt is therefore inferred, and proved by the method of exhaustion,that the segment itself is equal to %AJKqQ (Prop. 16).In order to enable the method to be applied, it has onlyto be proved that, by increasing the number of parts in Qqthe difference between the circumscribed andsufficiently,inscribed figures can be made as small as we please. Thiscan be seen thus. We have first to show that all the parts, asqF, into which qE is divided are equal.We have EîO^ = QO -.01^ = (n+1): 1,.e.or 0^ = -----. El 19whence also 2$ = --.OÊ2.

THE QUADRATURE OF THE PARABOLA 89And E 2 2:2 jK 2 = QO:0#2= (71+ 1): 2,22 2,^ +1 "' 2 '*'It follows that2 # = /S'JB 2 , and so on.Consequently 1 R IJ O^JR.2J 3 7i a ... are divided into 1, 2, 3 ...equal parts respectively by the lines from Q meeting qE.It follows that the difference between the circumscribed andinscribed figures is equal to the triangle FqQ,which can bemade as small as we please by increasing the number ofdivisions in Qqi.e. in qE.ySince the area of the segment is equal to %AEqQ, and it iseasily proved (Prop. 17) that AEqQ =: 4 (triangle with samebase and equal height with segment),it follows that the areaof the segment = -3times the latter triangle.It is easy to see that this solution isessentially the same asthat given in The Method (see pp. 29-30, above), only in a moreorthodox form (geometrically speaking). For there Archimedestook the sum of all the straight line*, as O lR l , 2R 2 >as making up the segment notwithstanding that there are aninfinite number of them and straight lines have no breadth.Here he takes inscribed and circumscribed trapezia proportionalto the straightlines and havingfinite breadth, and thencompresses the figures together into the segment itself byincreasing indefinitely the number of trapezia in each figure,i.e.diminishing their breadth indefinitely.The procedure is equivalent to an integration, thus :If X denote the area of the triangle FqQ, we have, if n bethe number of parts in (

90 ARCHIMEDESTaking the limit, we have, if A denote the area of thetriangle EqQ, so that A = nX,1 C Aarea of segment = -r^" JoJII. The purely geometrical method simply exhausts theparabolic segment by inscribing successive figures ' in therecognized manner' (see p. 79, above). For this purposeit is necessary to find, in terms of the triangle with the samebase and height, the area added to therespectively.Now QV* = 4/ilK2inscribed figure by doubling the number ofsides other than the base of the segment.Let QPq be the triangle inscribed cin therecognized manner', P being the point ofcontact of the tangent parallel to Qq, andPV the diameter bisecting Qq. If QV, Vqbe bisected in M, m, and RM, rm be drawnparallel to PV meeting the curve in R, r,the latter points are vertices of the nextfigure inscribed cin the recognized manner ',for RY, ry are diameters bisecting PQ, Pq,so that PV = 4PW, or RM = 3P\\ r .But YM = APF = 2P H', so that YM = 2 RY.Therefore A PRQ = | A PQM = A PQ ] r .SimilarlyAPr0 = JAPFg; whence (APRQ + &Prq)= %PQq. (Prop. 21.)In like manner it can be proved that the next additionto the inscribed figure adds % of the sum of AsPRQ, Prq,and so on.Therefore the area of the inscribed figure= { 1+ i + (J)a+...} AP

THE QUADRATURE OF THE PARABOLA 91the triangle PQq is half of the parallelogramand thereforemore than half the segment. And so on (Prop. 20).We now have to sum u terms of the above geometricalseries. Archimedes enunciates the problem in the form, Givena series of areas A, B, C, D . . .%, of which A is the greatest, andeach is equal to four times the next in order, then (Prop. 23)The algebraical equivalent of this is of courseTo find the area of the segment, Archimedes, instead oftaking the limit, as we should, uses the method of reductio adabswrdwin.Suppose K =-A7J (^/.(1) If possible, let the area of the segment be greater than K.We then inscribe a figure'in the recognized manner ' suchthat the segment exceeds it by an area less than the excess ofmust bethe segment over K. Therefore the inscribed figuregreater than Ar which is ,impossible sincewhere A = A7% (Prop. 23).(2) If possible, let the area of the segment be less than K.= J/J, and so on, until weFf then &PQq = A, K = %Atarrive at an area A" less than the excess of K over the area ofthe segment, we haveA +]* + (! + ... +X + $X = $A = K.Thus K exceeds A -f K + (J+ ...and exceeds the segment by an area greater+ X by an area less than A",than X.It follows that A +H + G+ ... +X> (the segment) ;whichis impossible (Prop. 22).Therefore the area of the segment, being neither greater norless than K, is equal to A' or %On Floating Bodies, I, II.In Book J of this treatise Archimedes lays down the fundamentalprinciples of the science of hydrostatics. These are

92 ARCHIMEDESdeduced from Postulates which are only two in number.Thefirst which begins Book I is this :'let itbe assumed that a fluid is of such a nature that, of theparts of it which lie evenly and are continuous, that which ispressed the less is driven along by that which is pressed themore; and each of its parts is pressed by the fluid whicli isperpendicularly above it except when the fluid is shut up inanything and pressed by something else ' ;the second, placed after Prop. 7,saysclet it be assumed that, of bodies which are borne upwards ina fluid, each is borne upwards along the perpendicular drawnthrough its centre of gravity '.Prop. 1 is a preliminary proposition about a sphere, andthen Archimedes plunges in med'Utu res with the theorem(Prop. 2) that 'the surface of any fluid at rest is a sjthere thecentre of which is the same as that of the earth ',and in thewhole of Book I the surface of the fluid isalways shown inthe diagrams as spherical. The method of proofis similar towhat we should expect in a modern elementary textbook, themain propositions established being the following. A solidwhich, size for size, is of equal weight with a fluid will, if letdown into the fluid, sink till it is just covered but not lower(Prop. 3) ; a solid lighter than a fluid will, if let down into it,be only partly immersed, in fact just so far that the weightof the solid is equal to the weight of the fluid displaced(Props. 4, 5), and, if it is forcibly immersed, it will be drivenupwards by a force equal to the difference between its weightand the weight of the fluid displaced (Prop. 6).The important proposition follows (Prop. 7) that a solidheavier than a fluid will, if placed in it, sink to the bottom ofthe fluid, arid the solid will, when weighed in the fluid, belighter than its true weight by the weightdisplaced.of the fluidThe problem of theCrown.This proposition gives a method of solving the famousproblem the discovery of which in his bath sent Archimedeshome naked crying evpijKa, tvprjKa, namely the problem of

ON FLOATING BODIES, I 93determining the proportions of goldand silver in a certaincrown.Let W be the weight of tho crown, w land u\ the 2 weights ofthe gold and silver in it respectively, so that W = w l+ w 2.(1) Take a weight PFof pure gold and itweigh in the fluid.The apparent loss of weight is then equal to the weight of thefluid displaced this is ascertained by weighing. Let it be F r;It follows that the weight of the fluid displaced by a weightM! of gold is ^.Fr(2)Take a weight W of silver, and perform the sameoperation. Let the weight of the fluid displaced be F 2 .Then the weight of the fluid displaced by a weight iv 2 ofsilver is ^ F2 .(3) Lastly weigh the crown itself in the fluid, and let F beloss of weight or the weight of the fluid displaced.We have then*. Fl+ TTJ?. F2= F,that is, '!F l -f w 2 F^ = (w i+ w>) F,whence = J 2 , TAccording to the author of the poem de etponderibus mensuriH(written 'probably about A.D. 500) Archimedes actuallyused a muthod of this kind. We first take, says our authority,two equal weights of gold and silver respectively and weighthem against each other when both are immersed in water;this gives the relation between their weights in water, andtherefore between their losses of weight in water. Next wetake the mixture of gold and silver and an equal weight ofsilver, and weigh them against each other in water in thesame way.Nevertheless I do not think it probable that this was theway in which the solution of the problem was discovered. Aswe are told that Archimedes discovered it in his bath, andthat he noticed that, ifthe bath was full when he entered it,so much water overflowed as was displaced by his body,he ismore likely to have discovered the solution by the alternative

94 ARCHIMEDESmethod attributed to him by Vitruvius, 1namely by measuringsuccessively the volumes of fluid displaced by three equalweights, (1) the crown, (2) an equal weight of gold, (3) anequal weight of silver respectively. Suppose, as before, thatthe weight of the 'crown is W and that it contains weightsw land w.j,of gold and silver respectively. Then(1) the crown displaces a certain volume of the fluid, V, say ;(2) the weight W of gold displaces a volume V v say,fluid;of the7/'therefore a weight ^v lof gold displaces a volume TJTV lofthe fluid ;(3) the weight W of silver displaces F 2,say, of the fluid;therefore a weight m 2of silver displaces -y^-F 2.It follows that V =--yj F, +W'.;i ufifWF,,,whence we derive (since W = iu l+ w> 2 ). _ ~ T/ v2~

ON FLOATING BODIES, I, II 95principal parameter of the generating parabola, is a veritabletour de force which must be read in full to be appreciated.Prop. 1 is preliminary, to the effect that, if a solid lighter thana fluid be at rest in the it, weight of the solid will be to thatof the same volume of the fluid as the immersed portion ofthe solid is to the whole. The results of the propositionsabout the segment of a paraboloid may be thus summarized.Let h be the axis or height of the segment, p the principalparameter of the generating parabola, s the ratio of thespecific gravity of the solid to that of the fluid (s always < 1).The segment is supposed to be always placed so that its baseis either entirely above, or entirely below, the surface of thefluid, and what Archimedes proves in each case is that, ifthe segment is so placed with its axis inclined to the verticalat any angle, it will not rest there but will return to theposition of stain lity.I. If h is not greater than f /?,the positionof stability is withthe axis vertical, whether the curved surface is downwards orupwards (Props. 2, 3).II.If h is greater than f p, then, in order that the position ofstability may be with the axis vertical,than (// f-/>)2//

96 ARCHIMEDESwhere k is the axis of the segment of the paraboloid cut oft' bythe surface of the fluid.)V. Prop. 10 investigates the positions of stability in the caseswhere h/%p>15/4, the base is entirely above the surface, ands has values lying between five pairs of ratios respectively.is theOnly in the case where s is not less than (h-^pY/h?that in which the axis is vertical.position of stabilityBAB lis a section of the paraboloid through the axis AM.G is a point on AM such that AC = 2 CM, K is a point on OAstoch that AM: OK = 15:4. CO is measured along CA suchthat CO = %p, and E is a point on AM such that MR = f CO.A 2 is the point in which the perpendicular to AM from Kmeets AH, and A 3is the middle point of AB. BA 2B 2,BA.^ Mare parabolic segments on A 2M^ A^M.^ (parallel to AM) as axesand similar to the original segment. (The parabolaisproved to pass through C by using the above relationAM: CK =15:4 and applying Prop. 4 of the Quadrature ofthe Parabola.) The perpendicular to AM from meets theparabola BA 2B 2in two points P 2, (

THE CATTLE-PROBLEM 97(ii)AM*2if s< A R* : but > Q, Q 3: AM*, the solid will not restwith its base touching the surface of the fluid in one pointonly, but in a position with the base entirely out of the fluidand the axis making with the surface an angle greaterthan U;(iiia) if s = Q^îAM 2 ,there is stable equilibriumwith onepoint of the base touching the surface and AM inclined to itat an angle equal to U\(iiib)= P if s : 1 P^ AAf 2 there is stable , equilibrium with onepoint of the base touching the surface and with AM inclinedto it at an angle equal to T;(iv) ittoPîAM* but

98 ARCHIMEDESSecondly, it is required thatW+X = a square, (6)Y + Z = a triangular number.(t)There is an ambiguity in the text which makes it just possibleof two whole numbersthat W+ X need only be the productinstead of a square as in (0). Jul. Fr. Wurm solved the problemin the simpler form to which this change reduces it. Thecomplete problem is discussed and partly solved by Amthor. 1The general solution of the first seven equationsisTF= 2.3.7.53.466771= 10366482^,Z= 2. 3 2 . 89. 4657?i = 7460514 n,,F= 3 4 . 11. 465771 = 414938771,Z 2 2 .5.79.4657n = 7358060??,w = 2\3. 5. 7. 23. 37371 = 720636071,X = 2. 32. 17. 15991 n = 4893246/1,y= 3 2 . 13.4648971 = 543921371,2 =2 2 . 3. 5. 7. 11. 761 71 = 3515820n.It is not difficultto find such a value of n that W+ X = asquare number; it is ri/ = 3 . 11 . 29 . 4657 2 = 4456 749 2 ,where isany integer. We then have to makea triangular number, i.e. a number of the form ^#(

ON SEMI-REGULAR POLYHEDRA 99angular, but not similar, polygons; those discovered byArchimedes were 13 in number. If we for conveniencedesignate a polyhedron contained by m regular polygonsof oc sides, n, regular polygons of ft sides, &c., by (?>ia , %...),the thirteen Archimedean polyhedra, which we will denote by1\, 1*, ... /| 3 , are as follows:Figure with 8 faces: 1\ = (4,, 4 C ).Figures with 14 faces: P 2 = (83 , 6 4 ), I\ = (6 4,8 G),I\ = (83, 8 8 ).Figures with 2(> faces: P 5 = (83 , 18 4 ),7 J 6 =(12 4 , 8 G,6 8 ).Figures with 32 faces: P 7 = (20 3,12 5 ), P 8= (12 5,20 6),P = (203 ,12 10).Figure with 38 faces: P 10 ~ (32.,, 6 4).Figures with 62 faces: P = n (20 3 , 30 4,12 5 ),P ]2 EE(30 4 ,20 G ,12 ]0).'Figure with 92 faces: P 1:J= (803,12 5 ).Kepler 1 showed how these figures can be obtained. Amethod of obtaining some of them is indicated in a fragmentof a scholium to the Vatican MS. of Pappus. If a solidangle of one of the regular solids be cut oft* symmetrically bya plane,i.e. in such a way that the plane cuts oft* the samelength from each of the edges meeting at the angle, thesection is a regular polygon which is a triangle, square orpentagon according as the solid angle is formed of three, four,or five plane angles. If certain equal portions be so cut off*from all the solid angles respectively, they will leave regularpolygons inscribed in the faces of the solid this ; happens(A) when the cutting planes bisect the sides of the faces andso leave in each face a polygon of the same kind, and (B) whenthe cutting planes cut off a smaller portion from each angle insuch a way that a regular polygon is left in each face whichhas double the number of sides (as when we make, say, anoctagon out of a square by cutting off the necessary portions,1Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6.H 2

100 ARCHIMEDESsymmetrically, from the corners). We have seen that, accordingto Heron, two of the semi-regular solids had already beendiscovered by Plato, and this would doubtless be his method.The methods (A) and (B) applied to the five regular solidsgive the following out of the 13 semi-regularsolids. Weobtain (1)from the tetrahedron, P l by cutting off anglesso as to leave hexagons in the faces ; (2) from the cube, P 2 byleaving squares, and P 4 by leaving octagons, in the faces ;(3) from the octahedron, P 2 by leaving triangles, and P 3 byleaving hexagons, in the faces ; (4) from the icosahedron,Pj by leaving triangles, and J^ by leaving hexagons, in thefaces; (5) from the dodecahedron, P 7 by leaving pentagons,and P 9 by leaving decagons in the faces.Of the remaining six, four arc obtained by cutting off allthe edges symmetrically and eqiially by planes parallel to theedges, and then cutting off angles. Take first the cube.(1) Cut off from each four parallel edges portions which leavean octagon as the section of the figure perpendicular to theedges ; then cut off equilateral triangles from the corners(see Fig. 1) this gives P ; 5 containing 8 equilateral trianglesand 18 squares. (P5is also obtained by bisecting all theedges of J^ and cutting off corners.) (2) Cut off from theedges of the cube a smaller portion so as to leave in eachface a square such that the octagon described in it has itsside equal to the breadth of the section in which each edge iscut ;then cut off hexagons from each angle (see Fig. 2) ; thisFIG. 1. FIG. 2.gives 6 octagons in the faces, 12squares under the edges andexactly8 hexagons at the corners; that is, we have P 6. An

ON SEMI-REGULAR POLYHEDRA 101similar procedure with the icosahedron and dodecahedronproduces P n and P n (see Figs. 3, 4 for the case of the icosahedron).FIG. 3. FIG. 4.The two remaining solids JJP, VAcannot be so simply produced.They are represented in Figs. 5, 6, which I haveFIG. 5. FIG. G.taken from Kepler. 1* is the snub cube in which eachsolid angle is formed by the angles of four equilateral trianglesarid one square; P riis the snub dodecahedron, each solidangle of which is formed by the angles of four equilateraltriangles and one regular pentagon.We are indebted to Arabian tradition for(y)The Liber Asswnptorum.Of the theorems contained in this collectionmany areso elegant as to afford a presumption that they may reallybe due to Archimedes. In three of them the figure appearswhich was called ap/Si/Ao?, a shoemaker's knife, consisting ofthree semicircles with a common diameter as shown in theannexed figure. If N be the point at which the diameters

102 ARCHIMEDESof the two smaller semicircles adjoin, and NP be drawn atright angles to AS meeting the external semicircle in P, thearea of the apjSijAos (included between the three semicirculararcs) is equal to the circle on PN as diameter (Prop. 4). InProp. 5 it is shown that, if a circle be described in the spacebetween the arcs AP> AN and the straight line PN touchingall three, and if a circle be similarly described in the spacebetween the arcs PB, NB and the straight line PN touchingall three, the two circles are equal. If one circle be describedin the #pj8r/Aoy touching all three semicircles, Prop. 6 showsthat, if the ratio of AN to NB be given,we can find therelation between the diameter of the circle inscribed to theapftrfXo? and the straight line AB] the proofis for the particularcase AN=%BN9 and shows that the diameter of theinscribed circle = ÂB.Prop. 8 is of interest in connexion with the problem oftrisecting any angle. If AB be any chord of a circle withcentre 0, and BG on AB produced be made equal to the radius,draw CO meeting the circle in /), E then will the arc BD be;one-third of the arc AE (or BF, if EF be the chord through Eparallel to AB). The problemisby this theorem reduced toa(cf. vol. i, p. 241).

THE LIBER ASSUMPTORUM 103Lastly, we may mention the elegant theorem about thearea of the Salinon (presumably 'salt-cellar') in Prop. 14.ACB is a semicircle on AS as diameter, AD, EB are equalmeasured from A and B 011 AB. Semicircles arelengthsdrawn with AD, EB as diameters on the side towards (7, anda semicircle with DE as diameter is drawn on the other side ofAB. CF is the perpendicular to AB through 0, the centreof the semicircles ACB, DFE. Then is the area bounded byall the semicircles (the tialiuon) equal to the circle on CFas diameter.The Arabians, through whom the ttook of Lemmas hasreadied us, attributed to Archimedes other works (1) on theCircle, (2) on the Heptagon in a Circle, (3)on Circles touchingone another, (4) on Parallel Lines, (5)on Triangles, (6) onthe properties of right-angled triangles, (7) a book of Data,(8) De clepsydris: statements which we are not in a positionto check.But the author of a book on the finding of chords1in a circle, Abu'l Raihan Muh. al-Birunl, quotes some alternativeproofs as coming from the first of these works.(8)Formula for area of triangle.More important, however, is the mention in this same workof Archimedes as the discoverer of two propositions hithertoattributed to Heron, the first being the problem of findingthe perpendiculars of a triangle when the sides are given, andthe second the famous formula for the area of a triangle interms of the sides,V{s(s-a)(s-b)(s-c)}.1See Bibliotheca mathematica, xi s , pp. 11-78.c

104 ERATOSTHENESLong as the present chapter is, it is nevertheless the mostappropriate place for ERATOSTHENES of Cyrene. It was to himthat Archimedes dedicated The Method, and the Cattle-Problempurports, by its heading, to have been sent through him tothe mathematicians of Alexandria. It is evident from thepreface to The Method that Archimedes thought highly of hismathematical ability. He was, indeed, recognized by his contemporariesas a man of great distinction in all branches ofknowledge, though in each subject he just fell short of thehighest place. On the latter ground he was called Beta, andanother nickname applied to him, Pentathlos, has the sameimplication, representing as it does an all-round athlete whowas not the first runner or wrestler but took the second prizein these contests as well as in others. He was verylittleyounger than Archimedes the date of his birth was;probably284 B.C. or thereabouts. He was a pupil of the philosopherAriston of Chios, the grammarian Lysanias of Cyrene, andthe poet Callimachus ;he is said also to have been a pupil ofZeno the Stoic, and he mayhave come under the influence ofArcesilaus at Athens, where he spent a considerable time.Invited, when about 40 years of age, by Ptolemy Euergetesto be tutor to his son (Philopator), he became librarian atAlexandria ;his obligation to Ptolemy he recognized by thecolumn which he erected with a graceful epigram inscribed onit. This is the epigram, with which we are already acquainted(vol. i, p. 260), relating to the solutions, discovered up to date,of the duplicationof the problemof the cube, and commendinghis own method by means of an appliance called /*eo-oAa/3oi>,itself represented in bronze on the column.Eratosthenes wrote a book with the title UAaroowKoy, and,whether it was a sort of commentary on the Timaeus ofPlato, or a dialogue in which the principal part was played byPlato, it evidently dealt with the fundamental notions ofmathematics in connexion with Plato's philosophy. It wasnaturally one of the important sources of Theon of Smyrna'swork on the mathematical matters which it was necessary forthe student of Plato to know ;and Theon cites the worktwice by name. It seems to have begun with the famousthe story quoted by Theon how theproblem of Delos, tellinggod required, as a means of stopping a plague, that the altar

PLATONIGUS AND ON MEANS 105there, "which was cubical in form, should be doubled in size.The book evidently contained a disquisition on proportion(dva\oyia), a quotation by Theon on this subject shows thatEratosthenes incidentally dealt with the fundamental definitionsof geometry and arithmetic. The principles of musicwere discussed in the same work.We have already described Eratosthenes' s solution of theproblem of Delos, and his contribution to the theory of arithmeticby means of his sieve (KQVKLVQV) for finding successiveprime numbers.He wrote also an independent work On means.This was intwo Books, and was important enough to be mentioned byPappus along with works by Euclid, Aristaeus and Apolloniusas forming part of the Treasury of Analysis 1 ;thisproves that it was a systematic geometrical treatise. Anotherpassage of Pappus speaks of certain loci which Eratosthenes2called 'loci with reference to means' (TOTTOL Trpoy /xeo-orTjray)these were presumably discussed in the treatise in question.What kind of loci these were is quite uncertain ;Pappus (if itis not an interpolator who speaks) merely says that these loci1belong to the aforesaid classes of loci ',but as the classes are''*'numerous (including plane solid ',',linear ',loci on surfaces ',&c.), we are none the wiser. Tannery conjectured that theywore loci of points such that their distances from three fixedstraight lines furnished a 'imnlitftd', i.e. loci (straight linesand conies) which wo should represent in trilinear coordinatesby such equations as 2y = x + 3, y/2= o:s, y(x + z)= 2 ore,x(x y) = z(y z), x(x y) = y(y z), the first three equationsrepresenting the arithmetic, geometric and harmonic means,while the last two represent the ' subcontraries ' to theharmonic and geometric means respectively. Zeutheii hasa different conjecture? He points out that, if QQ' be thepolar of a given point C with reference to a conic, and CPOP'be drawn through C meeting QQ' in and the conic in P, P',then CO is the harmonic mean to CP, CP' ;the locus of forall transversals CPP' is then the straight line QQ'. If A, ttare points on PP fsuch that CA is the arithmetic, and Cti the;1Pappus, vii, p. 636. 24. 2 lb. 9 p. 662. 15 sq.8Zeuthen, Die Lehre von den Keyelschnitten im Altertum, 1886, pp.320, 321.

106 ERATOSTHENESgeometric mean between CP, OP', the loci of A, G respectivelyare conies. Zeuthen therefore suggests that these loci andthe correspondingloci of the points on CPP* at a distancefrom equal to the subcontraries of the geometric andharmonic means between CP and GP' are the 'loci withreference to means ' of Eratosthenes ;the latter two loci are'linear', i.e. higher curves than conies. Needless to say, wehave no confirmation of this conjecture.Eratosthenes s measurement of the Earth.But the most famous scientific achievement of Eratostheneswas his measurement of the earth. Archimedes mentions, aswe have seen, that some had tried to prove that the circumferenceof the earth is about 300,000 stades. This wasevidently the measurement based on observations made atLysirnachia (on the Hellespont) and Syene. It was observedthat, while both these places were on one meridian, the headof Draco was in the zenith at Lysimachia, and Cancer in thezenith at Syeiie ;the arc of the meridian separating the twoin the heavens was taken to be I/ 15th of the complete circle.The distance between the two townswas estimated at 20,000 stades, andaccordingly the whole circumference ofthe earth was reckoned at 300,000stades. Eratosthenes improved on this.He observed (l) that at Syene, atnoon, at the summer solstice, thesun cast no shadow from an uprightgnomon (this was confirmed by theobservation that a well dug at thesame place was entirely lighted up atthe same time), while (2) at the same moment the gnomon fixedupright at Alexandria (taken to be on the same meridian withSyene) cast a shadow corresponding to an angle between thegnomon and the sun's rays of I/ 50th of a complete circle orfour right angles. The sun's rays are of course assumed to beparallel at the two places represented by S and A in theannexed figure. If a be the angle made at A by the sun's rayswith the gnomon (0A produced), the angle 80A is also equal to

MEASUREMENT OF THE EARTH 107a, or r/50th of four right angles. Now the distance from Sto A was known by measurement to be 5,000 stades; itfollowed that the circumference of the earth was 250,000stades. This is the figure given by Cleomedes, but Theon ofSmyrna and Strabo both giveit as 252,000 stades. Thereason of the discrepancyis not known ;it is possible thatEratosthenes corrected 250,000 to 252,000 for some reason,perhaps in order to get a figure divisible by 60 and, incidentally,a round number (700) of stades for one degree. IfPliny is right in saying that Eratosthenes made 40 stadesequal to the Egyptian crxo/oy, then, taking the oôo/cy at12,000 Royal cubits of 0-525 metres, we get 300 such cubits,or 157-5 metres, i.e. 516-73 feet, as the length of the stade.On this basis 252,000 stades works out to 24,662 miles, andthe diameter ofthe earth to about 7,850 miles, only 50 milesshorter than the true polar diameter, a surprisingly closeapproximation, however much it owes to happy accidentsin the calculation.We learn from Heron's Dioptra that the measurement ofthe earth by Eratosthenes was givon in a separate work Onthe Measurement of the Earth. According to Galen 1 this workdealt generally with astronomical or mathematical geography,treating of ' the size of the equator, the distance of the tropicand polar circles, the extent of the polar zone, the size anddistance of the sun and moon, total and partial eclipses ofthose heavenly bodies, changes in the length of the dayaccording to the different latitudes and seasons'. Severaldetails are preserved elsewhere of results obtained byEratosthenes, which were doubtless contained in this work.He is supposed to have estimated the distance between thetropic circles or twice the obliquity of the ecliptic at 1 l/83rdsof a complete circle or 47 42' 39"; but from Ptolemy'slanguage on this subjectit is not clear that this estimate wasnot Ptolemy's own. What Ptolemy saysis that he himselffound the distance between the tropic circles to lie alwaysbetween 47 40' and 47 45', 'from which we obtain about(vytMv) the same ratio as that of Eratosthenes, whichHipparchus also used. For the distance between the tropicsbecomes (or is found to be, yivtrai) very nearly 11 partsGalen, Instit. Logica, 12 (p. 26 Kalbfleiscb).

108 ERATOSTHENESout of 83 contained in the whole meridian circle'. 1 Themean of Ptolemy's estimates, 47 42' 30", is of course nearlyll/83rds of 360. It is consistent with Ptolemy's languageto suppose that Eratosthenes adhered to the value of theobliquity of the ecliptic discovered before Euclid's time,namely 24, and Hipparchus does, in his extant Commentaryon the Phaenomena of Aratus and Eudoxus, say that thesummer tropic is ' very nearly 24 north of the equator '. .The Doxographi state that Eratosthenes estimated thedistance of the moon from the earth at 780,000 stades andthe distance of thesun from the earth at 804,000,000 stades(the versions of Stobaeus and Joannes Lydus admit 4,080,000as an alternative for the latter figure, but this obviouslycannot be right). Macrobius 2 says that Eratosthenes madethe 'measure' of the sun to be 27 times that of the earth.It is not certain whether measure means ' solid content ' or'diameter ' in this case ;the other figures on record make theformer more probable, in which case the diameter of the sunwould be three times that of the earth. Macrobius also tellsus that Eratosthenes's estimates of the distances of the sunand moon were obtained by means of lunar eclipses.Another observation by Eratosthenes, namely that at Syene(which is under the summer tropic) and throughout a circleround it with a radius of 300 stades the upright gnomonthrows no shadow at noon, was afterwards made use of byPosidoiiius in his calculation of the size of the sun. Assumingthat the circle in which the sun apparently moves round theearth is10,000 times the size of a circular section of the earththrough its centre, and combining with this hypothesis thedatum just mentioned, Posidonius arrived at 3,000,000 stadesas the diameter of the sun.Eratosthenes wrote a poem called Hermes containing a gooddeal of descriptive astronomy; only fragments of this have'survived. The work Catasterismi (literally placings amongthe stars ')which is extant can hardly be genuine in the formin which it has reached us it ; goes back, however, to a genuinework by Eratosthenes which apparently bore the same name ;alternatively it is alluded to as KaraAoyoi or by the general1Ptolemy, Syntaxis, i. 12, pp. 67. 22-68. 6.2Macrobius, In Sown. Scip. i. 20. 9.

ASTRONOMY, ETC. 109word 'Acrrpovojiia (Suidas), which latter word is perhaps a mistakefor 'Aa-Tpodeo-ic, corresponding to the title 'Ao-rpoOeo-iaig

XIVCONIC SECTIONS.APOLLONIUS OF PERGAA. HISTORY OF CONICS UP TO APOLLONIUSDiscovery of the conic sections by Menaechmus.WE have seen that Menaechmus solved the problem of thetwo mean proportionals (and therefore the duplication ofthe cube) by means of conic sections, and that he is creditedwith the discovery of the three curves ;for the epigram ofEratosthenes speaks of ' the triads of Monaechmus ',whereasof course only two conies, the parabola and the rectangularhyperbola, actually appear in Menaechmus's solutions. Thequestion arises, how did Menaechmus come to think of obtainingcurves by cutting a cone On ? this we have no informationwhatever. Democritus had indeed spoken of a.section ofa cone parallel and very near to the base, which of coursewould be a circle, since the cone would certainly be the rightcircular cone. But it is probable enough that the attentionof the Greeks, whose observation nothing escaped, would beattracted to the shape of a section of a cone or a cylinder bya plane obliquely inclined to the axis when it occurred, as itoften would, in real life; the case where the solid was cutright through, which would show an ellipse, would presumablybe noticed first, and some attempt would be made toinvestigate the nature and geometrical measure of the elongationof the figure in relation to the circular sections of thesame solid ;these would in the first instance be most easilyascertained when the solid was a right cylinder; it wouldthen be a natural question to investigate whether the curvearrived at by cutting the cone had the same property as thatobtained by cutting the cylinder. As we have seen, the

DISCOVERY OF THE CONIC SECTIONS 111observation that an ellipsecan be obtained from a cylinderEuclid in his Phaeno-as well as a cone is actually made by'mena :if ', says Euclid, a cone * or a cylinder be cut bya plane not parallel to the base, the resulting section is asection of an acute-angled cone which issimilar to a Ovptos(shield)/ After this would doubtless follow the questionwhat sort of curves they are which are ifproduced wecut a cone by a plane which does not cut through thecone completely, but is either parallel or not paralleltoa generator of the cone, whether these curves have thesame property with the ellipse arid with one another, and,if not, what exactly are their fundamental properties respectively.As it is, however, we are only told how the first writers onconies obtained them in actual practice.We learn on thelauthority of Geminus that the ancients defined a cone as thesurface described by the revolution of a right-angled triangleabout one of the sides containing the right angle, and thatthey knew no cones other than right cones. Of these theydistinguished throe kinds ; according as the vertical angle ofthe cone was less than, equal to, or greater than a right angle,they called the cone acute-angled, right-angled, or obtuseangled,and from each of these kinds of cone they producedone and only one of the three sections, the section beingalways made perpendicular to one of the generating lines ofthe cone ;the curves were, on this basis, called ' section of anacute-angled cone* (= an ellipse), 'section of a right-angledcone' (= a parabola), and 'section of an obtuse-angled cone '(= a hyperbola) respectively. These names were still usedby Euclid and Archimedes.Menaechmuss probable procedure.Menaechmus's constructions for his curves would presumablybe the simplest and the most direct that would show thedesired properties, and for the parabola nothing could besimpler than a section of a right-angled cone by a plane at rightangles to one of its generators. Let OBG (Fig. 1) represent1Eutocius, Comm. on Conies of Apollonius.

112 CONIC SECTIONSa section through the axis OL of a right-angled cone, andconceive a section through AG (perpendicular to OA) and atright angles to the plane of the paper.FIG. 1.If P isany point on the curve, and PN perpendicular toAG, let J3(7be drawn through N perpendicular to the axis ofthe cone. Then P is on the circular section of the cone al>outBO as diameter.Draw AD parallel to EG, and DF, CG parallel to OL meetingAL produced in F, G. Then AD, AF are both bisectedKnow N = y, AN'=by OL.But jB, A , (7, G are concyclic, so thatBN.NC=AN.NGThereforey*= AN. 2AL. x,and 2 A L is the parameter ' of the principal ordinates y.In the case of the hyperbola Menaechmus had to obtain the

MENAECHMUSJS PROCEDURE 113particular hyperbola which we call rectangular or equilateral,and also to obtain its property with reference to its asymptotes,a considerable advance on what was necessary in theTwo methods of obtaining the particularcase of the parabola.hyperbola were possible, namely (1) to obtain the hyperbolaarising from the section of any obtuse-angled cone by a planeat right angles to a generator, and then to show how arectangular hyperbola can be obtained as a particular caseby finding the vertical angle which the cone must have togive a rectangular hyperbola when cut in the particular way,or (2) to obtain the rectangular hyperbola direct by cuttinganother kind of cone by a section not necessarily perpendicularto a generator.(1) Taking the first method, we draw (Fig. 2) a cone with itsvertical angle BO (1 obtuse. Imagine a section perpendicularto the plane of the paper and passing through AG which isperpendicular to OB. Let GA produced meet CO produced inA* ',and complete the same construction as in the case ofthe parabola.FIG. 2.In this case we havePN* = BN.= AN.NG.

114 CONIC SECTIONSBut, by similar triangles,NO:AF=NC:AD= A'N:AA'.Hence P^V 2 = A Jf . A'N.A FAA~,which is the property of the hyperbola, AA' being what wecall the transverse axis, and 2 AL the parameter of the principalordinates.Now, in order thatmust have 2 AL: A A' equal to 1.the hyperbola may be rectangular, weThe problem therefore nowis< given a straight line A A', and AL along AA producedequal to ÂA\ to find a cone such that L is on itsaxis andthe section through AL perpendicular to the generator throughInA is a rectangular hyperbola with A f A as transverseaxis.other words, we have to find a point on the straight linethrough A perpendicular to AA r such that OL bisects theangle which is the supplement of the angle A'OA.This is the case if A'Q :OA= A'L : LA = 3:1;therefore is on the circle which is the locus of all pointssuch that their distances from the two fixed points A', Aare in the ratio 3:1. This circle is the circle on KL asdiameter, where A'K : KA = A'L : LA = 3:1. Draw thiscircle, and is then determined as the point in which AOdrawn perpendicular to AA intersects the circle.It is to be observed, however, that this deduction of aparticular from a more general case is not usual in earlyGreek mathematics ;on the contrary, the particular usuallyled to the more general. Notwithstanding, therefore, that theorthodox method of producing conic sections is said to havebeen by cutting the generator of each cone perpendicularly,I am inclined to think that Menaechmus would get his rectangularhyperbola directly, and in an easier way, by means ofa different cone differently cut. Taking the right-angled cone,already used for obtaining a parabola, we have only to makea section parallel to the axis (instead of perpendicular to agenerator) to get a rectangular hyperbola.

MENAECHMUS'S PROCEDURE 115For, let the right-angled cone HOK (Fig. 3) be cut by aplane through A'AN parallelto the axis M and cutting thesides of the axial triangle HOKin A', A, N respectively. LetP be tho point on the curvefor which PN isthe principalordinate. Draw 00 parallelto HK. We have at once= CN*-CA*, since MK = OM, and MN= OC=CA.This is the property of the rectangular hyperbola having A'Aas axis. To obtain a particular rectangular hyperbola withaxis of given length we have only to choose the cutting planeso that the intercept A'A may have the given length.But Menaechmus had to prove the asymptote-property ofhis rectangular hyperbola. As he can hardly be supposed tohave got as far as Apollonius in investigating the relations ofthe hyperbola to its asymptotes,it isprobably safe to assumethat he obtained the particular property in the simplest way,i. e. directly from the property of the curve in relation, toits axes.RFIG. 4.If (Fig. 4) OR, OR' be the asymptotes (which are therefore

116 CONIC SECTIONSat right angles) and A'A the axis of a rectangular hyperbola,P any point on the curve, PN the principal ordinate, drawPK, PIC perpendicular to the asymptotes respectively. LetPN produced meet the asymptotes in U, R'.Now, by the axial property,= 2PK.PK', since /.PRK is half a right angle ;therefore PK .PK' =Works by Aristaeus and Euclid.If Menaechmus was really the discoverer of the three conicsections at a date which we must put at about 360 or 350 B.C.,the subject must have been developed very rapidly, for by theend of the century there were two considerable works onconies in existence, works which, as we learn from Pappus,were considered worthy of a place, alongside the (Ionics ofApollonius, in the Treasury of Analysis. Euclid flourishedabout 300 B.C., or perhaps 10 or 20 years earlier; but hisConies in four books was preceded by a work ofwhich was still extant in the time of Pappus,Aristaeuswho describes itas *five books of tiolid Loci connected (or continuous, crvvt^)with the conies'. Speaking of the relation of Euclid's Coniesin four books to this work, Pappus says (if the passage isgenuine) that Euclid gave credit to Aristaeus for his discoveriesin conies and did not attempt to anticipate him orwish to construct anew the same system. In particular,the three-Euclid, when dealing with what Apollonius calls'and four-line locus, wrote so much about the locus as waspossible by means of the conies of Aristaeus, without claimingcompleteness for his demonstrations We '.*gather from theseremarks that Euclid's Conies was a compilation and rearrangementof the geometry of the conies so far as known in his1Pappus, vii, p. 678. 4.

WORKS BY ARISTAEUS AND EUCLID 117time; whereas the work of Aristaeus was more specialized andmore original.'Solid loci 9 and 'solid problems'.'Solid loci ' are of course simply conies, but the use of thetitle ' Solid loci ' instead of ' conies ' seems to indicate thatthe work was in the main devoted to conies regarded as loci.'As we have seen, solid loci 'which are conies are distinguishedfrom ' plane loci on the one', hand, which are straight linesand circles, and from ' linear loci ' on the other, which arecurves higher than conies. There is some doubt as to thereal reason why the term ' solid loci ' was applied to the conicsections. We are told that ' plane ' loci are so called becausethey are generated in a plane (but so are some of the highercurves, such as the quadratrijc and the spiral of Archimedes),ft-nd that 'solid loci' derived their name from the fact thator tore).they arise as sections of solid figures (but so do some highercurves, e.g. the spiric curves which are sections of the onreipaBut some light is thrown on the subject by the correspondingdistinction which Pappus draws between 'plane','solid ' and ' linear ' problems.'Those problems', he says, 'which can be solved by meansof a straight line and a circumference of a circle may properlybe called />lanr for the lines; by means of which suchproblems are solved have their origin in a plane. Those,however, which are solved by using for their discovery one ormore of the sections of the cone have been called solid', fortheir construction requires the use of surfaces of solid figures,namely those of cones. There remains a third kind of problem,that which is called linear ;for other lines (curves)besides those mentioned are assumed for the construction, theorigin of which is more complicated and less natural, as theyare generated from more irregular surfaces and intricatemovements.' 1'The true significanceof the word plane ' as applied toproblems is evidently, not that straight lines and circles havetheir origin in a plane, but that the problems in question canbe solved by the ordinary plane methods of transformation of1Pappus, iv, p. 270. 5-17.

118 CONIC SECTIONSareas, manipulation of simple equations between areas and, inparticular, the application of areas in other; words, planeproblems were those which, if expressed algebraically, dependon equations of a degree not higher than the second.Problems, however, soon arose which did not yield to 'plane 'methods. One of the first was that of the duplication of thecube, which was a problem of geometry in three dimensions orsolid geometry. Consequently, when it was found that thisproblem could be solved by means of conies, and that nohigher curves were necessary, it would be natural to speak ofthem as * solid ' loci, especially as they were in fact producedfrom sections of a solid figure, the cone. The propriety of theterm would be only confirmed when it was found that, just asthe duplication of the cube depended on the solution of a purecubic equation, other problems such as the trisection of anangle, or the cutting of a sphere into two segments bearinga given ratio to one another, led to an equation betweenvolumes in one form or another, i.e. a mixed cubic equation,and that this equation, which was also a solid problem, couldlikewise be solved by means of conies.Aristaeus'sSolid LociThe Solid Loci of Aristaeus, then, presumably dealt withloci which proved to be conic sections. In particular, he musthave discussed, however imperfectly, the locus with respect tothree or four lines the synthesis of which Apollonius says thathe found inadequately worked out in Euclid's (Ionics. Thetheorems relating to this locus are enunciated by Pappus inthis way:If three straight lines be given in position and from one and'the same point straight lines be drawn to meet the threestraight lines at given angles, and if the ratio of the rectanglecontained by two of the straight lines so drawn to the squareon the remaining one be given, then the point will lie on asolid locus given in position, that is, on one of the three conicsections. And if straight lines be so drawn to meet, at givenangles, four straight lines given in position, and the ratio ofthe rectangle contained by two of the lines so drawn to thethen inrectangle contained by theremaining two be given,

ARISTAEUS'S SOLID LOCI 119the same way the point will lie on a conic section given inposition/ lThe reason why Apollonius referred in this connexion toEuclid and not to Aristaeus was probably that it was Euclid'swork that was on the same lines as his own.A very large proportion of the standard properties of coniesadmit of being stated in the form of locus-theorems; if acertain property holds with regard to a certain point, thenthat point lies on a conic section. But itmay be assumedthat Aristaeus's work was not merely a collection of theordinary propositions transformed in this wayit would deal;with new locus-theorems not implied in the fundamentaldefinitions and properties of the conies, such as those justmentioned, the theorems of the three- and four-line locus.But one (to us) ordinary property, the focus-directrix property,was, as it seems to me, in all probability included.Focus-directrix property known to Euclid.It is remarkable that the directrix does not appear at all inApollonius's great on conies. The focal treatise1 properties ofthe central conies are given by Apollonius, but the foci areobtained in a different way, without any reference to thedirectrix; the focus of the, parabola does not appearat all.We may perhaps conclude, that neither did Euclid's Coniescontain the focus-directrix property; for, according to Pappus,Apollonius based his first four books on Euclid's four books,while fillingthem out and adding to them. Yet Pappus givesthe proposition as a lemma to Euclid's Harfuce-Lwi, fromwhich we cannot but infer that it was assumed in thattreatise without proof. If, then, Euclid did not take it fromhis own ( 1 onic$, what more likely than that it was containedin Aristaeus's Solid Loei ?Pappus's enunciation of the theorem is to the etfect that thelocus of a point such that its distance, from a given pointis ina given ratio to its distance from a fixed straight line is a conicsection, and is an ellipse, a parabola, or a hyperbola accordingas the given ratio is less than, equal to, or greater than unity.1Pappus, vii, p. 678. 15-24.

120 CONIC SECTIONSProof from Pappus.The proof in the case where the given ratio is different fromunity is shortly as follows.Let S be the fixed point, SX the perpendicular from 8 onthe fixed line. Let P be any pointon the locus and PNpJvK AN SK' A 1 A' A Nperpendicular to SX, so that HP is to NX in the givenratio (e);thus e 2Take K on SX such thatthen, if K' be another point on SN, producedit'necessary,such that NK = NK',: NK2= PN*:XK.XK'.The positions of N, K, K' change with the position of P.If A, A' be the points on which N falls when K, K' coincidewith X respectively,we have'= SA' : A'X.Thereforewhence8X:BA = SK :SN=(l+e):e,(1 +e):e = (SX-SK)-.(SA-SN)= XK:AN.

FOCUS-DIRECTRIX PROPERTY 121Similarlyit can be shown that(1 ~e):e = XK':A'N.By multiplication, XK . XK': AN. A'N = (1- e2) e2 : ;and itfollows from above, ex aequuli, thatPN*: AN. A'N = (I 1, A andA' lie on opposite sides of X, while N lies on A'-A produced,and the conic is a hyperbola.The case where e, = 1 and the curve is a parabolais easyand need not be reproduced here.The treatise would doubtless contain other loci of typessimilar to that which, as Pappus says, was used for thetrisection of an angle: I refer to the proposition alreadyquoted (vol. i, p. 243) that, if A, B are the base angles ofa triangle with vertex P, and L 11 = 2 /.A, the locus of Pis a hyperbola with eccentricity2.PropositionsThat Euclid'sincluded in Euclid's Conies.Conic* covered much of the same ground asthe first three Books of Apolloniusis clear from the languageof Apollonius himself. Confirmation isforthcoming in thequotations by Archimedes of propositions (1) 'proved inthe elements of conies ', or (2) assumed without remark asalready known. The former class include the fundamentalordinate properties of the conies in the following forms :(1) for the ellipse,PN* : AN. A'N = P'N'* : AN'. A'N' = BU* : AC*;(2) for the hyperbola,PN* : AN. A'N = P'.V" 2 : AN'.A'N';(3) for the parabola, PN* = pa .AN',the principal tangent properties of the parabola ;the property that, if there are two tangents drawn from onepoint to any conic section whatever, and two intersecting

122 CONIC SECTIONSchords drawn parallelto the tangents respectively,the rectanglescontained by the segments of the chords respectivelyare to one another as the squaresof the parallel tangents ;the by no means easy proposition that, if in a parabola thediameter through P bisects the chord QQ' in F, and QD isdrawn perpendicular to PF, thenwhere pa is the parameter of the principal ordinatcs and p isthe parameter of the ordinates to the diameter P V.Conic sections in Archimedes.But we must equally regard Euclid's Conies as the sourcefrom which Archimedes took most of the other ordinaryproperties of conies which he assumes without proof.Beforesummarizing these it will be convenient to refer to Archimedes'sterminology. We have seen that the axes of anellipse are not called axes but diameters, greater and lesser ;the axis of a parabolais likewise its diameter and the otherdiameters are 'lines parallelto the diameter', although ina segment of a parabola the diameter bisectingthe base isthe ' diameter ' of the segment. The two * ' diameters ' (axes)In the case of the hyperbola theof an ellipse are conjugate.1diameter' (axis) is the portion of it within the (single-branch)hyperbola ; the centre is not culled the ' centre ',but the pointin which the ' nearest lines to the section of an obtuse-angledcone' (the asymptotes) meet; the half of the axis (OA) is1the line 'adjacent to the axis (of the hyperboloid of revolutionobtained by making the 'hyperbola revolve about its diameter '),and A'A is double of this line. SimilarlyCP is the lineJ'adjacent to the axis of a segment of the hyperboloid, andP'P double of this line. It is clear that Archimedes did notyet treat the two branches of a hyperbola as forming onecurve ;this was reserved for Apollonius.The main propertiesof conies assumed by Archimedes inaddition to those above mentioned may be summarized thus.Central Conies.1. The property of the ordinates to any diameter PP',

CONIC SECTIONS IN ARCHIMEDES 123In the case of the hyperbola Archimedes does not giveany expressionfor the constant ratios PN*:AN.A'N andQV*:PV.P'V respectively, whence we conclude that he hadno conception of diameters or radii of a hyperbola not meetingthe curve.2. The straight line drawn from the centre of an ellipse, orthe point of intersection of the asymptotes of a hyperbola,through the point of contact of any tangent, bisects all chordsparallel to the tangent.3. In the ellipse the tangents at the extremities of either of twoconjugate diameters are both parallel to the other diameter.4. If in a hyperbola the tangent at P meets the transverseaxis in r i\ and PN is the principal ordinate, AN > AT. (Itis not easy to see how this could be proved except by meansof the general property that, if PP' be any diameter ofa hyperbola, QV the ordinate to it from Q, and QT the tangentat Q meeting P*P in 7 7 ,then TP : TP' = PV: 1>'V.)5. If a cone, right or oblique, be cut by a plane meeting allthe generators, the section is either It circle or an ellipse.. If a line between the asymptotes meets a hyperbola andis bisected at the point of concourse, it will touch thehyperbola.7. If x, y are straight lines drawn, in fixed directions respectively,from a point on a hyperbola to meet the asymptotes,the rectangle xyis constant.8. If 7 J aV be the principal ordinate of P, a point on an ellipse,and if iYP be produced toratio pN:l*Nis constant.meet the auxiliary circle in p, the9. The criteria of similarity of conies and segments ofconies are assumed in practically the same form as Apolloniusgives them.The Parabola.1. The fundamental properties appear in the alternative forms:PN? /"JV= AN: AN', or P.V 2 = pa.AN,QV*:$V'*=PV:PV', or QV*=p.PV.Archimedes applies the term parameter (a wap9$LVSvvavrouat airo ray ro/zay) to the parameter of the principal ordinates

124 CONIC SECTIONSonly : p is simply the line to which the rectangle equal to QF 2and of width equal to PV is applied.2. Parallel chords are bisected by one straight line parallel tothe axis, which passes through the point of contact of thetangent parallel to the chords.3. If the tangent at Q meet the diameter P V in T, and QVP beV = PT.the ordinate to the diameter,By the aid of this proposition a tangent to the parabola canbe drawn (a) at a point on it, (6) parallel to a given chord.4. Another proposition assumed is equivalent to the propertyof the subnormal, NG = -|/>tt.5. If QQ' be a chord of a parabola perpendicularto the axisand meeting the axis in while MyQVq another chord parallelto the tangent at P meets the diameter through P in V, andRIIK is the principal ordinate of any point R on the curvemeeting PV in // and the axis in K, then PViPJI > or= MK:KA 'for this is; proved 1 (0/6 Floating Bodies, II. 6).Where it was proved we do not know; the proofaltogether easy. 1is not6. All parabolas are similar.As we have seen, Archimedes had to specialize in theparabola for the purpose of his treatises on the Quadratureof the Parabola, Conoids and tipherouls, Floating Bodies,Book II, and Plane Equilibriums, Book II ; consequently hehad to prove for himself a number of special propositions, whichhave already been given in their proper places.A few othersare assumed without proof, doubtless as being easy deductionsfrom the prgpositions which he does prove.They refer mainlyto similar parabolic segments so placed that their buses are inone straight line and have one common extremity.1. If any three similar and similarly situated parabolicsegments JBQ 19 Q 2,BQ.^ lying along the same straight lineas bases (BQ

CONIC SECTIONS IN ARCHIMEDES 1252. If two similar parabolic segments with bases BQ lt BQ 2beplaced as in the last proposition, and BR 1R if 2line through B meeting the segments in R l ,be any straight/J 2 respectively,These propositions are easily deduced from the theoremproved in the Quadrature of the Parabola, that, if throughEya point on the tangent at B, a straight line ERO be drawnparallel to the axis and meeting the curve in R and any chordBQ through B in 0, then3. On the strength of these propositions Archimedes assumesthe solution of the problem of placing, between two parabolicsegments similar to one, another and placed as in the abovepropositions, a straight lino of a given length and in a directionparallel to the diameters of either parabola.Euclid and Archimedes no doubt adhered to the old methodfrom sections of threeof regarding the three conies as arisingkinds of right circular cones (right-angled, obtuse-angled aridacute-angled) by planes drawn in each case at right angles toa generator of the cone. Yet neither Euclid nor Archimedeswas unaware that the 'section of an acute-angled cone', orellipse, could bo otherwise produced. Euclid actually says inhis Pkaenomena that 'if a cone or cylinder (presumably right)be cut by a plane not parallel to the base, the resulting sectionis a section of an acute-angled cone which is similar toa dvpeos (shield)'. Archimedes know that the non-circularsections even of an oblique circular cono made by planescutting all the generators are ellipses for he shows us how,;of whichgiven an ellipse, to draw a cone (in general oblique)it is a section and which has its vertex outside the planeof the on ellipse any straight line through the centre of theellipse in a plane at right angles to the ellipse and passingthrough one of its axes, whether the straight line is itselfperpendicular or not perpendicular to the plane of the ellipse ;drawing a cone in this case of course means finding the circularsections of the surface generated by a straight line alwayspassing through the given vertex and all the several points ofthe given ellipse.The method of proof would equally serve

126 APOLLONIUS OF PERGAfor the other two conies, the hyperbola and parabola, and wecan scarcely avoid the inference that Archimedes was equallyaware that the parabola and the hyperbola could be foundotherwise than by the old method.The first, however, to base the theory of conies on theproduction of all three in the most general way from anykind of circular cone, right or oblique, was Apollonius, towhose Work we now come.B. APOLLONIUS OF PERGAHardty anything is known of the life of Apollonius exceptthat he was born at Perga, in Pamphylia, that he wentwhen quite young to Alexandria, where ho studied with thesuccessors of Euclid and remained a long time, and thathe flourished (yeyoi/e) in the reign of Ptolemy Eucrgetes(247-222 B.C.). Ptolemaeus Chennus mentions an astronomerof the same name, who was famous during the reign ofPtolemy Philopator (222 205 B.C.),and it is clear that ourApolloniusis meant. AsApollonius dedicated the fourth andfollowing Books of his Comes to King Attains I (241-197 B.C.)we have a confirmation of his approximate date. He wasprobably born about 262 B.C., or 25 years after Archimedes.We hear of a visit to Pergamum, where he made the acquaintanceof Eudemus of Pergamum, to whom he dedicated thefirst two Books of the Conies in the form in which they havecome down to us ; they were the first two instalments of asecond edition of the work.The text of theComes.The Conies of Apollonius was at once recognized as theauthoritative treatise on the subject, and later writers regularlycited it when quoting propositions in conies. Pappuswrote a number of lemmas to it ;Serenus wrote a commentary,as also, according to Suidas, did Hypatia. EutociusA.D. (fl. 500) prepared an edition of the first four Books andwrote a commentary on them it is evident that he;had beforehim slightly differing versions of the completed work, and hemayalso have had the first unrevised edition which had gotinto premature circulation, as Apollonius himself complains inthe Preface to Book I.

THE TEXT OF THE CONICS 127The edition of Eutocius suffered interpolations which wereprobably made in the ninth century when, under the auspicesof Leon, mathematical studies were revived at Constantinople ;for it was at that date that the* uncial manuscripts werewritten, from which our best manuscripts, V (= Cod. Vat. gr.206 of the twelfth to thirteenth century) for the Conies, andW (= Cod. Vat. gr. 204 of the tenth century) for Eutocius,were copied.Only the first four Books survive in Greek; the eighthBook is altogether lost, but the three Books V-VII exist inArabic. It was Ahmad and al-Hasan, two sons of Muh. b.Musa b. Shakir, who first contemplated translating the Coniesinto Arabic. They were at first deterred by the bad state oftheir manuscripts; but afterwards Ahmad obtained in Syriaa copy of Kutocius's edition of Books 1-IV and had themtranslated by Hilal b. Abl Ililal al-Himsi (died 883/4).Books V-V1I were translated, also for Ahmad, by Thabitb.Qurra( 826 901) from another manuscript. Nasiraddin'srecension of this translation of the seven Books, made in 1248,is represented by two copies in the Bodleian, one of the year1301 (No. 943) and the other of 1626 containing Books V-VIIonly (No. 885).A Latin translation of Books I-IV was published byJohannes Baptista Memus at Venice in 1537;but the firstimportant edition was the translation by Commandinus(Bologna, 1566), which included the lemmas of Pappus andthe commentary of Eutocius, and was the first attempt tomake the book intelligible by means of explanatory notes.For the Greek text Commandinus used Cod. Marcianus 518and perhaps also Vat. gr. 205, both of which were copies of V,but not V itself.The first published version of Books V-VII was a Latintranslation by Abraham Echellensis and Giacomo AlfonsoBorelli (Florence, 1661) of a reproduction of the Books writtenFath al-Isfahiim.in 983 by Abu '1The editio .princeps of the Greek text is the monumentalwork of Halley (Oxford, 1710). The original intention wasthat Gregory should edit the four Books extant in Greek, withEutocius's commentary and a Latin translation, and thatHalleyshould translateBooks V-VI1 from the Arabic into

128 APOLLONIUS OF PERGALatin. Gregory, however, died while the work was proceeding,and Halley then undertook responsibility for the whole. TheGreek manuscripts used were two, one belonging to Savileand the other lent by D. Baynard their whereabouts cannot;apparently now be traced, but they were both copies of Paris,gr. 2356, which was copied in the sixteenth century from Paris,gr. 2357 of the sixteenth century, itself a copy of V. For thethree Books in Arabic Halley used the Bodleian MS. 885, butalso consulted (a) a compendium of the three Books by 'Abdelmelikal-Shlrazi (twelfth century), also in the Bodleian (913),(b) Borelli's edition, and (c)Bodl. 943 above mentioned, by meansof which he revised and corrected his translation when completed.Halley's edition is still, so far as I know, the onlyavailable source for Books V-VII, except for the beginning ofBook V (up to Prop. 7) which was edited by L. Nix (Leipzig,1889).The Greek text of Books I-IV is now available, with thecommentaries of Eutocius, the fragments of Apollonius, &c.,in the definitive edition of Heiberg (Teubner, 1891-3).Apollonius's own account of the Conies.A general account of the contents of the great work which,according to Geminus, earned for him the title of the ' greatgeometer' cannot be better given than in the words of thewriter himself. The prefaces to the several Books containinteresting historical details, and, like the prefaces of Archimedes,state quite plainly and simply in what way thetreatise differs from those of his predecessors, and how muchin it is claimed as original. The strictures of Pappus (ormore probably his interpolator), who accuses him of being abraggart and unfair towards his predecessors, are evidentlyunfounded. The prefaces are quoted byv. Wilamowitz-Moellendorff as specimens of admirable Greek, showing howperfect the style of the ^ great mathematicians could bewhen they were free from the trammels of mathematicalterminology.Book I. General Preface.Apollonius to Eudemus, greeting.If you are in good health and things are in other respectsas you wish, it is well ;with rne too things are moderately

THE CONIG8 129well. During the time I spent with you at PergamumI observed your eagerness to become acquainted with mywork in conies; I am therefore sending you the first book,which I have corrected, and I will forward the remainingbooks when I have finished them to my satisfaction. I daresay you have not forgotten my telling you that I undertookthe investigation of this subject at the request of Naucratesthe geometer, at the time when he came to Alexandria andstayed with me, and, when I had worked it out in eightbooks, I gave them to him at once, too hurriedly, because hewas on the point of sailing; they had therefore not beenthoroughly revised, indeed I had put down everything just asit occurred to me, postponing revision till the end. AccordinglyI now publish, as opportunities serve from time to time,instalments of the work as they are corrected. In the meantimeit has happened that some other persons also, amongthose whom I have met, have got the first and second booksbefore they were corrected do not be;surprised therefore ifyou come across them in a different shape.Now of the eight books the first four form an elementaryintroduction. The first contains the modes of producing thethree sections and the opposite branches (of the hyperbola),arid the fundamental properties subsisting in them, workedout more fully and generally than in th writings of others.The second book contains the properties of the diameters andtho axes of the sections as well as the asymptotes, with otherthings generally and necessarily used for determining limitsof possibility (& op and what I mean *07/01') by diametersand ; axes respectively you will learn from this book. Thethird book contains many remarkable theorems useful fortho syntheses of solid loci and for dioriumi ;the most andprettiest of those theorems are new, and it was their discoverywhich made me aware that Euclid did not work out thesynthesis of the locus with rospect to throe and four lines, butonly a chance portion of it, and that not successfully for it;was not possible for the said synthesis to be completed withoutthe aid of the additional theorems discovered by me: Thefourth book shows inthe sections of coneshow many wayscan meet one another and the circumference of a circle ;itcontains other things in addition,- none of which have beendiscussed by earlier writers, namely the questions in howmany points a section of a cone or a circumference of a circlecan meet [a double-branch hyperbola, or two double-branchhyperbolas can meet one another].The rest of the books are more by way of surplusageone of them deals somewhat fully with(7Tpiov(ria

130 APOLLONIUS OF PERGAminima and maxima, another with equal and similar sectionsof cones, another with theorems of the nature of determinationsof limits, and the last with determinate conic problems.But of course, when all of them are published,it will be open*to all who read them to form their own judgement about them,according to their own individual tastes. Farewell.The preface to Book II merely says that Apollonius issending the second Book to Eudemus by his son Apollomus,and begs Eudemus to communicate it to earnest students of thesubject, and in particular to Philonides the geometer whomApollonius had introduced to Eudemus at Ephesus. There isno preface to Book III as we have it, although the preface toBook IV records that it also was sent to Eudemus.Preface to Book IV.Apollonius to Attains, greeting.Some time ago I expoundedand sent to Eudemus of Pergamumthe first three books of my conies which I havecompiled in eight books, but, as he has passed away, I haveresolved to dedicate the remaining books to you because ofyour earnest desire to possess my works. I am sending youon this occasion the fourth book.It contains a discussion ofthe question,in how many points at most it is possible forsections of cones to meet one another and the circumferenceof a circle, on the assumption that they do not coincidethroughout, and further in how many points at most asection of a cone or the circumference of a circle can meet thehyperbola with two branches, [ortwo double-branch hyperbolascan meet one another]; and, besides these questions,the book considers a number of others of a similar kind.Now the first question Conon expounded to Thrasydaeus, without,however, showing proper mastery of the proofs, and onthis ground Nicoteles of Gyrene, not without reason, fell foulof him. The second matter has merely been mentioned byNicoteles, in connexion with his controversy with Conon,as one capable of demonstration; but I have not found itdemonstrated either by Nicoteles himself or by any one else.The third question and the others akin to it I have not foundso much as noticed by any one. All the matters referred to,which I have not found anywhere, required for their solutionmany and various novel theorems, most of which I have,as a matter of fact, set out in the first three books, while therest are contained in the present book. These theorems areof considerable use both for the syntheses of problems and for

THE CONWtt 131diorismi* Nicoteles indeed, on account of his controversywith Conon, will not have it that any use can be made of thediscoveries of Corion for the purpose of diorismi', he is,however, mistaken in this opinion, for, even if it is possible,without using them at all, to arrive at results in regard tolimits of possibility, yet they at all events afford a readiermeans of observing some things, e.g. that several or so manysolutions am possible, or again that no solution is possible;and such foreknowledge secures a satisfactory basis for investigations,while the theorems in question are again usefulfor the analyses of dioriami. And, even apart from suchusefulness, they will be found worthy of acceptance for thesake of the demonstrations themselves, just as we acceptmany other things in mathematics for this reason and forno other.The prefaces to Books V Vll now to be given are reproducedfor Book V from the translation of L. Nix and forBooks VI, VII from that of Halley.Preface to Book V.Apollonius to Attalus, greeting.In this fifth book I have laid down propositions relating tomaximum arid minimum straight lines. You must knowthat my predecessors and contemporaries have only superficiallytouched upon the investigation of the shortest lines,and have only proved what straight lines touch the sectionsand. conversely, what properties they have in virtue of whichthey are tangents. For my part, 1 have proved these propertiesin the first book (without however making any use, inthe proofs, of the doctrine of the shortest lines), inasmuch as1 wished to place them in close connexion with that partof the subject in which 1 treat of the production of the threeconic sections, in order to show at the same time that in eachof the three sections countless properties and necessary resultsappear, as they do with reference to the original (transverse)diameter. The propositions in which I discuss the shortestlines 1 have separated into classes, and I have dealt with eachindividual case by careful demonstration ;I have also connectedthe investigation of them with the investigation ofthe greatest lines above mentioned, because 1 considered thatthose who cultivate this science need them for obtaininga knowledge of the analysis, and determination of limits ofpossibility, of problems as well as for their synthesis: inaddition to which, the subject is one of those which seemworthy of study for their own sake. Farewell.

132 APOLLONIUS OF PERGAPreface to Book VI.Apollonius to Attalus, greeting.I send you the sixth book of the conies, which embracespropositions about conic sections and segments of conies equaland unequal, similar and dissimilar, besides some other mattersleft out by those who have preceded me. In particular, youwill find in this book how, in a given right cone, a section canbe cut which is equal to a given section, and how a right conecan be described similar to a given cone but such as to containa given conic section. And these matters in truth I havetreated somewhat more fully and clearly than those who wrotebefore my time on these subjects. Farewell.Preface to Book VII.Apollonius to Attalus, greeting.I send to you with this letter the seventh book on conicsections. In it are contained a large number of new propositionsconcerning diameters of sections and the figures describedupon them and all these, ;propositions have their uses in manykinds of problems, especially in the determination of thelimits of their possibility. Several examples of those occurin the determinate conic problems solved and demonstratedby me in the eighth book, which isby way of an appendix,and which I will make a point of sending to you as soonas possible. Farewell.Extent of claim to originality.We gather from these prefaces a very good idea of theplan followed by Apollonius in the arrangement of the subjectand of the extent to which he claims originality. Thefirst four Books form, as he says, an elementary introduction,by which he means an exposition of the elements of conies,that is, the definitions and the fundamental propositionswhich are of the most general use and application;the term'elements ' is in fact used with reference to conies in exactlythe same sense as Euclid uses it to describe his great work.The remaining Books beginning with Book V are devoted tomore specialized investigation of particular parts of the subject.It is only for a very small portion of the contend of thetreatise that Apollonius claims originalityin the first;threeBooks the claim is confined to certain propositions bearing onthe ' locus with respect to three or four lines ' ;and in thefourth Book (on the number of points at which two conies

THE CONICS 133may intersect, touch, or both) the part which is claimedas new is the extension to the intersections of the parabola,and circle with and ofellipse, the double-branch hyperbola,two double-branch hyperbolas with one another, of the investigationswhich had theretofore only taken account of thesingle-branch hyperbola. Even in Book V, the most remarkableof all, Apolloiiius does not say that normals as 'the shortestlines 'had not been considered before, but only that they hadbeen superficially touched upon, doubtless in connexion withpropositions dealing with the tangent properties. He explainsthat lie found it convenient to treat of the tangent properties,without any reference to normals, in the first Book in orderto connect them with the chord properties. It is clear, therefore,that in treating normals as maxima and minima, and bythemselves, without any reference to tangents, as he does inBook V, he was making an innovation ; and, in view of theextent to which the theory of normals as maxima and minimais developed by him (in 77 propositions), there is 110 wonderthat he should devote a whole Book to the subject. Apartfrom the developments in Books III, IV, V, just mentioned,and the numerous new propositions in Book VII with theproblems thereon which formed the lost Book VI11, Apolloniusonly claims to have treated the whole subject more fully andgenerally than his predecessors.Great generality of treatment from the beginning.So far from being a braggart and taking undue credithimself for the improvements which he made upon his predecessors,Apollonius is, if anything, too modest in his descriptionof his personal contributions to the theory of conicof his firstsections. For the ' more fully and generally'preface scarcely conveys an idea of the extreme generalitywith which the whole subject is worked out. This characteristicgenerality appears at the very outset.toAnalysisof the Conies.Book 1.Apollonius begins by describing a double oblique circularcone in the most general way. Given a circle and any pointoutside the plane of the circle and in general not lying on the

134 APOLLONIUS OP PERGAstraight line through the centre of the circle perpendicular toits plane, a straight line passing through the point and producedindefinitely in both directions is made to move, whilealways passing through the fixed point, so as to pass successivelythrough all the points of the circle ;the straight linethus describes a double cone which is in general oblique or, asApollonius calls it, scalene. Then, before proceeding to thegeometry of a cone, Apollonius gives a number of definitionswhich, though of course only required for conies, are stated usapplicable to any curve.1In any curve,' says Apollonius, 'I give the name diameter toany straight line which, drawn from the curve, bisects all thustraight lines drawn in the curve (chords) parallel to anystraight line, and I call the extremity of the straight line(i.e. the diameter) which is at the curve a vertex of the curveand each of the parallel straight lines (chords) an ordinate(lit. drawn ordinate- wise, Tray/xej>a>y Kar^\6ai) to thediameter/He then extends these terms to a pair of curves (the primaryreference being to the double-branch hyperbola), giving thename transverse diameter to any straight line bisectingall thechords in both curves which are parallel to a given straightline (this gives two vertices where the diameter meets thecurves respectively), and the name erect dmmeter (opOia) toany straight line which bisects all straight lines drawnbetween one curve and the other which are parallel to anystraight line; the ordinates to any diameter are again theparallel straight lines bisected by it. Conjugate diameters inany curve or pair of curves are straight lines each of whichbisects chords parallel to the other. Axes are the particulardiameters which cut at right angles the parallel chords whichthey bisect and; conjugate axes are related in the same wayas conjugate diameters. Here we have practically our moderndefinitions, and there is a great advance on Archimedes'sterminology.The conies obtained in theoblique cone.most general way from anHaving described a cone (in general oblique), Apolloniusdefines the axis as the straight line drawn from the vertex to

THE CONICti, BOOK I 135the centre of the circular base. After proving that allsections parallel to the base are also circles, and that thereis another set of circular sections subcontrary to these, heproceeds to consider sections of the cone drawn in anymanner. Taking any triangle through the axis (the base ofthe triangle being consequently a diameter of the circle whichis the base of the cone), he is careful to make his section cutthe base in a straight line perpendicular to the particulardiameter which is the base of the axial triangle. (There is110 loss of generality in this, for, ifany section is taken,without reference to any axial triangle, we have only toselect the particular axial triangle the base of which is thatdiameter ofthe circular base which isat right angles to the straight line inwhich the section of the cone cuts thebase.) Let ABC be any axial triangle,and letany section whatever cut thebase in a straight line ])E at rightangles to BC\ if then PM be the intersectionof the cutting plane and theaxial triangle, and ifQQ' be any chordin the section parallel to DE, ApolloniusInproves that QQ' is bisected by PM.other words, PM is a diameter of the section. Apollonius iscareful to explain that,if the cone is a right cone, the straight line in the base (DK)'will be at right angles to the common section (PM)of thecutting plane, and the triangle through the axis, but, if thecone is scalene, it will not in general be at right angles to PM,but will be at right angles to it only when the plane throughthe axis (i.e.the axial triangle) is at right angles to the baseof the cone ' (I. 7).That is to say, Apollonius works out the properties of theconies in the most general way with reference to a diameterwhich is not one of the principal diameters or axes, but ingeneral has its ordinates obliquely inclined to it. The axes donot appear in his expositiontill much later, after it has beenshown that each conic has the same property with referenceto any diameter as it has with reference to the originaldiameter arising out of the construction ;the axes then appear

136 APOLLONIUS OF PERGAas particularcases of the new diameter of reference, xneare madethree sections, the parabola, hyperbola, and ellipsein the manner shown in the figures. In each case they passthrough a DE straight line in the plane of the base whichis at right angles to BO, the base of the axial triangle, orto EG produced. The diameter PM is in the case of the

THE CONICS, BOOK I 137parabola parallel to AC] in the case of the hyperbolait meetsthe other half of the double cone in P' ;and in the case of theellipse it meets the cone itself again in P'. We draw, inthe cases of the hyperbola and ellipse,AF parallel to PMto meet BC or ]$C produced in F.Apollonius expresses the properties of the three curves bymeans of a certain straight line PL drawn at right anglesto PM in the plane of the section.KIn the case of the parabola, PL is taken such thatPL: PA = BW-.BA.AC;and in the case of the hyperbola and ellipsesuch thatIn the latter two cases we join P'L, and then drparallel to PL to meet P'L, produced if necessary, in h.If UK be drawn through V parallel to J1C and meetingAB, AC in //, K respectively, IIK is the diameter of the circularsection of the cone made by a plane parallel to the base.Therefore Q V 2 = 11 V .VK.Then (1)triangles,andfor the parabola we have, by parallels and similar11V:PV=BC:CA,VK:PA = B

138 APOLLONIUS OF PERGATherefore QV*:PV .whence QV* = PL .PA = HV. VK : PV. PA= BC* :. BA . AC= PL: PA, by hypothesis,= PL.PV:PV.PA,PV.(2) In the case of the hyperbola and ellipse,UV:PV=BF:FA,VK:P'V=FC:AF.Therefore QV' 2 : PV.P'V = HV . VK:= BF.FO:AF*= PL :PV. P'VPP', by hypothesis,= RV:P'V= PV.VR:PV.P'V,whence QV* = PV . VR.New names, 'parabola', 'ellipse9,'hyperbola'.Accordingly, in the case of the parabola, the square of theordinate (QV 2 )is equal to the rectangle applied to PL andwith width equal to the abscissa (PV) ;in the case of the hyperbola the rectangle applied to PLand has its width equal to the abscissawhich is equal to QV 2PV overlaps or exceeds (irrrepfidXXti) by the small rectangle LRwhich is similar and similarly situated to the rectangle containedby PL, PP' ;in the case of the ellipse the corresponding rectangle fallsshort (eXXeiTTti) by a rectangle similar and similarly situatedto the rectangle contained by PL, PP'.Here then we have the properties of the three curvesexpressed in the precise language of the Pythagorean applicationof areas, and the curves are named accordingly:^>ara6o/ct(irapa/3o\rj)where the rectangle is exactly applied, hyperbola(irTrepfJoXri) where it exceeds, and ellipse (eAAet^/rts') where itfalls short.

THE CONICS, BOOK I 139PL is called the latus rectum (opOia)or the parameter of9the orditiates (irap $v Svvavrai al Karayofjitvat reray/zli/a)?)ineach case. In the case of the central conies, the diameter PP'is the transverse (17 irXayia] or transverse diameter ; while,even more commonly, Apollonius speaks of the diameter andthe corresponding parameter together, calling the latter thelatus rectum or erect side (opOia nXtvpd) and the formerthe transverse side of tiiefigurr (elSos) on, or applied to, thediameter.Fundamental properties equivalent to Cartesian equations.If p is the parameter, and d the corresponding diameter,the propertiesof the curves are the equivalent of the Cartesianequations, referred to the diameter and the tangentextremity as axes (in general oblique),at itsy 2-= px (the parabola),y 2 = px-f 1 x~ (the hyperbola and ellipse respectively).aThus Apollonius expresses the fundamental property of thecentral conies, like that of the parabola, as an equationbetween areas, whereas in Archimedes itappears as aproportionwhich, however, is equivalent to the Cartesian equationreferred to axes with the centre as origin. The latter propertywith reference to the original diameter is separatelyproved in I. 21, to the effect that QV 2 varies as PF.P'F, asis really evident from the fact that :QV* PV.P'V = PL :PP',seeing that PL : PP' is constant for any fixed diameter PP'.Apollonius has a separate proposition (1. 14) to prove thatthe opposite branches of a hyperbola have the same diameterand equal latera recta corresponding thereto. As he was thefirst to treat the double-branch hyperbola fully, he generallydiscusses the hyperbola (i.e.the single branch) along withthe ellipse, and the opposites, as he calls the double-branchhyperbola, separately. The propertiesof the single-branchhyperbola are, where possible, included in one enunciationwith those of the ellipseand circle, the enunciation beginning,

140 APOLLONIUS OP PERGAcIf in a hyperbola, anellipse, or the circumference of a circle ' ;sometimes, however, the double-branch hyperbola and theellipse come in one proposition, e.g. in I. 30: 'If in an ellipseor the opposites (i. e. the double hyperbola) a straight line bedrawn through the centre meeting the curve on both sides ofthe centre, it will be bisected at the centre/ The property ofconjugate diameter9 in an ellipse is proved in relation tothe original diameter of reference and its conjugate in I. 15,where it is shown that, if DD' is the diameter conjugate toPP' (i.e.the diameter drawn ordinate-wise to PP'), just asPP' bisects all chords parallel to /)/)', so DD' bisects all chordsparallel to PP' ; also, if DL' be drawn at right angles to DD'and such that DJJ'.DD' = PP /2 (or DL' is a third proportionalin rela-to DD', PP 7 then the ), ellipse has the same propertytion to DD' as diameter and DL' as parameter that ithas inrelation to PP' as diameter and PL as the corresponding parameter.Incidentally it appears that PL . PP' = DD'*, or PL isa third proportional to PP',/)//, as indeed, is obvious from theproperty of the curve Q V* : PV. P7' = PL: PP' = DD'* : PP'*.The next proposition,I. 16, introduces the secondai^y diameterof the double-branch hyperbola (i.e.the diameter conjugate tothe transverse diameter of reference), which does not meet thecurve; this diameter is defined as that straight line drawnto the ordinates of the transversethrough the centre paralleldiameter which is bisected at the centre and is of length equalto the mean proportional between the ' sides of the figure ',i.e. the transverse diameter PP' and the corresponding parameterPL. The centre is defined as the middle point of thediameter of reference, and it is provedare bisected at it (1. 30).that all other diametersProps. 17-19, 22-9, 31-40 are propositions leading up toand containing the tangent properties. On lines exactly likethose ofEucl. III. 16 for the circle, Apollonius proves that, ifa straight line is drawn through the vertex (i.e. the extremityof the diameter of reference) parallel to the ordjnates to thediameter, it will fall outside the conic, and no other straightline can fall between the said straight line and the conic ;therefore the said straight line touches the conic (1.17, 32).Props. I. 33, 35 contain the property of the tangent at anypoint on the parabola, and Props. I. 34, 36 the property of

THE CONICS, BOOK I 141the tangent at any point of a central conic, in relationto the original diameter of reference ;if Qis the point ofcontact, QV the ordinate to the diameter 'through P, andif QT, the tangent at Q, meets the diameter produced in Tythen (1) for the parabola PV FT, and (2) for the centralconic TP:TP' = PV: VP'. The method of proofis to take apoint T on the diameter produced satisfying the respectiverelations, and to prove that, if TQ be joined and produced,any point on TQ on either side of Q is outside the curve : theform of proofisby reductio ad absnrdum, and in eachcase it is again proved that no other straightline can fallbetween TQ and the curve. The fundamental propertyTP:TP'= PV\VP' for the central conic is then used toprove that UV.

142 APOLLONIUS OF PERGAeFT U P\W w'

THE CONICS, BOOK I 143(2) in the hyperbola or Aellipse?,RUW = thebetween the triangles GFW and CPE.difference(1) In the parabola &RUW: &QTV = RW* :QV*= PW:PVBut, since TV=2PV, AQTV =ARUW = C3EW.therefore(2) The proof of the propositionwith reference to thecentral conic depends on a Lemma, proved in I. 41, to the effectthat, if PX VY be similar9parallelograms on Cl\ C'Fas bases,and if VX be an equiangular parallelogram on QFas base andsuch that, if the ratio of (JP to the other side of PX is m, theratio of QV to the other side of VZ is m.p/Pl*,to the difference between FFand PX. Thethen VZ isequal proof of theLernma by Apollonius is difficult, but the truth of it can boeasily seen thus.By the property of the curve, QV' 2 : ( !V* - (T 2 = y>: PP'therefore(! V* ^ ( ?P* = PPP. OV*.;Now C3PX= p. (!!**/ m, where p is a constant dependingon the angle of the parallelogram.Similarlya VY - n.CT'Ym,and Q VZ = p.-^QV2/m.It follows thatD VY * DAY = D VZ.Taking now the triangles OF\\ r 9CPEand RVW in theare similar, and(ellipse or hyperbola, we see that CF]\ r CPE,RUW has one angle (at W) e([tial or supplementaryangles at P and V in the other two triangles,to thewhile we havewhence QV: VT =r (p:PP') (CV: QV),.and, by parallels,RW: WU = (p:PP') . (OP : PE).

144 APOLLONIUS OF PERGATherefore RUW, CPE, CFWsuce the halves of parallelogramsrelated as in the lemma ;thereforeARUW = AGFW - A CPE.The same property with reference to the diameter secondaryto

THE CON1CS, BOOK I 145or CQ in the case of the central coriic, bisects all chords asRR' parallel to the tangent at Q.Consequently EQ and CQare diameters of the respective conies.In order to refer the conic to the new diameter and thecorresponding ordinates, we have only to determine the parameterof these ordinates and to show that the property of theconic with reference to the new parameter and diameter is inthe same form as that originally found.The propositions I. 49, 50 do this, and show that the newparameter is in all the cases p', where (if is the point ofintersection of the tangents at I* and Q)OQ:QE = p':2QT.(1) In the case of the parabola, we have TP = PF= EQ,whenceA EOQ = A POT.Add to each the figure POQF'W;thereforeQTW'F' = a EW = bR'UW,whence, subtracting AlUW'F' from both, we haveTherefore KM .But KM : MF'therefcms KM* : li'M . MF'MF' = 2 .QT QM.= OQ:QE = p': 2 QT, by hypothesis ;:= p.QM 2 .QT QM.And KM . MF' = 2QT. QM, from above ;thereforewhich is the desired property.1KW^tf.QM,1The proposition that, in the case of the parabola, ifp be the parameterof the ordinates to the diameter through Q, then (see the first figureon p. 142)has an interesting application;for it enables us to prove the proposition,assumed without proof by Archimedes (but not easy to prove otherwise),that, if in a parabola the diameter through P bisects the chord QQ' in K,and QD is drawn perpendicular to PF, then

146 APOLLONIUS OF PERGA(2) In the case of the central conic, we haveAR'UW = ACF'W ~ ACPE.(Apollonius here assumes what he does not prove till III. 1,namely that ACPE = AGQT. This is proved thus.We have CV: CT = GV* : CPtherefore , ACQV: ACQT = ACQV: ACPE,Z; (I. 37, 39.)so thatThereforeACQT = ACPE.)A E'UW = ACF'W - ACQT,and it is easy to prove that in all casesAR'MF'=QTUM.Therefore R'M . MF' = QM(QT +MU).Let QL be drawn at right angles to tJQ and equal to p'.in K.in 11 and MK in N.Join Q'L and draw MK parallel to QL to meet Q'LDraw CH parallel to Q'L to meet QLNow R'M: MF' = OQ:QE= QL:But QT:so that (QH + MN):MU= Q1I:QT.= CQ:2QT, by hypothesis,CM= QII:MN,(QT + MU) = QIt:QT= R f M:MF', from above.where pa is the parameter of the principal ordinates and p the parameterof the ordinates to the diameterFV. ,If the tangent at the vertex A meetsVP produced in E, and PT, the tangentat P, in 0, the proposition of Apolloniusproves that OP:PE=p:2PT.ButthereforeOP=%PT;PT" = p.PEThus QV* QD* = PT 2 : :PN*, by similar triangles,= p.AN:pa. AN

It follows thatTHE CONICS, BOOK I 147:QM(QH+ MN) QM(QT+ MU) = KM* : R'Mbut, from above, QM(QT + Mlf)= R'M . MF'thereforeli'M* = QM(QH+ MN)= QM. MK,;.MF';the desired property.In the case of the hyperbola, the same propertyis true forwliicli isthe opposite branch.These important propositions show that the ordinate propertyof the three conies is of the same form whatever diameter istaken as the diameter of reference. It is therefore a matterof indifference to which particular diameter and ordinates the* .conic is referred. Tins is stated by Apollonius in a summarywhich follows I. 50.Firstapiwinmcc of ^r^je^wi

148 APOLLONIUS OF PERGAIt has been my object, by means of the above detailedaccount of Book I, to show not merely what results areobtained by Apollonius, but the way in which he went towork ;and it will have been realized how entirely scientificand general the method is. When the foundation is thus laid,and the fundamental properties established, Apolloniusis ableto develop the rest of the subject on lines more similar tothose followed in our text-books. My description of the restof the work can therefore for the most part be confined to asummary of the contents.Book II begins with a section devoted to the properties ofthe asymptotes. They are constructed in II. 1 in this way.Beginning, as usual, with any diameter of reference and thecorresponding parameter and inclination of ordimites, Apolloniusdraws at P the vertex (the extremity of the diameter)a tangent to the hyperbola and sets off along it lengths PL, PL'on either side of P such that PL*=PL'*= t) . PP' [= M> a l>where p is the parameter. He then proves that (!L, (Hf producedwill not meet the curve in any finite point and are thereforeasymptotes. II. 2 proves further that no straight linecan itselfthrough G within the angle between the asymptotesbe an asymptote.II. 3 proves that the intercept made by theasymptotes on the tangent at any point P is bisected at P, andthat the square on each half of the intercept is equal to onefourthof the ''figure corresponding to the diameter throughP (i.e.one-fourth of the rectangle contained by the 'erect'side, the latus rectum or parameter corresponding to thediameter, and the diameter itself) this ;isproperty used as ameans of drawing a hyperbola when the asymptotes and onepoint on the curve are given (II. 4). II. 5-7 are propositionsabout a tangent at the extremity of a diameter being parallelto the chords bisected by it. Apollonius returns to theasymptotes in II. 8, and II. 8-14 give the other ordinaryproperties with reference to the asymptotes (II. 9 is a converseof II. 3), the equality of the intercepts between theasymptotes and the curve of any chord (II. 8), the equality ofthe rectangle contained by the distances between either pointin which the chord meets the curve and the points of intersectionwith the asymptotes to the square on the parallelsemi-diameter (II. 10), the latter propertywith reference to

THE CONICS, BOOK II 149the portions of the asymptotes which include between thema branch of the conjugate hyperbola (II. 11), the constancy ofthe rectangle contained by the straight lines drawn from anypoint of the curve in fixed directions to meet the asymptotes(equivalent to the Cartesian equation with reference to theasymptotes, xy = const.) (II. 12), and the fact that the curveand the asymptotes proceed to infinity and approach continuallynearer to one another, so that the distance separatingthem can be made smaller than any given length (II. 14). II. 15proves that the two opposite branches of a hyperbola have thesame asymptotes and II. 16 proves for the chord connectingpoints on two branches the property of II. 8. II. 1 7 shows that''conjugate opposites (two conjugate double-branch hyperbolas)have the same asymptotes. Propositions follow aboutcoftjugate hyperbolas; any tangent to the conjugate hyperbolawill meet both branches of the original hyperbolaand will be bisected at the point of contact (II. 19); if Q beany point on a hyperbola, and (IE parallel to the tangentat Q meets the conjugate hyperbola in E, the tangent atE will be parallel to CQ and CQ, CE will be conjugatediameters (II. 20), while the tangents at Q, K will meet on oneof the asymptotes (II. 21) if a chord ; Qq in one branch ofa hyperbola meet the asymptotes in R, r and the conjugatehyperbola in Q',

150 APOLLONIUS OF PERGAthe last problem, proving that, ifthe tangent to an ellipse atany point P meets the major axis in T, the angle OPT is notgreater than the angle ABA', where JB is one extremity of theminor axis.Book III begins with a series of propositions about theequality of certain areas, propositions of the same kind as, andeasily derived from, the propositions (I. 41-50) by means ofwhich, as already shown, the transformation of coordinates iseffected. We have first the proposition that, if the tangentsat any points P, Q of a conic meet in 0, and if they meetthe diameters through Q, P respectively in E, T, thenA OPT = AOQE (III.] ,4) ; and, if P, Q be points on adjacentbranches of conjugate hyperbolas,= AOP7^T &CQT (III. 13).With the same notation, if R be any other point on the conic,and if we draw RU parallel to the tangent at Q meeting thediameter through P in 'U and the diameter ^hrough Q in M,and RW parallel to the tangent at P meeting QT in II andthe diameters through Q, P in F, W, then AHQF = quadrilateralHTUR (III. 2. 6) ;this isproved at once from the factthat &RMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6above) by subtracting or adding the area HRMQ on eachside. Next take any other point R', and draw R'U', F'lI'R'Win the same way as before it is then;proved that, if RU, R'Wmeet in / and R'U', R W in J, the quadrilaterals F'IRF, I U U'lt'are equal, and also the quadrilaterals FJR'F', JU'UH (III. 3,7, 9, 10). The proof varies according to the actual positionsof the points in the figures.In Figs. 1, 2 AHFQ = quadrilateral UTUR,By subtraction, FHH'F'= WU'R' + (IU)\whence, if IH be added or subtracted, F'IRF= IVU'R',and again,if IJ be added to both,FJR'F' = JU'UR.In Fig. 3so thatbR'U'W = &CF'W'-&CQT,ACQT = CU'R'F'.

THE CONICS, BOOK III 151E F 1 FFia. 1.FIG. 2.Fm. 3.

152 APOLLONIUS OF PERGAAdding the quadrilateral CF'H'T, we haveAH'F'Q = H'TU'tt,and similarlyBy subtraction,AHFQ = HTUR.F'H'HF= H'TU'R'-IITUR.Adding H'IRH to each side, we haveIf each of these quadrilateralsis subtracted from /,/",FJR'F' = ./IThe corresponding results are proved in III. 5, 11, 12, 14for the case where the ordinates through RR' are drawn toa secondary diameter, and in III. 15 for the case where P, Qare on the original hyperbola and R, R' on the conjugatehyperbola.The importance of these propositions lies in the fact thatthey are immediately used to prove the well-known theoremsabout the rectangles contained by the segments of intersectingchords and the harmonic properties of the pole and polar.The former questionis dealt with in III. 16-23, which givea great variety of particular cases. We will give the proofof one case, to the effect that, if OP, ()Q be two tangentsto any conic and Rr, R'r' be any two chords parallel tothem respectively and intersecting in an internal or externalJypoint,then RJ . Jr : R'JWe have.Jr' = OP 2 :()Q 2 = (const.).RJ.Jr = RW**JW*> and RW 2 : JW* = &RUW: AJÎF;thereforeRJ .Jr : RW2 = (RW2 - JW*): RW* = JU'VR : &RUW.But RW 2 : OPtherefore, ex aequali, RJ .2 = ARUW :Jr : OPA'OPT;= 2 JU'UR : A OPT.

Similarly KM'* :THE CONICS, BOOK III 153JM' = 2 kll'F'M' :&JFM',whence R'J . Jr' : R'M' = 2 FJR'F' : A R'F'M'.But R'M'* :OQ2 =-- A R'F'M' : A OQE ;therefore, ex aeqwili, R'J . Jr' :OQ = 2 FJR'F' :AOQfl.It follows, since FJR'F' = JU'UR, and A07T = AOQA T ,that III . Jr : OP' = 2 #',/. ,/r' :Of/ 2 ,or RJ . Jr : R'J. Jr' = OP 2 :OQ*.If we had taken chords /h'j, /i'r/ parallel respectively toOQ, OP and intersecting in /, an internal or external point,we should have in like mannerft/. /> H'T. Trf = OQ^: OP 2 .As a particular case, if PP' IH a diameter, and /i? 1 R'r' he,chords parallel respectively to the tangent at P and thediameter PP' and intersecting in 7, then (as is separatelyproved)RI.Ir:R'l.Tr' = p:PP'.The corresponding results are provedin the cases where certainof the points lie on the conjugate hyperbola.The six following propositions about the segments of intersectingchords (III. 24- 9) refer to two chords in conjugatehyperbolas or in an ellipse drawn parallel respectively to twoconjugate diameters PP\ /)//, and the results in modern formare perhaps worth quoting. If lii\ R'r' be two chords sodrawn and intersecting in 0, then()the ellipse_ ~~CD*

154 APOLLONIUS OF PERGAThe general propositions containing the harmonic propertiesof the pole and polar of a conic are III. 37-40, which provethat in any conic, if TQ, Tq be tangents, and if Qq the chordof contact be bisected in F, then(1) if any straight line through T meet the conic in R', R andTR' = RI :Qq in /, then (Fig. 1)RT :IR' ;(2) if any straight line through Fmeet the conic in R, R'the parallel through T to Qq in 0, then (Fig. 2)FIG. 2.The above figures represent theorem (1) for the parabola andtheorem (2) for the ellipse.

To prove (1)R'L* : lR* = H'(f:QH*=THE CONICS, BOOK III 155we havebll'F'Q : AHFQAlso XfT* : TR* = R'U'* : UR>and ,R'T* : TR- = TW* : TW*= H'TU'R' : HTUR(III. 2,= kR'U'W :ARUW,= AT7/'1F : A THW,3, &c.).so that R'T-:TR~ = bTH'W - AR'U'W: A.THW ~ &RUW= U'TU'R'-.HTUR= R'l* :IR\ from above.To prove (2)we haveand alsoRV-.VR'- = RU*: R'U'* = &RUW :AR'U'}\",= HQ Z :Qll'*=:AHFQ AJI'F'Q = HTUR*: H'TU'R',KO that^:VR' 2 = UTllR + A RUW-Jl'TU'li' + bR'U'W= HO* : OH'2 .Props, 111. 30-6 deal separately with the particular casesin which (a) the transversal is parallel to an asymptote of thehyperbola or (b) the chord of contact is parallel to an asymptote,i.e. where one of the tangentsis an asymptote, which isthe tangent at infinity.Next we have propositions about intercepts made by twotangents on a third: If the tangents at three points of aparabola form a triangle, allthree tangents will be cut by thepoints of contact in the same proportion (III. 41); if the tangentsat the extremities of a diameter PP' of a central conicare cut in r, r' by any other tangent, Pr .ifPV= CD 2 (III. 42) ;the tangents at P, Q to a hyperbola meet the asymptotes in* Where a quadrilateral, as HTUR in the figure, is a cross-quadrilateral,the area is of course the difference between the two triangleswhich it forms, as HTW ^ RUW.

156 APOLLONIUS OF PERGALy IS and My M' respectively, then L'M, ZJlf'are both parallelto PQ (III. 44).The first of these propositions asserts that, if the tangents atthree points P, Q, R of a parabola form a triangle pqr, thenPr = :rq rQ Qp = qp pR.: :From this property it is easyto deduce the Cartesianequation of a parabola referred to two fixed tangents ascoordinate axes. Taking qR, qP as fixed coordinate axes, wefind the locus of Q thus. Let x, y be the coordinates of Q.Then, if qp = x lt qr = y l , qR= h, qP = k, we haveFrom these equations we derivei #1 2/i t . *also, since = t/1 3 we have h -- = 1.2/1 -y î 2/1By substituting for o^, 2/1the values V(ltx), V(ky) weobtain'+ (!)'='The focal properties of central conies are proved inIII. 45-52 without any reference to the directrix ;there isno mention of the focus of a parabola. The foci aro called'the points arising out ofthe application'(ra 4/c rr/y Trapa-are takenfloXfjs yivonsva crrjfjLtTa), the meaning being that >V, ASy/on the axis AA' such that A8.SA' = Atf .tfA' = \l\L .AA'or 6 r 2 that, is, in the phraseology of application of areas,to one-fourtha rectangle is applied to A A' as base equalpart of the 'figure', and in the case of the hyperbola exceeding,but in the case of the ellipse falling short, by asquare figure. The foci being thus found, it is proved that,if the tangents Ar, A'r' at the extremities of the axis are metby the tangent at any point P in r, ?' respectively, rr' subtendsa right angle at $, &', and the angles rrOS f A'r'S',are equal, asalso are the angles r'r&', ArS (III. 45, 46)." It is next shownthat, if be the intersection of r x r>Sy/ >S f then OP is , ,perpendicularto the tangent at P (III. 47).These propositions are

THE CONICS, BOOK III 157used to prove that the focal distances of P make equal angleswith the tangent at P (III. 48).In III. 49-52 follow theother ordinary properties, that, if SY be perpendicular tothe tangent at P, the locus of F is the circle on A A' asdiameter, that the lines from C drawn parallel to the focaldistances to meet the tangent at P are equal to CA, and thatthe sum or difference of the focal distances of any point isequal to A A'.The last propositions of Book III are of use 'with referenceto the locus with respect to three or four lines.They are asfollows.1. If PP / be a diameter of a central conic, and if PQ, P'Qdrawn to any other point Q of the conic meet the tangents at1", P in JK', li respectively, then PR.P'R' = 4r'D 2 (III. 53).2. If TQ, TQ' be two tangents to a conic, V the middle pointof QQ', P thepoint of contact of the tangent parallel to QQ',and liany other point on the conic, let Qr parallel to TQ'meet (// in r, and Q'/ parallel to TQ meet QR in / then;Qr CJV . :QQ'* = (PV*: .PT*) (TQ TQ': QV*). . (III. 54, 56.)3. If the tangents are tangents to oppositebranches of ahyperbola and meet in /,and if R, /*,/ are taken as before,while t([is half the chord through t parallel to QQ', thenQr.Q'r' QQ? = : tQ.tQ' : tq*. (III. 55.)The second of these propositions leads at once to the threelinelocus, and from this we easily obtain the Cartesianequation to a conic with reference to two fixed tangents asaxes, where the lengths of the tangents are h, ,viz.Book IV ison the whole dull, and need not be noticed atlength. Props. 1-23 prove the converse of the propositions inBook III about the harmonic properties of the pole and polarfor a large number of particular cafees. One of the propositions(IV. 9) gives a method of drawing two tangents toa conic from an external point T. Draw any two straightlines through T cutting the conic in Q, Q' and in R, R' respec-

158 APOLLONIUS OP PERGfAtively. Take on QQ' and 0' on RR' so that TQ', TR' areharmonically divided. The intersections of 00' produced withthe conic give the two points of contact required.The remainder of the Book (IV. 24-57) deals with intersectingconies, and the number of points in which, in particular cases,they can intersect or touch. IV. 24 proves that no two coniescan meet in such a way that part of one of them is commonto both, while the rest is not. The rest of the propositionscan be" divided into five groups, three of which can be broughtunder one general enunciation. Group I consists of particularcases depending on the more elementary considerations affectingconies:e.g. two conies having their concavities in oppositedirections will not meet in more than two points (IV. 35);if a conic meet one branch of a hyperbola,it will not meetthe other branch in more points than two (IV. 37); a conictouching one branch of a hyperbola with its concave sidewill not meet the opposite branch (IV. 39). IV. 36, 41, 42, 45,54 belong to this group. Group II contains propositions(IV. 25, 38, 43, 44, 46, 55) showing that no two conies(including in the term the double-branch hyperbola) canintersect iiimore than four points. Group III (IV. 26, 47 )t48,49, 50, 56) are particular cases of the proposition that twoconies which touch at one point cannot intersect at move thantwo other points. Group IV (IV. 27, 28, 29, 40, 51, 52, 53, 57)are cases of the proposition that no two conies which toucheach other at two points can intersect at any other point.Group V consists of propositions about double contact. Aparabola cannot touch another parabola in more points thanone (IV. 30); this follows from the property TP = PV. Aparabola, if it fall outside a hyperbola,cannot have doublecontact with it (IV. 31); it is shown that for the hyperbolaPV>PT, while for the parabola P'V = PT; therefore thehyperbola would fall outside the parabola, which is impossible.A parabola cannot have internal double contact with an ellipseor circle (IV. 32). A hyperbola cannot have double contactwith another hyperbola having the same centre (IV. 33) ;If an ellipse have doubleproved by means of GV. GT = GP 2 .contact with an ellipse or a circle, the chord of contact willpass through the centre (IV. 34).Book V is of an entirely different order, indeed it is the

THE CONICS, BOOKS IV-V 159most remarkable of the extant Books.It deals with normalsto conies regarded as maximum and minimum straight linesdrawn from particular points to the curve. Included in it area series of propositions which, though worked out by thepurest geometrical methods, actually lead immediately to thedetermination of the evolute of each of the three conies ;thatis to say, the Cartesian equations to the eyolutes can be easilydeduced from the results obtained by Apollonius. There canbe no doubt that the Book is almost wholly original,and it isa veritable geometrical tour deforce.Apollonius in this Book considers various points and classesof points with reference to the maximum or minimum straightlines which it is possible to draw from them to the conies,i.e. as the feet of normals to the curve. He begins naturallywith points on the axis, and he takes first the point E whereAE measured along the axis from the vertex A is ^>, p beingthe principal parameter. The first throe propositions provegenerally and for certain particular cases that, if in an ellipseor a hyperbola AM be drawn at right angles to A A' and equaland if CM/ moot the ordinate PN of any point P of theto I y>,curve in 7/, then PjV = 2 2 (quadrilateral MANIl) this is a;lemma used in the proofs of later propositions, V. 5, 6, &c.Next, in V. 4, 5, 6, he proves that, if AE = |^, then AE is theminimum straight line from E to the curve, and if P be anyother point on it, PE increases as P moves farther away fromA on either side ;from 1\he proves in fact that, if PN be the ordinate(1) in the case of the parabolaPE* = AE*(2) in the case of the hyperbola or ellipsePE* = AN 2 + AN*where of course p = BB' 2 /AA' y and therefore (AA' p)/AA'is equivalent to what we call e2 ,the square of the eccentricity.line fromIt is also proved that KA' is the maxivmim ^straightE to the curve. It is next proved that, if be any point onthe axis between A and E, OA isfromthe minimum straight lineto the curve and, if P isany other point on the curve,OP increases as P moves farther from A (V, 7).

160 APOLLONIUS OF PERGANext Apollonius takes points G on the axis at a distancefrom A greater than ^p, and he proves that the minimumstraight line from G to the curve (i.e. the normal) is Gl\where P is such a point that(1) in the case of the parabola NG = \p ;(2):in the case of the central conic NGON = :y>AA' ;and, if P' isany other point on the conic, P'G increases as P'moves away from P on either side this is ; proved by showingthat(1) for the parabola P'G 2 = PGP + NN'* ;(2) for the central conic P'G* = PG a + JViV /a ~-.vAAAs these propositions contain the fundamental properties ofthe subnormals, it is worth while to reproduce Apollonius' Bproofs.(1) In the parabola, if G be any pointon the axis such thatAG > %p, measure GN towards A equal to -Jyê^ 1*^ ^ >ethe ordinate throughNt P' any other point on the curve.Then shall PG be the minimum line from G to the curve, &c.

THE CONICS,'BOOK V 161We have P'iV' 2 = p. AN' = 2NG. AN';and N'G* = NN' 2 + NG 2 2NG. NN',according to the position of N'.ThereforeP'G 2 = 2NG.AN+ NG 2 + NN'*and the proposition is proved.(2) In the case of the central conic, take G on the axis suchthat AG > %p, and measure GN towards A such thatNG:CN = p:AA'.Draw the ordinate PN through N, and also the ordinate P'N'from any other point lv .We have first to prove the lemma (V. 1, 2, 3) that, if AM bedrawn perpendicular to AA' and equal to %p, and if CM,produced if necessary, meet P^Vin //, thenPiV = 2 2 (quadrilateral MANll).This is easy, for, if AL(= 2AM) be the parameter, and A'Lmeet PN inR, then, by the property of the curve,= AN (Nil + AM)= 2 (quadrilateral MANH).Let GU, produced if necessary, meet P'N' in 11'. From Hdraw 111 perpendicular to P'H'.Now, since, by hypothesis, NG : (>N = />: ANH = NG, whence alsoThereforeAndH'N' = N'G.A'= AM:AC= I1N:NC,NG* = 2 AUNG, N'G' = 2AI1'N'G.PN* = 2(MANH);therefore PG* = NG 2 + PiV a = 2(AMHG).

162 APOLLONIUS OF PEEGASimilarly, if CM meets P'N' in K,= 2 AH'N'G + 2(AMKN')Therefore, by subtraction,P'G*-PG 2 == HI.(H'IIK)= HI.(HT1K)CAwhich proves the proposition.If be any point on PG, OP is the minimum straight linefrom to the curve, and OP' increases as P' moves awayP fromon either side; this is proved in V. 12. (Since P'G > 'PG,L GPP' > L GP'P ; therefore, a fortiori, L OPP' > L OP'P,and OP' > OP.)Apolloiiius next proves the corresponding propositions withreference to points on the minor axis of an ellipse.If p' bethe parameter of the ordinates to the minor axis, 2>'=AA ' 2/BB',or ij/= CA Z /CB.If now E' be so taken that BE'=\p',then BE' is the maximum straight line from E' to the curveand, if P be any other point on it, E'P diminishes as P movesfarther from B on either side, and E'B' is the minimumstraightline from E' to the curve. It(V. 16-18). If be any pointwhere Bti isis, in fact, proved thatthe abscissa of Pon the minor axis such thatBO > BE', then OB is the maximum straight line from tothe curve, &c. (V. 19).If g be a point on the minor axis such that Bg > BG, butBg < % p',and if Gn be measured towards B so thatthen n is the foot of the ordinates of two points P such thatPC/ is the maximum straight line from gto the curve. Also,

THE CONICS, BOOK V 163if P' be any other point on it, P'gfarther from P on either side to B or B', andIfP> > '2-P = nu 2 . or nndiminishes as P' movesbe any point on Pg produced beyond the minor axis, POis the maximum straight line from to the same part of theellipse, for which *Pgr is a maximum, i.e.the semi-ellipse BPB',&c. (V. 20-2).In V. 23 it isproved that, if (j is on the minor axis, and gPa maximum straight line to the curve, and ifPg meets AA'in G then GP is the minimumy straight line from G to theone normalcurve ;this is proved by similar triangles. Onlycan be drawn from any one point on a conic (V. 24-6). Thenormal at any point P of a conic, whether regarded as aminimum straight line from G on the major axis or (in thecase of the ellipse) as a maximum straight line from g on theminor axis, is perpendicular to the tangent at P (V. 27-30);in general (1) if () be any point within a conic, and OP bea maximum or a minimum straight line from to the conic,the straight line through P perpendicular to PO touches theconic, and (2)if(Y be any point on OP producedoutside theconic, O'P is the minimum straight line from 0' to the conic,&c. (V. 31-4).Number of normals from a point.We now come to propositions about two or more normalsmeeting at a point.If the normal at P meet the axis ofa parabola or the axis A A' of a hyperbola or ellipse in G, theangle PGA increases as P or (i moves farther away from A,but in the case of the hyperbola the angle will always be lessthan the complement of half the angle between the asymptotes.Two normals at points on the same side of A A' will meet onthe opposite side of that axis ;and two normals at points onthe same quadrant of an ellipse as AB will meet at a pointwithin the angle ACB' (V. 35-40). In a parabola or anellipse any normal PG will meet the curve again; in thehyperbola, (1) if AA' be not greater than p, no normal canmeet the curve at a vsecond point on the same branch, but

164 APOLLONIUS OF PERGAA (2) if A' > p some normals will meet the same branch} againand others not (V. 41-3).If P 1G V P 2r 2b normals at points on one side of the axis ofa conic meeting in 0, and if be joined to any other point Pon the conic (it being further supposed in the case of theellipse that all three lines OP OPlf 2,OP cut the same half ofthe axis), then(1) OP cannot be a normal to the curve ;OP (2) if meet the axis in K, and PG be the normal at P, AGis less or greater than AK according as P does or does not liebetween P land P .2From this propositionit isproved that (1)three normals atpoints on one quadrant of an ellipse cannot meet at one point,and (2) four normals at points on one semi-ellipse bounded bythe major axis cannot meet at one point (V. In 44-8).any M conic, if be any point on the axis such that AMis not greater than \ y, no normal can be drawn through which cutsthe axis ; but, OP if be any straight line drawn to the curvecutting the axis in K, NK

THE CONICS, BOOK V 165greater or less than NG according as OP is or is not intermediatebetween the two normals (V. 51, 52).The proofs are of course long and complicated.The lengthy is determined in this :way(1) In the case of the parabola, measure MH towards thevertex equal to/;, and divided// at-A^ so that HNl = 2NÂ.The length yis then taken such thatwhere P lN lis the ordinate passing through N^ ;(2) In the case of the hyperbola and ellipse,we haveAM>%p, so that OA \AAI

166 APOLLONIUS OF PERGAThe propositions V. 53, 54 are particular cases of* the precedingpropositions.Construction of normals.The next section of the Book (V. 55-63) relates to the constructionof normals through various points according to theirposition within or without the conic and in relation to theaxes. It isproved that one normal can be drawn through anyinternal point and through any external point which is noton the axis through the vertex A . In particular, if isanypoint below the axis A A' of an ellipse, and OM is perpendicularto A A', then, if AM>ACy one normal can always bedrawn through cutting the axis between A and (7, but nevermore than one such normal (V. 55-7). The points on thecurve at which the straight lines through are normals aredetermined as the intersections of the conic with a certainrectangular hyperbola. The procedureof Apollonius is equivalent to the followinganalytical method. Let AM bethe axis of a conic, PGO one of thenormals which passes through the givenpoint 0, PN the ordinate at P and let;OM be drawn perpendicular to the axis.Take as axes of coordinates the axes in the central conic and,in the case of the parabola, the axis and the tangent at thevertex.If then (x, y) be the coordinates of P and (x lt y those ofv )we have y NO~~'2/1Therefore (1) for the parabolax^xNG-J .or^-(ai-*#)y--yi.*2> = ;(1)(2) in the ellipse or hyperbolaXy T a*)~l2/51 ' y&= '*< 2 >The intersections of these rectangular hyperbolas respec-

THE CONIC3, BOOKS V, VI 167lively with the conies give the points at which the normalspassing through are normals.Pappus criticizes the use of the rectangular hyperbola inthe case of the parabola as an unnecessary resort to a 'solidlocus'; the meaning evidently is that the same points ofintersection can be got by means of a certain circle takingthe place of the rectangular hyperbola. We can, in fact, from2the equation (1)above combined with y= px, obtain thecircle= o.The Book concludes with other propositions about maximaand minima. In particular V. 68-71 compare the lengths oftangents TQ, TQ', where Q is nearer to the axis than Q'.V. 72, 74 compare the lengths of two normals from a pointfrom which only two can be drawn and the lengths of otherstraight lines from to the curve ;V. 75-7 compare thelengths of throe normals to an ellipse drawn from a pointbelow the major axis, in relation to the lengths of otherstraight lines from to the curve.Book "VI is of much less interest. The first part (VI. 1-27)relates to equal (i.e. congruent) or similar conies and segmentsof conies ;it is naturally preceded by some definitions includingthose of ' equal ' and similar ' as applied to conies and'segments of conies. Conies are said to be similar if, the samenumber of ordinates being drawn to the axis at proportionaldistances from the vertices, all the ordinates are respectivelyproportional to the corresponding abscissae. The definition ofsimilar issegments the same with diameter substituted foraxis, and with the additional condition that the anglesbetween the base and diameter in each are equal. Twoparabolas are equal if the ordinates to a diameter in each areinclined to the respective diameters at equal angles and thecorresponding parameters are equal; two ellipses or hyperbolasare equalif the ordinates to a diameter in each areequally inclined to the respective diameters and the diametersas well as the corresponding parameters are equal (VI. 1. 2).Hyperbolas or ellipses are similar when the 'figure' on a'ondiameter of one is similar (instead of equal) to the cfigurea diameter of the other, and the ordinates to the diameters in

168 APOLLONIUS OF PERGAeach make equal angles with them ; all parabolasare similar(VI. 11, 12, 13).No conic of one of the three kinds (parabolas,hyperbolas or ellipses) can be equal or similar to a conicof either of the other two kinds (VI. 3, 14, 15). Let QPQ',qpq' be two segments of similar conies in which QQ', q

THE CONWS, BOOKS VI, VII 169The same is true if A A' isthe minor axis of an ellipseand pthe corresponding parameter (VII. 2, 3).If AA' be divided at //' as well as H (internally for thehyperbola and externally for the ellipse) so that H is adjacentto A and //' to A', and if A'II : All = AH': A'H' = AA' :p,the lines AH, A'11' (corresponding to p in the proportion) arecalled by Apollonius komoloyues,and lie makes considerableuse of the auxiliary points //, //' in later propositions fromVII. G onwards. Meantime he proves two more propositions,which, like VII. 1 3, are by way of lemmas. First, if CD bethe semi-diameter parallelto the tangent at P to a centralconic, and if the tangent meet the axis AA' in T, thenPT* : CD* CX. (VII. 4.)Draw AE, TF at right angles to CA to meet CP, and let AEmeet PT in 0. Then, if p' be the parameter of the ordinatesto GP, we have= NT :$p':PT=OP:PE (1.49,50.)orTherefore PT* : CD*= |?/. PF: $p'. CP= PF: CP

170 APOLLONIUS OF PERGASecondly, Apollonius proves that, if PN be a principalordinate in a parabola, p the principal parameter, p' theparameter of the ordinates to the diameter through P, thenp'=p + AN (VII. 5); this isproved by means of the sameproperty as VII. 4, namely %p' PT = OP : : PE.Much use is made in the remainder of the Book of twopoints Q and M, where AQ is drawn parallel to the conjugatediameter CD to meet the curve in Q, and M is the foot ofthe principal ordinate at Q;since the diameter OP bisectsto OP.both AA' and QA, it follows that A'Q is parallelMany ratios between functions of PP', DD' are expressedinterms of AM, A'M, MH, MH', AH, A'H,&c. The first propositionsof the Book proper (VII. 6, 7) prove, for instance,that PP' 2 : DD"* = MH': MH.For PT 2 : CD = 3 NT: ON = AM: A'M, by similar triangles.Also CP i : PT = 2 A'Q*: ATherefore,, ex aequaU,CP 2 : CD* = (AM:Q*.A'M) x (A'Q 2 :AQ2 )=2(AM: A'M) x (A'Q A'M. :MH')x (A'M. MH': AM. MH) x :(AM.MH AQ= (AM:A'M) x (AA': All') x (A'M: AM)x (MH': MH) x (A'H A :Therefore PP' 2 : DD'2 = MH ' : MH.2 )A'), by aid of VII. 2, 3.Next (VII. 8, 9, 10, 11) the following relations are proved,namely(1)AA f2 :(PP' + DDJ=A f II.MH': {MH' + V(MH.MH')} 2 ,(2) AA' 2 : PP'JID' = A'H :V(MH. MH')~(3) AA'* :(PP 1 *+ DD' = 2 )A'H : MH+ MH'.The steps by which these results are obtained are as follows.First, A A' 2 : PP' 2 =A'H: MH' (a)= A'H.MH':MH\(Tliis is proved thus :AA' 2 :PP'*=CA 2 :CP 2= CN.CT:CP Z= A'M. A'A :A'Q 2 .

THE VONICS, BOOK VII 171But A'Q*:A'M.MH'=AA':AH' (VII. 2, 3)= AA':A'H= A'M. AA': A'M. A'H,so that, alternately,A'M. A A': A'ip = A'M. A'H := A'11 -.Mil'.)A'M . ME'Next, PP'- : DD"whence PP': DD' = Mil': V(MH.and PI "* :(PP' + DD') = Mil'* := MH ' :Mil, as above, (/3)= MH'*'.MH.MU' IMH'), (y){ Mil ' + V(MH .*Mil') }(1) above follows from this relation and (ex) ex aequali;(2) follows from (a)nnd (y) ex aequali, and (3)from (a)and (/?).We now obtain immediately the important proposition thatPP"* + DD' 2 is constant, whatever be the position of P on anellipse or hyperbola (the upper sijfn referring to the ellipse),and is equal to AA U + J3B'' Z (VII. 12, 13, 29, 30).For AA* :BB'* = AA':>p= A'H :All = A'H :A'H',;therefore A A" 2 : A A'* + BB' = Z A'H : HH'also, from (a) above,;by construction :and, by means of (/3),'+ MilEx aequali, from the last two relations, we haveAA' 2 : (PP'* + DD" 2 )= A'H : HH'= AA'*:AA'*BB"\ from above,whence PP>* DD'* = A A' 2 BB' 2 .

.VII.172 APOLLONIUS OF PERGAA number of other ratios are expressed in terms of thestraight lines terminating at A, A', H, H', M, M' as follows(VII. 14-20).In the ellipse AA'* : PP'* * DD' 2 = A'H:2 CM,and in the hyperbola or ellipseordinates to PP')(if p be the parameterAA'*:p> = A'H.MH':MH*,A A' 2 :(PP' + pf = A'11 . Mir:A A'* : PP' .p = A'H :MH,(MHMH')*>and A A'* :(PP' 2 + = F2A'II MH':) . (Mil'* + MW).of theApollonius is now in a position, bymeans of all theserelations, resting on the use of the auxiliary points H, H', M,to compare different functions of any conjugate diameterswith the same functions of the axes, and to show how theformer vary (by way of increase or diminution) as P movesaway from A. The followingis a list of the functions compared,where for brevity I shall use a, b to represent AA' yBE' \a', I/ to represent PP', DD' ;and p, p' to represent the parametersof the ordinates to AA', PP /respectively.In a hyperbola, according as a > or < 6, of > or < &', and theratio a':// decreases or increases as P moves from A oneither side; also, if a = b, a'=b' (VII. 21-3); in an ellipsea:b>af:b', and the latter ratio diminishes as P moves fromA to B (VII. 24).In a hyperbola or ellipse a + b

THE CONICS, BOOK VII 173conjugate diameters in an ellipse or conjugate hyperbolas, andif the tangents at thoir extremities form the parallelogramLL'MM', thenthe parallelogram LL'MM' = rcct. A A'. BB'.The proof is interesting. Let the tangents at P, D respectivelymeet the major or transverse axis in T, T',Now (by VII. 4) PT* : VIP= NT : ONtherefore 2 A CPT :2&T'DC = NT :;CN.But 2 A CPT = :(GL) PT := CP :CD,DT', by similar triangles,That is, (CL)is a mean proportional between 2 &CPT and'DC.Therefore, since '/(NT. CN) is a mean proportionalNT betweenand CN,

174 APOLLONIUS OF PERGA2 ACPT: (GL)= V(GN. NT): GN(1.37,39)therefore (GL) = CA .The remaining propositions of the Book trace the variationsof different functions of the conjugate diameters, distinguishingthe maximum values, &c. The functions treated are thefollowing :p', the parameter of the ordinates to PP' in the hyperbola,according as A A' is (1) not less than p, the parameter correspondingto A A* ',(2) less than p but not less than \ p, (3) lessthan |p (VII. 33-5).PP'p' 9 as compared with AA'^p in the hyperbola (VII. 36)or the ellipse (VII. 37).PP'+p' AA'+p in the hyperbola (VII.38-40) or the ellipse (VII. 41).PP'./ AA'.p in the hyperbola (VII. 42)or the ellipse (VII. 43).., AA"~+p 2 in the hyperbola, accordingas (1)A A' is not less thanp, or (2) AA'< p, but A A'* notless than ^(AA^pY or9 (3)PP' 2 +// 2 AA'^+yP in the ellipse, accordingas AA' 2 is not greater, or isPl**fPgreater, than (AA' + p)247, 48).(VII.A A'* ~p* in the hyperbola, accordingas AA f > or

THE CONICU, BOOK VII 175As we have said, Book VIII is lost. The nature of itscontents can only be conjectured from Apollonius's ownremark that it contained determinate conic problems forwhich Book VII waS useful, particularly in determininglimits of possibility. Unfortunately, the lemmas of Pappusdo not enable us to form any clearer idea. But it is probableenough that the Book contained a number of problems havingfor their object the finding of conjugate diameters in a givenconic such that certain functions of their lengths have givenvalues. It was on this assumption that Halley attempteda restoration of the Book.If it be thought that the above account of the Conies isdisproportionately long for a work of this kind, it must beremembered that the treatise is a great classic which deservesto be more known than it is. What militates against itsbeing read in its original form is the greatextent of theexposition (it contains 387 separate propositions), due partlyto the Greek habit of proving particular cases of a generalproposition separately from the proposition itself, but more tothe cumbrousness of the enunciations of complicated propositionsin general terms (without the help of letters to denoteparticular points) and to the elaborateness of the Euclideanform, to which Apollonius adheres throughout.Other works by Apollonius.Pappus mentions and gives a short indication of the contentsof six other works of Apollonius which formed part of the1Treasury of A tialysis. Three of these should be mentionedin close connexion with the (Joules.(a) Om the Cutting-off of

176 APOLLONIUS OF PERGAa given point a straight line which shall cut off segments fromeach line (measured from the fixed points) bearing a givenratio to one another. 9 Thus, let A, B be fixed points on thetwo given straight lines AC, BK, arid let be the givenpoint. It is required to draw through a straight linecutting the given straight lines in points M, N respectivelysuch that AM is to UN in a given ratio. The two Books ofthe treatise discussed the various possible cases of this problemwhich arise according to the relative positions of thegiven straight lines and points, and also the necessary conditionsand limits of possibility in cases where a solution is notalways possible. The first Book begins by supposing thegiven lines to be parallel, and discusses the different caseswhich arise ; Apollonius then passes to the cases in which thestraight lines intersect, but one of the given points, A or B, isat the intersection of the two lines. Book II proceeds to thegeneral case shown in the above figure, and first proves thatthe general case can be reduced to the case in Book I whereone of the given points, A or J5, is at the intersection of thetwo lines. The reduction is easy. For join OB meeting A(Jin J3', and draw ffN' parallel to BN to meet OM in N'. Thenthe ratio ffN' :BN, being equal to the ratio OB' OB, is con-:stant. Since, therefore, BN: AM is a given ratio, the ratio11'N' :AM is also given.Apollonius proceeds in all cases by the orthodox method ofanalysis and synthesis. Suppose the problem solved andOMN drawn through in such a way that B'N'iAM is agiven ratio = A, say.

O.V THE CUTTJNG-OFF OF A RATIO 177Draw 0(7 parallel to ZLV or WN' to meet AM in V.D on AM such that OC : AD = X = B'N' : 4 J/.Then AM:AD = ffN':OUTaketherefore MD :AD = B'C : CM,or CM .MD= AD. B'C, a given rectangle..Henco tlieproblem is reduced to one of applyingto CD arectcnu/le (CM . MD) equal to

178 APOLLONIUS OF PERGAthe value of X in order that the solution may be possible.Apollonius begins by stating the limiting case, saying that weobtain a solution in a specialmanner in the case where M isthe middle point of CD, so that the rectangle CM . MI) orCB' . AD has its maximum value.The corresponding limiting value of \ is determined byfinding the corresponding positionof D or M.We have B'C : MD= CM: AD, as before,= B'M: MA',whence, sinceso thatMD = CM,B'C :B'M = CM: MA= B'M: B'A,B'M* = B'C. B'A.Thus M is found and therefore D also.According, therefore, as X is less or greater than the particularvalue of OG-.AD thus determined, Apollonius finds nosolution or two solutions.Further, we have >AD = B'A + B'C- (B'D + B'C)= B'A + B'C-2B'M= B'A + B'C- 2 VB'A .B'C.If then we refer the various points to a system of coordinates in which B'A, B'N' are the axes of x and y,and iiwe denote by (x, y) and the length B'A by h,X = OC/AD = y/(h + x-2

O.Y THE CUTTING-OFF OF A RATIO 179Further, we it' put for A the ratio between the lengthstwo fixed tangents, then if h, k be those lengths,of thek__-ywhich can easily be reduced tothe equation of the parabola referred to the two fixed tangentsas axes.(ft)On the cuttiny-off of an area (\piov dnoro^),two Books.This work, also in two Books, dealt with a similar problem,with the difference that the intercepts on the given straightlines measured from the given points are required, not tohave a given ratio, but to contain a given rectangle. Halleyincluded an attempted restoration of this work in his editionof the De sectione rationis.The general case can here again be reduced to the morespecial one in which one of the fixed pointsis at the intersectionof the two given straight lines. Using the samefigure as before, but with D taking the position shown by (D)iu the figure,we take that point such thator0(J .A I) = the given rectangle.We have then to draw ON'M throughB'N' .AM=OC.AD,&N'\()G=AD:AM.But, by parallels, B'N' : OG = B'M: CM',therefore AM : CM = AD: B'Msuch thatso that'B fM .MD = AD. B'C.Hence, as before, the problem is reduced to an applicationof a rectangle in the well-known manner. The complete

180 APOLLONIUS OF PERGAtreatment of tins problem in all its particular cases with their8iopt

ON DETERMINATE SECTION 181pletc Theory of Involution. Pappus says that the separatecases were dealt with in which the given ratio was that ofeither (1) the square of one abscissa measured from therequired point or (2) the rectangle contained by two. suchabscissae to any one of the following: (1) the square of oneabscissa, (2) the rectangle contained by one abscissa andanother separate line of given length independent of theposition of the required point, (3) the rectangle contained bytwo abscissae. We learn also that maxima and minima wereinvestigated. From the lemmas, too, we may draw otherconclusions, e. g.(1) that, in the case where X = 1, or AP.VP =Apollonius used the relation III* : DP = AK . Il( ' : AD .DC,(2) that Apollonius probably obtained a double point E of theinvolution determined by the point-pairs A

182 APOLLONIUS OF PERGAa circle, (8) two circles and a straight line, (9) a point, a circleand a straight line, (10) three circles. Of these varieties thefirst two are treated in Eucl. IV ;Book I of Apollonius'streatise treated of (3), (4), (5), (6), (8), (9),while (7),the case oftwo straight lines and a circle, and (10), that of the threethe whole of Book II.circles, occupiedThe last problem (10), where the data are three circles,has exercised the ingenuity of many distinguished geometers,including Vieta and Newton. Vieta (1540-1603) set the problemto Adrianus llomanus (van Roomeu, 1561-1615) whosolved itby means of a hyperbola. Vieta was not satisfiedwith this, and rejoined with his Apollonius Galhis (1600) inwhich he solved the problem by plane methods. A solutionof the same kind is given by Newton in his ArithmeticaUniversalis (Prob. xlvii), while an equivalent problemissolved by means of two hyperbolas in the Principia, Lemmaxvi. The problem is quite capable of a plane 'solution, and,'as a matter of fact, it is not difficult to restore the actualsolution of Apollonius (which of course used the 'plane' methoddepending on the straight line and circle only), by means ofthe lemmas given by Pappus. Three things are necessary tothe solution. (1)A proposition, used by Pappus elsewhere 1and easily proved, that, if two circles touch internally orexternally, any straight lino through the point of contactdivides the circles into segments respectively similar. (2) Theproposition that, given three circles, their six centres of similitude(external and internal) lie three by three on four straightlines. This proposition, though not proved in Pappus, wascertainly known to the ancient geometersit is even; possiblethat Pappus omitted to prove it because it was actually provedby Apollonius in his treatise. (3)An auxiliary problem solvedby Pappus and enunciated by him as follows. 2 Given a circleABC, and given three points D, E, F in a straight line, toinflect (the broken line)DAE (to the circle) so as to make BGin a straight line with CF\ in other words, to inscribe in thecircle a triangle the sides of which, when produced, passrespectively through three given points lying in a straightline. This problem is interesting as a typical example of theancient analysis followed by synthesis. Suppose the problem1Pappus, iv, pp. 194-6. 2 lb. vii, p. 848.

CONTACTS OR TANGENCIES 183solved, i.e.suppose DA, EA drawn to the circle cuttingpoints B, C such that BC produced passes through F.Draw BG parallel to DF; join GCand produceit to meet DE in H.ThenLBAQ=LRGCit intherefore A, Dy= supplement of Z CHD ;//, C lie on a circle, andDE.EH=AE.EC. o H K e FNow AE.E(J is given, being equal to the squareon thetangent from E to the circle ;and DE is given ; therefore HEis given, and therefore the point //.But F is also given ; therefore the problemis reduced todrawing HC, FC to meet the circle in such a way that, ifHC, FC produced meet the circle again in G, B, the straightline BG is parallel to IIF: a problem which Pappus haspreviously solved. 1Suppose this done, and draw BK the tangent at B meetingHF'mK. ThenZ KBC = Z BGC, in the alternate segment,Also the angle (JFK is common to the two triangles KBF>CHF\ therefore the triangles are similar, andorNow BF .FC is given, and so is HF\therefore FK is given, and therefore K is given.The synthesisis as follows. Take a point H on DE suchthat DE .the circle.EH is equal to the square on the tangent from E toNext take K on IIF such that HF.FK = the square on thetangent from F to the circle.Draw the tangent to the Circle from K, and let B be thepoint of contact. Join BF meeting the circle in (7, and join1Pappus, vii, pp. 830-2.

184 APOLLONIUS OF PERGAHO meeting the circle again in (?. It is then easy to provethat BG is parallelto DF.Now join EC, and produceit to meet the circle again at A ;join AB.We have only to prove that A B, BD are in one straightline.Since DE.EH = AE.EC, the points A, D, II, are concyclic.Now the angle CHF, which is the supplement of the angleCHD, is equal to the angle JiGC, and therefore to tlieangle BAG.Therefore the angle BAC isequal to the supplement ofangle DEC, so that the angle BAG is equal to the angle DAG,and AB, BD are in a straight line.The problem of Apolloniusis now easy.We will take thecase in which the required circle touches all the three givencircles externally as shown in the figure. Let the radii of the

OX CONTACTS OR TANGENOIEH 185given circles be a, ft, c and their centres A, B, C. Let D, #, Fbe the external centres of similitude so that BD: 7)C'=&:c, &c.Suppose the problem solved, and let P, Q, R be the pointsof contact. Let PQ produced meet the circles with centresA By again in K, L. Then, by the proposition (1) above, thesegments KGP, QHL are both similar to the segment PYQ ;therefore theyare similar to one another. Itfollows that PQproduced beyond L passes through F. Similarly QR, PRproduced pass respectively through />, E.Let PE, QD meet the circle with centre M (! again in N.yThen, the segments PQR, RXM being similar, the anglesPQR, RNM are equal, and therefore MN is parallel to PQ.Produce XM to meet EF in V.Then E V :therefore the pointEF = EM: EP = EC : EAV is given.= c : a ;Accordingly the problem reduces itself to this: Given threerpoints J , E, D in a straight line, it is required to draw DR, ERto a point R on the circle with centre (! so that, if DR, ER meetthe circle again in iV, M, XM produced shall pass through V.This is the problem of Pappus just solved.Thus R is found, and DR, ER produced meet the circleswith centres B and A in the other required points Q, Prespectively.(e)Plane loci, two Books.Pappus gives a pretty full account of the contents of thiswork, which has sufficed to enable restorations of it tobe made by three distinguished geometers, Fermat, vanSchooten, and (most completely) by Robert Simson. Pappusprefaces his account by a classification of loci on twodifferent plans. Under the first classification loci are of threekinds: (1) tfaKTiKOL, lioldin

186 APOLLON1US OF PERGAof a line. 1The second classification is the familiar division intoplane, solid, and linear loci, plane loci being straight linesand circles only, solid loci conic sections only, and linear locithose which are not straight lines nor circles nor any of theconic sections. The loci dealt with in our treatise are accordinglyall straight lines or circles. The proof of the propositionsis of course enormously facilitated by the use ofCartesian coordinates, and many of the loci are really thegeometrical equivalent of fundamental theorems in analyticalor algebraical geometry. Pappus begins with a compositeenunciation, including a number of propositions, in theseterms, which, though apparently confused, are not difficultto follow out:'If two straight lines be drawn, from one given point or fromtwo, which are (a) in a straight line or(/>) parallel or(c) include a given angle, and either (a) bear a given ratio toone another or (]8)contain a given rectangle, then, if the locusof the extremity of one of the lines is a plane locus given inposition, the locus of the extremity of the other will also be aplane locus given in position, which will sometimes be of thesame kind as the former, sometimes of the other kind, andwill sometimes be similarly situated with reference to thestraight line, and sometimes contrarily, according to the2particular differences in the suppositions/(The words ' with reference to the straight line ' are obscure, butthe straight line is presumably some obvious straight line ineach figure, e.g., when there are two given points, the straightline joining them.) After quoting thrue obvious loci ' addedby Charmaridrus ',Pappus gives three loci which, though containingan unnecessary restriction in the third case, amountto the statement that any equation of the first degree betweencoordinates inclined at fixed angles to (a) two axes perpendicularor oblique, (/>) to any number of axes, represents astraight line.The enunciations (5-7)are as follows/'5. ' If, when a straight line is given in magnitude and isInoved so as always to be parallel to a certain straight linegiven in position, one of the extremities (of the movingstraight line), lies on a straight line given in position, the1Pappus, vii, pp. 660. 18-662. 5. 2 Ib. vii, pp. 662. 25-664. 7.3Ib., pp. 664. 20-666. 6.

PLANE LOCI 187other extremity will also lie on a straight line given inposition/(That is, x = a or y = 6 in Cartesian coordinatesstraight line.)represents a6. 'If from any point straight lines be drawn to meet at givenangles two straight lines either parallel or intersecting, and ifthe straight lines so drawn have a given ratio to one anotheror if the sum of one of them and a line to which the other hasa given ratio be given (in length), then the point will lie on astraight line given in position/(This includes the equivalent of saying that, if x, ybe thecoordinates of the point, each of the equations x = my,= c represents a straight line.)7. 'If any number of straight lines be given in position, andstraight lines be drawn from a point to meet them at givenangles, and if the straight lines so drawn be such that therectangle contained by one of them and a given straight lineadded to the rectangle contained by another of them and(another) given straight line is equal to the rectangle containedby a third and a (third) giveil straight line, and similarlywith the others, the point will lie on a straight line givenin position/(Here we have trilinear or multilinear coordinates proportionalto the distances of the variable point from each of thethree or more fixed lines. When there are three fixed lines,the statement is that ax + by= cz represents a straight line.The precise meaning of the words 'and similarly with thethe others' or 'of the others 1 KOI r>v \onrSw 6/io/coyisuncertain; the words seem to imply that, when there weremore than three rectangles a,i\ lnj.cz ..., two of them weretaken to be equal to the sum of all the others ; but it is quitepossible that Pappus meant that any linear equation betweenthese rectangles represented a straight line. Precisely howfar Apollonius went in generality we are not in ajudge.)The last enunciation (8) of Pappus referringstates that,position toto Book IIf from any point (two) straight lines be drawn to meet (two)'parallel straight lines given in position at given angles, and

188 APOLLONIUS OF PERGAcut off from the parallels straight lines measured from givenpoints on them such that (a) they have a given ratio or(6) they contain a given rectangle or the sum or difference(c)of figures of given species described on them respectively isequal to a given "area, the point will lie on a straight linegiven in position/ 1The contents of Book II are equally interesting.Some ofthe enunciations shall for brevity be given by means of lettersinstead of in general terms. If from two given points A, Btwo straight lines be ' inflected ' (K\aorQSxnv) to a point P, then(1), if AP* * BP* is given, the locus of P is a straight line ;(2) if AP, BP are in a given ratio, the locus is a straight lineor a circle [thisis the proposition quoted by Eutocius in hiscommentary on the Conies, but already known to Aristotle];(4) if AP2 is 'greater by a given area than in a given ratio 'to J9P 2 ,i.e. if AP* = d z + m . BP' 1 the locus is a circle , given inposition. An interesting propositionis (5) that, 'If from anynumber of given points whatever straight lines be inflected toone point, and the figures (givenin species) described on all ofthem be together equal to a given area, the point will lie on'a circumference (circle) given in position that is to; say, ifa.AP* + p. BP* + y.G'P 2 +... = a given area (where a,/3, y ...are constants), the locus of P is a circle. (3) states that, ifAN be a fixed straight line and A a fixed point on it, and it'line drawn to a point P such that, if PNAP be any straightis perpendicular to AN, AP 2 = a . ANor a .BN, where a is agiven length and B is another fixed point on AN, then thelocus of P is a circle given in position ; this is equivalentto the fact that, if A be the origin, AN the axis of x, andx = A N, y = PN bo the coordinates of P, the locus ,/: 2+ y'1= axor

PLANE LOCI 189inflected lies on a .circle given in position/ The meaningseems to be this : Given two fixed points A, J3, a length a,a straight line OX with a point fixed upon it, and a directionrepresented, say, l>y any straight line OZ through 0, then,if AP, BP be drawn to P, and PM parallel toin M, the locus of P will b(3 a circle given in position ifOZ meets OXwhore a, f$ are constants. The last two loci are againobscurely expressed, but the sense is this : (7) If PQ be anychord of a circle passing through a fixed internal point 0, andIf be an external point on PQ produced(a) OE = 2 PR .JtQ or Ofi*(/>) + PO .OQ= PR .RQsuch that eitherythe locusof Ji is a straight line given in position. (8) is the reciprocalof this: Given tho fixed point (), the straight line which isthe locus of R, and also the; relation (a)or (6),the locus ofP, Q is a circle.() Neva-eis (Verylnys or Inclinations),two Books.As we have seen, the problem in a i/cvcrt? is to placebetween two straight lines, a straight line and a curve, ortwo curves, a straight line of given length in such a waythat it rertfe* towards a fixed point,i.e. it will, if produced,pass through a fixed point. Pappus observes that,when we conic* to particular cases, the problem will be'plane', * solid' or 'linear', according to the nature of theparticular hypotheses; but a selection had been made fromthe class which could be solved by plane methods, i.e. bymeans of the straightline*, and circle, the object being to givethose which wore more generally useful in geometry. Thefollowing were the cases thus selected and proved. 1I. Given (a) a semicircle and u straight line at right anglesto the base, or (/;)two semicircles with their bases in a straightline, to insert a straight line of given length verging to anangle of the semicircle [or of one of the semicircles].II. Given a rhombus with one side produced,a straight line of given length in the external angleto insertso that itverges to the opposite angle.1Pappus, vii, pp. 670-2.

190 APOLLONIUS OF PERGAIII.Given a circle, to insert a chord of given length vergingto a given point.In Book I of Apollonius's work there were four cases ofI (a),two cases of III, and two of II ;the second Book containedten cases of I (b).Restorations were attempted by Marino Ghetaldi (Apolloniusredivivus, Venice, 1607, and Apollonius redivivus . . . Libersecundus, Venice, 1613), Alexander Anderson (in a tiupplemeutumApollo mi redivivi, 1612), and Samuel Horsley(Oxford, 1770); the last is much the most complete.In the case of the rhombus (II) the construction of Apolloniuscan be restored with certainty. It depends on a lemma givenby Pappus, which is as follows: Given a rhombus AD withdiagonal BC produced to E, if F be taken on EC such that EFis a mean proportional between BE and EC, and if a circle bedescribed with E as centre and EF as radius cutting CDin K and AC produced in //, then shall J8,KyHbe in onestraight line. 1Let the circle cut AC in L, join LK meeting BC in M, andjoin HE, LE, KE.Since now CL, CK are equally inclined to the diameter ofthe circle, CL = CK. Also EL = EK and it follows that theytriangles ECK, ECL are equal in all respects,LCKE = LCLE = LCIIE.so thatorBy hypothesis,EB:EF=EF: EC,EB:EK = EK:EC.1Pappus, vii, pp. 778-80.

NETSEIS (VERGING^ OR INCLINATIONS) 191Therefore the triangles BEK, KEC, which have the angleBEK common, are similar, andZ CBK = Z GKK = Z G## (from above).ButZ 7/C'# = Z AGE = ZTherefore in the triangles G'-fi/f, C7/JF two angles arerespectively equal,so that Z (JElf = Z (7/i 7i also.But since LGKK = LCHK (from above), K, G f , E, II areconcyclic.Hence Z

192 APOLLONIUS OF PERGABut, by similar triangles EKH, DOB,EK:KH=J)G:GB,and, since the ratio DG:GB, as well as KH, is given, EKisgiven.The construction then is as follows.If k be the given length, take a straight line /;such thatEG equal to p 2 and exceeding bya square ;then with E as centre and radius equal to />describe acircle cutting A(! produced in II and (H) in K. UK is thenequal to k and, by Pappus's lemma, verges towards B.Pappus adds an interesting solution of the same problemwith reference to a square instead of a rhombus ;the solutionapply to BG a rectangle BE .isby one Heraclitus and depends on a lemma which Pappusalso gives.1We hear of yet other lost works by Apollonius.A(77) Comparison of the dodecahedron with the icosahedron.This is mentioned by Hypsicles in the preface to the so-calledin twoBook XIV of Euclid. Like the Conies, it appearededitions, the second of which contained the proposition that,if there be a dodecahedron and an icosahedron inscribed inone and the same sphere, the surfaces of the solids are in thesame ratio as their volumes ;this was established by showing'that the perpendiculars from the centre of the sphere toa pentagonal face of the dodecahedron and to a triangularface of the icosahedron are equal.(&} Marinus on Euclid's Data speaks of a General Treatise(KaOoXov TTpay/jiaTeia) in which Apollonius used the wordassigned (rtrayfjitvov) as a comprehensive term to describe thedatum in general. It would appear that this work musthave dealt with the fundamental principles of mathematics,definitions, axioms, &c., and that to it must be referred thevarious remarks on such subjects attributed to Apollonius byProclus, the elucidation of the notion of a line, the definition1Pappus, vii, pp. 780-4.

OTHER LOST WORKS 193of plane and solid angles, and his attempts to prove the axioms ;it must also have included the three definitions (13-15) inEuclid's Data which, according to a scholium, were due toApollonius and must therefore have been interpolated (theyare definitions of Karrj-yfievr], avr\y/jL^vqand the, ellipticalphrase wapa 0cre*, which means ' parallel to a straight linegiven in position '). Probably the same work also containedApollonius' s alternative constructions for the problems ofEucl. I. 10, 11 and 23 given by Proclus. Pappus speaksof a mention by Apollonius 'before his own elements' of theclass of locus called e^/crj/coy, and it may be that the treatisenow in question is referred to rather than the Plane Lociitself.(i) The work On the Cochliau was on the cylindrical helix.It included the theoretical generationof the curve on thesurface of the cylinder, and the proof that the curve ishomoeomeric or uniform, i.e. such that any part will fit uponor coincide with any other.A work (K) on Unordered Irrationals is mentioned byProclus, and a scholium on Eucl. X. 1 extracted from Pappus'scommentary remarks that c Euclid did not deal with allrationals and irrationals, but only with the simplest kinds bythe combination of which an infinite- number of irrationalsare formed, of which latter Apollonius also gave some'.To a like effect is a passage of the fragment of Pappus'scommentary on Eucl. X discovered in an Arabic translationby Woepcke: 'it was Apollonius who, besides the orderedirrational magnitudes, showed the existence of the unordered,and by accurate methods set forth a great number of them'.The hints given by the author of the commentary seem to implythat Apollonius's extensions of the theory of irrationals tooktwo directions, (1) generalizing the medial straight line ofEuclid, 011 the basis that, between two lines commensurable insquare (only), we may take not only one sole medial line butthree or four, and so on ad infinitwin, since we can take,between any two given straight lines, asmanylines aswe please in continued proportion, (2) forming compoundirrationals by the addition and subtraction of more than twoterms of the sort composing the binomials, (tpotomes, &c.1523.2

194 APOLLONIUS OF PERGA(A) On the burning-miwor (rrpi rov isirvptov) the title ofanother work of Apollonius mentioned by the author of theFragmentum mathematicum Boliense, which is attributed byHeiberg to Anthemius but is more likely (judging by its survivalsof antiquated terminology) to belong to a much earlierdate. The fragment shows that Apollonius discussed thespherical form of mirror among others. Moreover, the extantfragment by Anthemius himself (on burning mirrors) proves theproperty of mirrors of parabolic section, using the properties ofthe parabola (a) that the tangent at any point makes equalangles with the axis and with the focal distance of the point,and (&) that the distance of any point on the curve from thefocus is equal to its distance from a certain straight line(our ' directrix ') ;and we can well believe that the parabolicform of mirror was also considered in Apollonius's work, andthat he was fully aware of the focal properties of the parabola,notwithstanding the omission from the Conies of all mentionof the focus of a parabola.(/*) In a work called &KVTOKLOV ( quick-delivery ') ApolloniuHis said to have found an approximation to the value of TT ' bya different calculation (from that of Archimedes), bringing itwithin closer limits '} Whatever these closer limits may havebeen, they were considered to be less suitable for practical usethan those of Archimedes.It is a moot question whether Apollonius's system of arithmeticalnotation (by tetrads) for expressing large numbersand performing the usual arithmetical operations with them,as described by Pappus, was included in this same work.Heiberg thinks it probable, but there does not seem to be anynecessary reason why the notation for large numbers, classifyingthem into myriads, double myriads, triple myriads, &c.,i.e. according to powers of 10,000, need have been connectedwith the calculation of the value of TT, unless indeed the numbersused in the calculation were so large as to require thetetradic system for the handling of them.We have seen that Apolloniusis credited with a solutionof the problem of the two mean proportionals (vol. i,pp. 262-3).1y. Eutocius on Archimedes, Measurement of a Circle,

OTHER LOST WORKS 195Astronomy.We are told lby Ptolemaeus Chennus that Apollouiua wasfamed for his astronomy, and was called (Epsilon) becausethe form of that letter is associated with that of the moon, towhich his accurate researches principally related. Hippolytussays he made the distance of the moon's circle from the surfaceof the, earth to be 500 myriads of stades. 2 This figure'can hardly be right, for, the diameter of the earth being,according to Eratosthenes's evaluation, about eight myriads ofstades, this would make the distance of the moon from theearth about 125 times the earth's radius. This is an unlikelyfigure, seeing that Aristarchus had given limits for the ratiosbetween the distance of the moon and its diameter, andbetween the diameters of the moon and the earth, which leadto about 19 as the ratio of the moon's distance to the earth'sradius. Tannery suggests that perhaps Hippolytus made amistake in copying from his source and took the figure of5,000,000 stades to be the length of the radius instead of thediameter of the moon's orbit.But we have better evidence of the achievements of Apolloniusin astronomy. In Ptolemy's tiynt(u:is:}he appears asan authority upon the hypotheses of epicycles and eccentricsdesigned to account for the apparent motions of the planets.The propositions of Apollonius quoted by Ptolemy containexact statements ofthe alternative hypotheses, and from thisfact it was at one time concluded that Apollonius inventedthe two hypotheses. This, however, is not the case. Thehypothesis of epicycles was already involved, though withrestricted application, in the theory of Heraclides of Pontusthat the two inferior planets, Mercury and Venus, revolve incircles like satellites round the sun, while the sun itselfrevolves in a circle round the earth ;that is, the two planetsdescribe epicycles about the material suji as moving centre.In order to explain the motions of the superior planets bymeans of epicycles it was necessary to conceive of an epicycleabout a point as moving centre which is not a material buta mathematical point. It was some time before this extensionof the theory of epicycles took place, and in the meantime12apud Phottunt, Cod. cxc, p. 151 b 18, ed. Bekker.3Hippol. Befitt. iv. 8, p. 66. ed, Duncker. Ptolemv. Suntaxis. xii. 1.

196 APOLLONIUS OF PERGAanother hypothesis, that of eccentrics, was invented to accountfor the movements of the superior planets only.We are at thisstage when we come to Apollonius. His enunciations showthat he understood the tlieory of epicyclesin all its generality,but he states specifically that the theory of eccentrics can onlybe applied to the three planets which can be at any distancefrom the sun. The reason why he says that the eccentrichypothesis will not serve for the inferior planets is that, inorder to make it serve, we should have to suppose the circledescribed by the centre of the eccentric circle to be greaterthan the eccentric circle itself. (Even this generalization wasmade later, at or before the time of Hipparchus.) Apolloniusfurther says in his enunciation about the eccentric that 'thecentre of the eccentric circle moves about the centre of thezodiac in the direct order of the signs and at a speed equal tothat of the sun, while the star moves on the eccentric aboutits centre in the inverse order of the signs and at a speedequal to the anomaly It '. is clear from this that the theoryof eccentrics was invented for the specific purpose of explainingthe movements of Mars, Jupiter, and Saturn about thesun and for that purpose alone. This explanation, combinedwith the use of epicycles about the sun as centre to accountfor the motions of Venus and Mercury, amounted to thesystem of Tycho Brahe that ; system was therefore anticipatedby some one intermediate in date between Heraclides andApollonius and probably nearer to the latter, or itmayhave been Apollonius himself who took this important step.If it was, then Apollonius, coining after Aristarchus ofSamos, would be exactly the Tycho lirahe of the Copernicusof antiquity. The actual propositions quoted by Ptolemy asproved by Apollonius among others show mathematically atwhat points, under each of the two -hypotheses, the apparentforward motion changes into apparent retrogradation andvice versa, or the planet appears to be stationfiw/.

XVTHE SUCCESSORS OF THE GREAT GEOMETERSWITH Archimedes and Apollonius Greek geometry reachedits culminating point. There remained details to be filledin, and no doubt ina work such as, for instance, the Cuiiicsgeometers of the requisite calibre could have found propositionscontaining the germ of theories which were capable ofindependent development. But, speaking generally, the furtherprogress of geometry on general lines was practicallybarred by the restrictions of method and form which wereinseparable from the classical Greek geometry. True, it wasopeit to geometers to discover and investigate curves of ahigher order than conies, such as spirals, conchoids, and thelike. Bat the Greeks could not get very far even on theselines in the absence of some system of coordinates and withoutfreer means of manipulation such as are afforded by modernalgebra, in contrast to the geometrical algebra, which couldonly deal with equations connecting lines, areas, and volumes,but involving no higher dimensions than three, except in sofar as the use of proportions allowed a very partial exemptionfrom this limitation. The theoretical methods availableenabled quadratic, cubic and bi-quadratic equations or theirequivalents to be solved. But all the solutions were (jeometrieal;in other words, quantities could only be represented bylines, areas and volumes, or ratios between them. There wasnothing corresponding to operations with general algebraicalquantities irrespective of what they represented. There wereno symbols for such quantities. In particular, the irrationalwas discovered in the form of incommensurable lines ;henceirrationals came to be represented by straight lines as theyare in Euclid, Book X, and the Greeks had no other way ofrepresenting them. It followed that a product of two irrationalscould only be represented by a rectangle, and so on.Even when Diophantus came to use a symbol for an unkngwii

198 SUCCESSORS OF THE GREAT GEOMETERSquantity, it was only an abbreviation for the wordwith the meaning of an 'undetermined multitude of units ',not a general quantity. The restriction then of the algebraemployed by geometers to the geometrical form of algebraoperated as an insuperable obstacle to any really new departurein theoretical geometry.It might be thought that there was room for further extensionsin the region of solid geometry. But the fundamentalprinciples of solid geometry had also been laid down in Euclid,Books XI XIII ;the theoretical geometry of the sphere hadbeen fully treated in the ancient sphaeric ;and any furtherapplication of solid geometry, or of loci in three dimensions,was hampered by the same restrictions of method whichhindered the further progress of plane geometry.Theoretical geometry being thus practicallyat the end ofits resources, it was natural that mathematicians, seeking foran opening, should turn to the applications of geometry. Oneobvious branch remaining to be worked out wavS the geometryof measurement, or mensuration, in its widest sense, which ofcourse had to wait on pure theory and to be based on itsresults. One species of mensuration was immediately requiredfor astronomy, namely the measurement of triangles, especiallyspherical triangles; in other words, trigonometry plane andspherical. Another species of mensuration was that in whichan example had already been set by Archimedes, namely themeasurement of areas and volumes of different shapes, andarithmetical approximations to their true values in caseswhere they involved surds or the ratio (IT)between thecircumference of a circle and its diameter ;the object of suchmensuration was largely practical.0f these two kinds ofmensuration, the first (trigonometry) is represented by Hipparchus,Menelaus and Ptolemy the second; by Heron ofAlexandria. These mathematicians will be dealt with in laterchapters this chapter will be devoted to the successors of the;great geometers who worked on the same lines as the latter.Unfortunately we have only very meagre information as towhat these geometers actually accomplished in keeping up thetradition. No geometrical works by them have come downto us in their entirety,and we are dependent on isolatedextracts or scraps of information furnished by commen-

NICOMEDES 199tators, and especially by Pappus and Eutocius. Some ofthese are very interesting, and it is evident from theextracts from the works of such writers as Diocles andDionysodorus that, for some time after Archimedes andApollonius, mathematicians had a thorough grasp of thecontents of the works of the great geometers, and were ableto use the principles and methods laid down therein withease and skill.Two geometers properly belonging to this chapter havealready been dealt with. The first is NICOMEDES, the inventorof the conchoid, who was about intermediate in date betweenEratosthenes and Apollonius. The conchoid lias already beendescribed above (vol. i, pp. 238-40). It gave a general methodof solving any vtv

200 SUCCESSORS OF THE GREAT GEOMETERSThe second name is that of DIOCLES. We have already(vol. i, pp. 264-6) seen him as the discoverer of the curveknown as the cissoid, which he used to solve the problemof the two mean proportionals, and also (pp. 47-9 above)as the author of a method of solving the equivalent ofa certain cubic equation by means of the intersectionof an ellipseand a hyperbola. We are indebted for ourinformation on both these subjects to Eutocius, 1 who tellsus that the fragments which he quotes came From Diocles'swork ?T/oi wvpefcw, On burning-mirrors. The connexion ofthe two things with the subject of this treatise is not obvious,and we may perhaps infer that it was a work of considerableWhat exactly were the forms of the burning-mirrorsscope.discussed in the treatise it is not possible* to say, but it isprobably safe to assume that among them were concavemirrors in the forms (1) of a sphere, (2) of a paraboloid, and(3) of the surface described by the revolution of an ellipseabout its major axis. The author of the Frayment'iim nuithematicumBoliense says that Apollonius in his book On liteburniny-mirror disclosed the case of the concave sphericalmirror, showing about what point ignition would take place ;and it is certain that Apollonius was aware that an ellipse hasthe property of reflecting all rays throughone focus to theother focus. Nor is it likely that the corresponding propertyof a parabola with reference to rays parallel to the axis wasunknown to Apollonius. Diodes therefore, writing a centuryor more later than Apollonius, could hardly have failed todeal with all three cases. True, Anthemius (died aboutA.D. 534) in his fragment on burning-mirrors says that theancients, while mentioning the usual burning-mirrors andsaying that such figures are conic sections, omitted to specifywhich conic sections, and how produced, and gave no geometricalproofs of their properties. But if the propertieswere commonly known and quoted, it is obvious that theymust have been proved by the ancients, and the explanationof Anthemius's remark ispresumably that the original worksin which they were proved (e.g. those of Apollonius andDiodes) were already lost when he wrote. There appears tobe no trace of Diocles's work left either in Greek or Arabic,1Eutocius. loc. cit.. n. 6G. 8 so., n. 160. 3 so.

DIOCLES 201unless we have a fragment from it in theiMithematicum Jiobiense. But Moslem writers regarded Dioclesas the discoverer of the parabolic burning-mirror; 'the ancients',says al Singrirl (Sachawl, Ansari), made * mirrors of planesurfaces. Some made them concave (i.e. spherical) untilDiocles (Diuklis) showed and proved that,if the surface ofthese mirrors has its curvature in the form of a parabola, theythen have the greatest power and burn most strongly. Thereis a work on this subject composed by Ibn al-Haitham/ Thiswork survives in Arabic and in Latin translations, and isreproduced by Heiberg and Wiedemann 1 : it does not, however,mention the name of Diocles, but only those of Archimedesand Aiithemius. Urn al-Haitham says that famousmen like Archimedes and Anthemius had used mirrors madeup of a number of spherical rings; afterwards, he adds, theyconsidered the form of curves which would reflect rays to onepoint, and found that the concave surface of a paraboloid ofrevolution has this property. It is curious to find Ibn al-Haitham saying that the ancients had not set out the proofssufficiently, nor the method by which they discovered them,words which almost exactly recall those of Antheniius himself.Nevertheless the whole course of Ibn al-Haitham's proofsison the CJreek model, Apollonius being actually quoted by namein tin* proof of the main property of the parabola required,namely that the tangent at any pointof the curve makesequal angles with the focal distance of the, point and thestraight line, drawn through it parallel to the axis. A proofof the property actually survives in the (ireek Fratjinentiifiimtithmnitwimi Hohicnê, which evidently came from sometreatise on the parabolic burning-mirror; but Ibn al-Haithamdoes not seem to have had even this fragment at his disposal,since his proof takes a different course, distinguishing threedifferent cases, reducing the property by analysisto theknown property A X = A 2\ and then working out the synthesis.Tim proof in the Fnuj'nwntutH is worth giving. It issubstantially as follows, beginning with the preliminary lemmathat, if PT} the tangent at any point P, meets the axis at r l\and if AM be measured along the axis from the vertex Aequal to %AL, where A L is the parameter, then &P = ST.1Bibliotheca mathematics, x 3 , 1910, pp. 201-37,

202 SUCCESSORS OF THE GREAT GEOMETERSLet PN be the ordinate from P; draw AY at right anglesto the axis meeting PT in Y, and join HY.NowPN*= 4 AH. AN= 4AH. AT (since AN =But PN = ZAY (since 4JV = AT) ;therefore 4 F 2 = TA .AH,and the angle TYH is right.The triangles 8YT, HYP being right-angled, and 2T beingequal to YP, it follows that SP = AT.With the same figure, let #7' be a ray parallel to AXimpinging on the curve at P. It is required to prove thatthe angles of incidence and reflection (to H) are equal.We have HP = &1\ so that < the angles at the points 7 f ,Pare equal. So', says the author, 'arc the angles TPA, KPli[the angles between the tangent and the curve, on each side, ofthe point of contact]. Let the difference between the anglesbe taken; therefore the angles HP A, RPIt which remain[again * mixed* angles] are equal. Similarly we shall showthat all the lines drawn parallel to AH will be reflected atequal angles to the point H.'The author then proceeds: 'Thus burning-mirrors constructedwith the surface of impact (in the form) of thein the mannersection of a right-angled cone may easily,

DIOCLES. PERSEUS 203above shown, be proved to bring about ignition atindicated/the pointHeiberg held that the style of this fragment is By/antineand that it is probably by Anthemius. Cantor conjecturedthat here we might, after all, have an extract from Diocles'swork. Heiberg's supposition seems to me untenable becauseof the author's use (1) of the ancient terms 'section ofa right-angled cone' for parabola and 'diameter' for axis(to say nothing of tin* use of the parameter, of which there isno word in the genuine fragment of Anthemius), and (2) ofthe mixed 'angles of contact*. Nor does it seem likely thateven Diodes, living a century after Apollonius, would have4spoken of the 'section mixedof a right-angled cone' instead of aparabola, or used theangle of which there is only the'merest survival in Euclid. The assumption of the equalityof the two angles made by the curve with the tangent onboth sides of the point of contact reminds us of Aristotle'sassumption of the equality of the angles * nf a segment 1 ofa eirde as prior to the truth provedin Eucl. I. 5. I aminclined, therefore, to date the fragment much earlier oventhan Diodes. Zeuthen suggested that the property of theparaholoidal mirror may him? been discovered by Archimedes,who, according to a Greek tradition, wrote (\ito^tncn. This,however, does not receive any confirmation in Ibn al-Haithamor in Anthemius, and we can only say that the fragment atleast goes back to an original which was probablythan 'Apollonius.not laterPKUSKI'S isonly known, from allusions to him in Proclus, 1as the. discoverer mid investigator of the >/>/?'/

204 SUCCESSORS OF THE GREAT GEOMETERSlike renown as discoverers of other curves to be obtained bycutting well-known solid figures other than the cone andcylinder. A particular case of one such solid figure, theorrerpa, had already been employed by Archytas, and the moregeneral form of it would not unnaturally be thought of aslikely to give sections worthy of investigation. Since Geininusis Proclus's authority, Perseus may have lived at any date fromEuclid's time to (say) 75 B.C., but the most probable suppositionseems to be that he came before Apollonius and near toEuclid in date.The spire in one of its forms is what we calla tore, or ananchor-ring. It is generated by the revolution of a circleabout a straight line in its plane in such a w$ty that the planeof the circle always passes through the axis of revolution. Ittakes three forms according as the axis of revolution is(a) altogether outside the circle, when the .spireisopen(8i\r)s), (b) a tangent to the circle, when the .surface, is continuous((rvv*\ri$), or (c)a chord of the circle, when it is interlaced(c/ZTreTrAcy/zli/Ty),or crossing- itself (cTraAAarrovo-a) ;analternative name for the surface was *pt'/coy,a riiuj.celebrated his discovery in an epigramcPerseusto the effect thatPerseus on his discovery of three lines (curves) upon fivesections gave thanks to the gods therefor'. 1 There is omodoubt about the meaning of * three JinevS u/xm five sections'(rpcf? ypa/jipas TTI rrli/re Topa^s). We gather from Proclus'saccount of three sections distinguished by Perseus that theplane of section was always parallel to the axis of revolutionor perpendicular to the plane which cuts the tore symmetri-It is difficult to inter-cally like the division in a split-ring.pret the phrase if it means three curves made by five differentsections. Proclus indeed implies that the three curves weresections of the three kinds of torerespectively (the open, theclosed, and the interlaced), but this isevidently a slip.Tannery interprets the phrase as meaning 'three curves in,addition to five sections a '. Of these the five sections belongto the open tore, in which the distance (c)of the centre of thegenerating circle from the axis of revolution is greater thanthe radius (a) of the generating circle. If d be the perpen-1Proclus on Eucl. I, p. 112. 2.2 See Tannery, Mtmoirett scientijiques, II, pp. 24-8.

PERSEUS 205dicular distance of the plane of section from the axi*s of rotation,we can distinguish the following cases :(1)c + a>d>c. Here the curve is an oval.(2) d = c: transition from case (1) to the next case.(3) c>J>c a. The curve is now a closed curve narrowestin the middle.= (4) eZ r In ft. this case the curve is the hipi>opede(horse-fetter), a curve in the shape of the figure, of 8. Thelemniscate of Bernoulli is a particular case of this curve, thatnamely in which c = 2a.(5) r a>iZ>0. In this case the section consists of twoovals symmetrical with one another.The three curves specified l>y Prcx-lus are those correspondingto (1), (3) and (4).When the ton* is 'continuous 7 or closed, c =,and we havesections corresponding to (1), (2) and (3) only; (4) reduces totwo cireles touching one another.But Tannery finds in the third, the interlaced, form of torethree new sections corresponding to(l) (2) (3),each with anoval in the middle. This would make three curves in additionto the five sections, or eight curves in all. We cannot hecertain that this is the true explanation of the phrase in theepigram: but it seems to l>e the l>est suggestionbeen made.that hasAccording to Proelus, Perseus worked out the property ofhis curves, as Nicomedes did that of the conchoid, Hippiasthat of the ij-mulrutrix,and Apollonius those of the threeconic sections. That is, Perseus must have given, in someform, the equivalent of the Cartesian equation by which wecan represent the different curves in question.If we refer thetore to three axes of coordinates at right angles to one anotherwith the, centre of the tore as origin, the axis of y being takenI/O be the axis of revolution, and those of z, scbeing perpendicularto it in the plane bisecting the tore (making it a splitring),the equation of the tore is

206 SUCCESSORS OF THE GREAT GEOMETERSwhere c, a have the same meaning as above.. The differentsections parallel to the axis of revolution are obtained bygiving (say) z any value between and c + a. For the valuez = a the curve is the oval of Cassini which has the propertythat, if r, r' be the distances of any point on the curve fromtwo fixed points as poles, rr'= const. For, = if z' a, the equationbecomes(#* + y* + c 2 )- = 4 ca a 2 + 4 c 2 a 2 ,or Jc a"and this is equivalent to rr'= + 2ra if ./,?/ are the coordinatesof any point on the curve referred to Ox, Oy as axes, whewthe twois the middle point of the line (2v in length) joiningpoles, and Ox lies along that line in either direction, while Oyis perpendicular to it. Whether Perseus discussed this caseand arrived at the property in relation to the two polesis ofcourse quite uncertain.Isoperirnetricfigures.The subject of isoperimetric figures, that is to say, the comparisonof the areas of figures having different shapes but thesame perimeter, was one which would naturally appeal to theearly Greek mathematicians. We gather from Proclus's noteson Eucl. I. 36, 37 that those theorems, proving that parallelogramsor triangles on the same or equal bases and betweenthe same parallels are equal in area, appeared to the ordinaryperson paradoxical because they meant that, by moving theside opposite to the base in the parallelogram, or the vertexof the triangle, to the right or left as far as we please, we mayincrease the perimeter of the figure to any extent while keepingthe same area. Thus the perimeter in parallelograms ortriangles is in itself no criterion as to their area. Misconceptionon this subject was rifeamong non-mathematicians.Proclus tells us of describers of countries who inferredthe size of cities from their perimeters; he mentions alsocertain members of communistic societies in his own time whocheated their fellow-members by giving them land of greaterperimeter but less area than the plots which they took

ISOPERIMETRIC FIGURES. ZENODORUS 207themselves, so that, while they got a reputation for greaterhonesty, they in fact took more than their share of theproduce. 1 Several remarks by ancient authors show theprevalence of the same misconception, Thucydides estimatesthe size of Sicily according to the time required for circumnavigatingit. 2 About 130 B.C. Poly bias observed that therewere people who could not understand that camps of the sameperiphery might have different capacities. 3Quintilian has asimilar remark, and Cantor thinks he may have had in hismind the calculations of Pliny, who compares the size ofdifferent parts of the earth by adding their lengthsbreadths. 4to their/ENonours wrote, at some date lx*tween (Ray) 200 B.C. andA.I). 90, a treatise ncpi iero/*rpan> o-xâra)!/,On isometricjiyure*. A number of propositions from it are preserved inthe commentary of Theon of Alexandria on Book I ofPtolemy's tfyutaxix; and they are reproduced in Latin in thethird volume of Hultsch's edition of Pappus, for the purposeof comparison with Pappus's own exposition of the samepropositions at the beginning of his B(x>k V, where he appearsto have followed Zenodorus pretty elosely while making somechanges in detail. 5 From the closeness with which the styleof Zenodorus follows that ofEuclid and Archimedes we mayjudge that his date was not much later than that of Archimedes,whom he mentions as the author of the proposition(Afeamrenient

208 SUCCESSORS OF THE GREAT GEOMETERSarea. Zenodorus's treatise was not confined to propositionsabout plane figures, but gave also the theorem that Of allsolid figures the surfaces of which are equal, the sphere isgreatest in solid content.We will briefly indicate Zenodorus's method of proof. Tobegin with (1) ;thelet ABC, DBF be equilateral and equiangularpolygons of the same perimeter, DEF having more anglesthan ABC. Let G, H be the centres of the circumscribingcircles, OK, HL the perpendiculars from

ZENODORUS 209The triangles NMK, HDL are now equal in all respects,andNK is equal to IIL, so that OK < HL.But the area of the polygon ABC is half the rectanglecontained by GK and the perimeter, while the area of thepolygon DEF is half the rectangle contained by HL andthe same perimeter. Therefore the area of the polygon DEFis the greater.(2) The proof that a circle is greater than any regularpolygon with the same perimeter is deduced immediately fromArchimedes'** proposition that the area of a circle is equalto the right-angled triangle with perpendicular side equal tothe radius and base equal to the perimeter of the circle;Zenodorus inserts a proof in exteiiso of Archimedes's proposition,with preliminary lemma. The perpendicular fromthe centre of the circle circumscribing the polygon is easilyproved to IK; less than the radius of the given circle withperimeter equal to that of the polygon whence the :propositionfollows.(3) The proof of this proposition depends on some preliminarylemmas. The first proves that, if there be twotriangles on the same base and with thepuinc perimeter, one being isosceles andthe other scalene, the isosceles trianglehas the greater area. (Given the scalenetriangle BDC on the base B(\ it is easy todraw on BC as base the isosceles trianglehaving the same perimeter. We haveonly to take BU equal to (/M)+/>f f ).bisect li(! at fa\ and erect at K the perpendicularAK such that AK* = Bll*-BK\)Produce BA to Fso that BA = AF, and join AD, DF.1Then Jil) + DF> BF< i.e.BA +A( i.e. BD,+ DC, by hypothesis;therefore DF > DC, whence in the triangles FAD,CAD the angle FA I) > the angle CAD.Therefore L FA 1) > \ L FA ( '> LBVA.Make the angle FAG equal to the angle BCA or ABC, sothat AG is parallel tojBC; let BD produced meet AG in G,and join GC.1BSS.SP

210 SUCCESSORS OF THE GREAT GEOMETERSThen> &DBC.The second lemrna is to the effect that, given two isoscelestriangles not similar to one another, if we construct on thesame bases two triangles similar to one another such that thesum of their perimeters isequal to the sum of the perimetersof the first two triangles, then the sum of the areas of thethan the sum of the areas ofsimilar triangles is greaterthe non-similar triangles. (The easy construction of thesimilar triangles is given in a separate lemma.)Let the bases of the isosceles triangles, KB, BC, bo placet Ione straight line, BC being greater than KB.inLet ABC, DKB be the similar isoscelestriangles, and FttC,GKB the non-similar, the triangles being such thatBA+AC + ED + DB = BF+ FC+EG + GB.Produce AF, GD to meet the bases in K, L.AK, GL bisect BC, EB at right angles at K, L.Produce GL to //, making LH equal to GL.Join HB and produceit to N\ join IIF.Then clearlyNow, since the triangles ABC, DEB are similar,ABC is equal to the angle DEB or DBE.the angleTherefore Z NBC (= L HBE = Z GBE) > Z DBE or Z ABC;therefore the angle ABH, and a fortiori the angle FBH, isless than two right angles, and HF meets BK in some point M.

it isproved that I>L + AK > GL + FK,ZENODORUS 211Now, by hypothesis, DB + BA = GB + BF:thereforeDB + BA = HB + BF> HF.By an easy lemma, since the triangles DEB, A BC are similar,(DB + BA) = 2(DL + AK)* + (BL + BK)*Therefore (DL + AK)* + LK* :> HF*whenceDL + A K > GL + FK,and it follows that AF > GD.But BK > BL] therefore AF.BK > GD.BL.Hence the * hollow-angled (figure)' (KoiXoywvtov)greater than the hollow-angled (figure)(fEDB.Adding A DEB + & fi/*Vto each, we haveA DKH + & A B( > A HEB+A FBC.1ABFC isThe above is the only case taken by Zenodorus. The proofstill holds if Kit = B(\ so that BK = BL. But it fails in thecase in which EB > IK 1 and the vertex G of the triangle EBbelonging to the non-similar pairis still above /) and notbelow it (as F is below A in the preceding case). This wasno doubt the reason why Pappus gave a pnx)f intended toapply to all the cases without distinction. This isproof thesame as the above proof by Zenodorus up to the point wherebut then* diverges. Unfortunately the text is bad, and givesno sufKcient indication of the course of the proof; but it wouldseem that Pappus used the relationsandDL :AK :GLAl\*:DLt2= A DEB : A GEB,FK = AABC:AFB(\=AAB(':&DEB,combined of course with the fact that GB+ BF = DB + BA,in order to prove the proposition that,according asDL + AK > or < GL + FK,A DEB + & ABC > or < A GEB+& FBC.

212 SUCCESSORS OF THE GREAT GEOMETERSThe proof of his proposition,whatever it was, Pappusbut in the text as we have itindicates that he will give later ;the promiseis not fulfilled.Then follows the proof that the maximum polygon of givenperimeter is both equilateral andequiangular.(1) It is equilateral.For, if not, let two sides of themaximum polygon, as AB, lid, beunequal. Join A(\ and on AC asbase draw the isosceles triangle AFCsuch that AF+ FC= AB + BC. Thearea of the triangle AFC is thengreater than the area of the triangle ARC, and the area ofthe whole polygon has been increased bythe construction:which is impossible, as by hypothesismaximum.the area is aSimilarly it can be proved that no other side is unequalto any other.(2) It is also equiangular.For, if possible, let the maximum polygon ABCJ)E (whichwe have proved to be equilateral)have the angle at B greater thanDEC.BAC, DECmvthe angle at I). Thennon-similar isosceles triangles. OnAC, CE as bases describe the twoisosceles triangles FAG, GKC similarto one another which have the sumof their perimeters equalto thesum of the perimeters of BA(\Then the sum of the areas of the two similar isoscelestriangles is greater than the sum of the areas of the trianglesBACy DEC] the area of the polygon is therefore increased,which is contrary to the hypothesis.Hence no two angles of the polygon can be unequal.The maximum polygon of given perimeteris therefore bothequilateral and equiangular.Dealing with the sphere in relation to other solids having

ZENODORUS. HYPSICLES 213their surfaces equal to that of the sphere, Zenodorus confinedhimself to proving (1) that the sphere is greaterif the othersolid with surface equal to that of the sphereis a solid formedby the . revolution of a regular polygon about a diameterbisecting it as in Archimedes, On the Sphere and (lylinder,Book I,and (2) that the sphereisgreater than any ofthe regular solids having its surface equalto that of thesphere.Pappus's treatment of the subject is more complete in thathe proves that the sphere is greater than the cone or cylinderthe surface of which is equal to that of the sphere, and furtherthat of the five regular solids which have the same surfacethat which has more faces is the greater.1HYPSICLKS (second half of second century B.C.) has alreadybeen mentioned (vol. i, pp. 419 20) as the author of the continuationof the JJlententa known as 15ook XIV. Me is quotedby Diophantus as having given a definition of a polygonalnumber as follows:4If there are as many numbers as we please beginning from1 and increasing by the same common difference, then, whenthe common difference is 1, the sum of all the numbers ina triangular number when; 2, a square: when 3, a pentagonalnumber [and so on].And the number of isangles calledafter the number which exceeds the common difference by 2,and the side after the number of terms including1.'This definition amounts to saying that the /

214 SUCCESSORS OF THE GREAT GEOMETERSwork in which we find the division of the eclipticcircle into360 ' parts ' or degrees. The author says, after the preliminarypropositions,*The circle of the zodiac having been divided into 360 equalcircumferences (arcs),let each of the latter be called a degree'in space (polpa TOTTIKTJ, local ' or ' spatial part And similarly,supposing that the time in which the zodiac circle').returns to any position it has left is divided into 360 equaltimes, let each of these be called a degree in time (fioipaFrom the word KaXeiada) ('let it be called ')we may perhapsinfer that the terms were new in Greece. This brings us tothe question of the origin of the division (1) of the circle ofthe zodiac, (2)of the circle in general, into 360 parts. On thisquestion innumerable suggestions have been made. Withreference to (1)it was suggested as long ago as 1788 (by Formaleoiii)that the division was meant to correspond to thenumber of days in the year. Another suggestionis that itwould early be discovered that, in the case of any circle theinscribed hexagon dividing the circumference into six partshas each of its sides equal to the radius, and that this wouldnaturally lead to the circle being regularly divided into sixparts after this, the very ancient sexagesimal system would;naturally come into operation and each of the parts would bedivided into 60 subdivisions, giving 360 of these for the wholecircle.Again, there is an explanation which is not evengeometrical, namely that in the Babylonian numeral system,which combined the use of 6 and 10 as base**, the numbers 6,60, 360, 3600 were fundamental round numbers, and thesenumbers were transferred from arithmetic to the heavens.The obvious objection to the first of these explanationsthe solar year) is(referring the 360 to the number of days inthat the Babylonians were well acquainted, as far back as themonuments go, with 365-2 as the number of days in the year.A variant of the hexagon-theory is the suggestion .that anatural angle to be discovered, and to serve as a measure ofothers, is the angle of an equilateral triangle, found by drawinga star * like a six-spoked wheel without anythe base of a sundial was so divided into six angles, it would becircle. If

HYPSICLES 215natural to divide each of the sixth parts into either 10 or 60parts; the former division would account for the attesteddivision of the day into 60 hours, while the latter division onthe sexagesimal system would give the 360 time-degrees (eachof 4 minutes) making up the day of 24 hours. The purelyarithmetical explanation is defective in that the series ofnumbers for which the Babylonians had special names is not60, 360, 3600 but 60 (Soss), COO (Ner), and 3600 or 60 2 (Sar).On the whole, after all that has been said, I know of nobetter suggestion than that of Tannery. 1 It is certain thatboth the division of the ecliptic into 360 degrees and that ofthe vv\Qrinepov into 360 time-degrees were adopted by theGreeks from Babylon. Now the earliest division of theecliptic was into 12 parts, the signs, and the question is, howwere the signs subdivided? Tannery observes that, accordingto tho cuneiform inscriptions, as well as the testimony ofGreek authors, the sign was divided into parts one of which(tlarfjatu) was double of the other (wwnm), the former beingl/30th, the other (called stadium by Manilius) l/60th, of thesign the former division would give 360 ; parts, the latter 720parts for the whole circle. The latter division was morenatural, in view of the long-established system of sexagesimalfractions; it also had the advantage of corresponding tolerablyclosely to the apparent diameter of the sun in comparisonwith the circumference of the sun's apparent circle. But, onthe other hand, the double fraction, the l/30th, was containedin the circle of the zodiac approximately the same number oftimes as there are days in the year, and consequently correspondednearly to the distance described by the sun along thexodiac in one day. It would seem that this advantage wassufficient to turn the scale in favour of dividing each sign ofthe zodiac into 30 parts, giving 360 parts for the wholecircle. While the Chaldaeans thus divided the ecliptic into360 parts, it does not appear that they applied the same divisionto the equator or any other circle. They measured anglesin general by dls yan ell representing 2, so that the completecircle contained 180, not 360, parts,which they called ells.The explanation may perhaps be that the Chald&eans divided1Tannery. La ' coudee aatronomique et les anciennes divisions ducercle ' (IKmoircs scieHtifiquea, ii, pp. 256-68).

216 SUCCESSORS OF THE GREAT GEOMETERSthe diameter of the circle into 60 ells in accordance with theirusual sexagesimal division, and then came to divide the circumferenceinto 180 such ells on the ground that the circumferenceis roughly three times the diameter. The measurementin ells and dactyli (of which there were 24 to the ell)survives in Hipparchus (On the Phaeiiometia ofEudoxus andAratus), and some measurements in terms of the same unitsare given by Ptolemy. It was Hipparchus who first dividedthe circle in general into 360 parts or degrees, and theintroduction of this division coincides with his invention oftrigonometry.The contents of Hypsicles's tract need not detain us long.The problem is : If we know the ratio which the length of thelongest day bears to the length of the shortest day at anygiven place, to find how many time-degrees it takes any givensign to rise ; and, after this has been found, the author tindswhat length of time it takes any given degree in any sign torise, i.e. the interval between the rising of one degree-point onthe ecliptic and that of the next following. It is explainedthat the longest clay is the time during which one half of thezodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises,and the shortest day the time during which the other half(Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises.Now at Alexandria the longest dayis to the shortest as 7to 6; the longest therefore contains 210 ' time-degrees', theshortest 150. The two quadrants Cancer- Virgo and Libra-Sagittarius take the same time to rise, namely 105 timedegrees,and the two quadrants Capricornus -Pisees and Aries-Gemini each take the same time, namely 75 time-degrees.It is further assumed that the times taken by Virgo, Leo,Cancer, Gemini, Taurus, Aries are in descending arithmeticalprogression, while the times taken by Libra, Scorpio, Sagittarius,Capricornus, Aquarius, Pisces continue the same descendingarithmetical series. Theand proved:following lemmas are usedI. If a lsa 2...a n,aw + 1, a n+2... a 2n is a descending arithmeticalprogression of 2n terms with 8 (= aâ^ = a 2-a 3= ...)as common difference,

HYPSICLES 217II. If MJ,rt 2'-- a n-*- a 2n-i *s a Descending arithmetical progressionof 2 /i-- 1 termsis the middle term, thenwith 5 as common difference and a ni+ 2+ ... + 2n-i= (2u l)a n.III. If a tt T , a...a n,^n+1 ...a 2nis a descending arithmeticalprogression of 2 jt,terms, thenNow let ;1, 5, (/ ho the descending series the sum of whichis 105, and /), K, Fthe next three terms in the same seriesthe sum of which is 75, the common difference being

218 SUCCESSORS OF THE GREAT GEOMETERSand' from these and the common difference 0< 0'13"20'" allthe times corresponding to all the degrees in the circle can befound,The procedure was probably, as Tannery thinks, takendirect from the Babylonians, who would no doubt use it forthe purpose of enabling the time to be determined at anyhour of the night. Another view is that the object wasastrological rather than astronomical (Manitius). In eithercase the method was exceedingly rough, and the assumedincreases and decreases in the times of the risings of the signsin arithmetical progressionare not in accordance with thefacts. The book could only have been written before the inventionof trigonometry by Hipparchus, for the problem offinding the times of rising of the signs is reallyone ofspherical trigonometry, and these times were actually calculatedby Hipparchus and Ptolemy by means of tables ofchords.DIONYSODORUS is known in the first place as the author ofa solution of the cubic equation subsidiary to the problem ofArchimedes, On the Spliere and Cylinder, II. 4, To cut a givensphere by a plane so that the volumes of the segments have toone another a given ratio (see above, p. 46). Up to recentlythis Dionysodorus was supposed to be Dionysodorus of Amisenein Pontus, whom Suidas describes as 'a mathematician worthyof mention in the field of education'. But we now learn froma fragment of the Herculaneum Roll, No. 1044, that 'Philonideswas a pupil,first of Eudeinus, and afterwards of Dionysodorus,the son of Dionysodorus the Caunian \ Now Eudeiuus isevidently Eudemus of Pergamum to whom Apolloriiu.s dedicatedthe first two Books of his Co ivies, and Apollonius actuallyasks him to show Book II to Philonides. In another fragmentPhilonides is said to have published some lectures byDionysodorus. Hence our Dionysodonis may be Dionysodorusof Caunus and a contemporary of Apollonius, or very littlelater. 1 A Dionysodorusis also mentioned by Heron 2 as theauthor of a tract On the Spire (or tore) in which he provedthat, if d be the diameter of the revolving circle1W. Schmidt in BiUiotheca mathematics iv s , pp. 321-5.8Heron, Metrica, ii. 13, p. 128. 3.

DIONYSODORUS 219generates the tore, and c the distance of its centre from theaxis of revolution,(volume of tore) : nc2 . d = %ird*:that is, (volume of tore) = %n 2 .cd*,^ccZ,which is of course the product of the area of the generatingcircle and the length o the path of its centre of gravity. Theform in which the result is stated, namely that the tore is tothe cylinder with height(/ and radius c as the generatingcircle of the tore is to half the parallelogram cd tindicatesquite clearly that Dionysodorus proved his result by the sameprocedure as that employed by Archimedes in the Method andin the book On Conoids and Spheroids and\indeed the proofon Archimedean lines is not difficult.Before passing to the mathematicians who are identifiedwith the discovery and development of trigonometry,it willbe convenient, I think, to dispose of two more mathematiciansbelonging to the last century B.C., although this involvesa slight departure. from chronological order ;I mean Posidoniusand Geminus.POSIDONIUS, a Stoic, the teacher of Cicero, is known asPosidonius of Apamea (where, he was born) or of Rhodes(where he taught) his date may be taken as approximately;135-51 iu_'. In pure mathematics he is mainly quoted as theauthor of certain definitions, or for views on technical terms,e.g.'theorem' and 'problem', and subjects belonging to elementarygeometry. More important were his contributionsto mathematical geography and astronomy. He gave hisgreat work on geography the title On the Ocean, using theword which had always had such a fascination for the Greeks ;its contents are, known to us through the copious quotationsfrom it in Strabo; it dealt with physical as well as mathematicalgeography, the zones, the tides and their connexionwith the moon, ethnography and all sorts of observations madeduring extensive travels. His astronomical book bore thetitle Meteoroloyica or irepl /Lierea>pa>i>, -and, while Geminuswrote a commentary on or exposition of this work, we mayfrom Posidonius inassign to it a number of views quoted

220 SUCCESSORS OF THE GREAT GEOMETERSCleomedes's work De motu circulars corporum caelestium.Posidonius also wrote a separate tract on the size of the sun.The two things which are sufficiently important to deservemention here are (1) Posidonius's measurement of the circumferenceof the earth, his (2) hypothesis as to the distance andsize of the sun.(1) He estimated the circumference of the earth in thisway. He assumed (according to Cleomedes ] ) that, whereasthe star Canopus, invisible in Greece, was just seen to graze thehorizon at Rhodes, rising and setting again immediately, the*meridian altitude of the same star at Alexandria was a fourthpart of a sign, that is, one forty-eighth partof the zodiaccircle' (= 7^); and he observed that the distance betweenthe two places (supposed to lie on the same meridian)*wasconsidered to be 5,000 stades'. The circumference of theearth was thus made out to be 240,000 stades. Unfortunatelythe estimate of the difference of latitude, 7^, was very farfrom correct, the true difference being 5^ only moreover;the estimate of 5,000 stades for the distance was incorrect,by mariners,being only the maximum estimate put upon itwhile some putit at 4.000 and Eratosthenes, by observationsof the shadows of gnomons, found it to be 3,750 studes only.'Strabo, on the other hand, says that Posidonius favoured thelatest of the measurements which gave the smallest dimensionsto the earth, namely about 180,000 stades 1 2.Thisevidently 48 times 3,750, so that Posidonius combined Eratosthenes'sfigureof 3,750 stades with the incorrect estimateof 7| for the difference of latitude, although Eratosthenespresumably obtained the figure of 3,750 stades from his ownestimate (250,000 or 252,000) of the circumference of the earthcombined with an estimate of the difference of latitude whichwas about 5f and therefore near the truth.(2) Cleomedes rj tells us that Posidonius supposed the circlein which the sun apparently moves round the earth to be10,000 times the size of a circular section of the earth throughits centre, and that with this assumption he combined theis1Cleomedes, De motu circular},i. 10, pp. 92-4.2Strabo, ii. c. 95.8Cleomedes, op. cit. ii. 1, pp. 144-6, p. 98. 1-5.

POSIDONIUS 221statement of Eratosthenes (based apparently upon hearsay)that at Syene, which is under the summer tropic, andthroughout a circle round it of 300 stades in diameter, theupright gnomon throws no shadow at noon. It follows fromthis that the diameter of the sun occupies a portion of thesun's circle 3,000,000 stades in length ; in other words, thediameter of the sun is 3,000,000 stades. The assumption thatthe sun's circle is10,000 times as large as a great circle of theearth was presumably taken from Archimedes, who had provedin the tfaitd-reckoiwr that the diameter of the sun's orbit isle#8 than 10,000 times that of the earth; Posidonius in facttook the maximum value to be the true value ;but his estimateof the sun's size is far nearer the truth than the estimatesof Aristarchus, Hipparchus, and Ptolemy. Expressed in termsof the mean diameter of the earth, the estimates of theseastronomers give for the diameter of the sun the figures 6|,12^, and 5J respectively; Posidonius's estimate gives 39J, thetrue figure being 108-9.In elementary geometry Posidonius is credited by Procluswith certain definitions. He defined 'figure' as 'confininglimit' (ntpas ) }mid 'parallels' as 'those lines which,being in one plane, neither converge nor diverge, but have allthe perpendiculars equal which are drawn from the points ofone line to the other'.- (Both these definitions are includedin the Dejinitio'HN of Heron.) He also distinguished sevenspecies of quadrilaterals, and had views on the distinctionbetween theorem and problem. Another indication of hisinterest in the fundamentals of elementary geometryis thefact 3 that he wrote a separate work in refutation of theEpicurean Zeno of Sidon, who had objected to the very beginningsof the Elements on the ground that they contained unprovedassumptions. Thus, said Zeno, even Eucl.1. 1 requires itto be admitted that ftwo straight lines cannot have a commonsegment ' ; and, as regards the ' proof ' of this fact deducedfrom the bisection of a circle by its diameter, he would objectthat it has to be assumed that two arcs of circles cannot havea common part.Zeno argued generally that, even if weadmit the fundamental principles of geometry, the deductions1Proclus on Eucl. I, p. 143. 8.376., pp. 199. 14-200. 3.2/&., p. 176. 6-10.

222 SUCCESSORS OF THE GREAT GEOMETERSfrom them cannot be proved without the admission of somethingelseas well which has not been included in the saidprinciples, and he intended by means of these criticisms todestroy the whole of geometry. 1We can understand, therefore,that the tract of Posidonius was a serious work.A definition of the centre of gravity by one 'Posidonius aStoic 'isquoted in Heron's Mechanics, but, as the writer goeson to say that Archimedes introduced a further distinction, wemay fairly assume that the Posidonius in isquestion notPosidonius of Rhodes, but another, perhaps Posidonius ofAlexandria, a pupil of Zeno of Cittium in the third centuryB.C.We now come to GEMINUS, a very important authority onmany questions belonging to the history of mathematics, as isshown by the numerous quotations from him in Proclus'sCommentai-y on Euclid, Book I. His date and birthplace areuncertain, and the discussions on the subject now form a wholeliterature for which reference must be made to Manitius'sedition of the so-called Gemini elementa astronomiae (Teubner,1898) and the article 'Geminus' in Pauly-Wissowa's Real-Encyclopddie. The doubts begin with his name. Petau, whoincluded the treatise mentioned in his Umnologion (Paris,1630), took it to be the Latin Gemlnus. Manitius, the latesteditor, satisfied himself that it was Gernlnus, a Greek name,judging from the fa^t that it consistently appears with theproperispomenon accent in Greek (Ptfjiivos),while it is alsofound in inscriptions with the spelling re/*/oy; Manitiussuggests the derivation from yc//, as '/jyôy from spy, andMXeô/oy from d\$; he compares also the unmistakablyGreek names 'Ifcro/oy, KpaTivos. Now, however, we are told(by Tittel) that the name is, after all, the Latin GAnlnus,that T*IJUVO$ came to be so written through false analogywith i4A6|u/oy, &c., and that re[/i]ea/oy,if the readingiscorrect, is also wrongly formed on the model of 'Avrwcivos,'Aypnrirttva. The occurrence of a Latin name in a centreof Greek culture need not surprise us, since Romans settled insuch centres in large numbers during the last century B.C.Geminus, however, in spite of his name, was thoroughly Greek.1Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.

GEMINUS 223An upper limit for his date is furnished by the fact that hewrote a corhmentary on or exposition of Posidonius's workTTtpl /zerccopcoi/on the other hand, Alexander Aphrodisiensis;(about A.D. 210) quotes an important passage from an 'epitome*of this 6^177170-19 by Ocminus. The view most generallyaccepted is that he was a Stoic philosopher, born probablyin the island of Rhodes, and a pupil of Posidonius, and thathe wrote about 73-67 B.C.Of Geminus's works that which has most interest for usis a comprehensive work on mathematics. Proclus, thoughhe makes great use of it, does not mention its title, unlessindeed, in the passage where, after quoting from Geminus'a classification of lines which never meet, he says theseremarks I have selected from the i\oKa\taof Geminus', 1the word iXoKa\iais a title or an alternative title. Tappus,however, quotes a work of Geminus ' on the classification ofthe mathematics' (tv rS> irepl TTJS rS>v fjia()r)fjLaTa>i' ra^6o>y),while Eutocius quotes from 'the sixth book of the doctrine ofthe mathematics ' (iv r ZKTW rfjs rS>v /jLaOrjfJidTwv 0ci>pias).The former title?corresponds well enough to the long extracton the division of the mathematical sciences into arithmetic,geometry, mechanics, astronomy, optics, geodesy, canonic(musical harmony) and logistic which Proclus givesin hisfirst prologue, and also to the fragmentsAnonym i variae coUectioues published bycontained in theHultsch in hisedition of Heron; but it does not suit most of the otherpassages borrowed by Proclus. The correct title was mostprobably that given by Eutocius, The Doctrhw, or Tlieory,of the Mathematics', and Pappus probably refers to oneparticular section of the work, say the first Book. If theconclude fromsixth Book treated of conies, as we mayEutocius's reference, there must have been more Books tofollow; for Proclus has preserved us details about highercurves, which must have come later. If again Geminusfinished his work and wrote with the same fullness about theother branches of mathematics as he did about geometry,there must have been a considerable number of Booksaltogether. It seems to have been designed to give a completeview of the whole science of mathematics, and in fact1Proclus on Eucl. I, p. 177. 24.

224 SUCCESSORS OF THE GREAT GEOMETERSto have been a sort of encyclopaedia of the subject.Thequotations of Proclus from Geminus's work do not standalone; we have other collections of extracts, some more andsome less extensive, and showing varieties of tradition accordingto the channel through which they came down. Thescholia to Euclid's Elements, Book I, contain a considerablepart of the commentary on the Definitions of Book I, and arevaluable in that they give Geminus pure and simple, whereasProclus includes extracts from other authors. Extracts fromGeminus of considerable length are included in the Arabiccommentary by an-Nairizi (about A.D. 900) who got themthrough the medium of Greek commentaries on Euclid,especially that of Simplicius. It does not appear to bedoubted any longer that 'Aganis' in an-NahizI is reallyGeminus ;this is inferred from the close agreement betweenan-NairizI's quotations from c Aganis' and the correspondingpassages in Proclus; the caused difficulty by the factthat Simplicius calls Aganis 'socius nostcr' is met by thesuggestion that the particular word socius is either theresult of the double translation from the Greek or meansnothing more, in the mouth of Simplicius, than ' colleague 'in the sense of a worker in the same field, or ' authorityA '.few extracts again are included in the Aitonymi variaecollectiones in Hultsch's Heron. Nos. 5-14 give definitions ofgeometry, logistic, geodesy and their subject-matter, remarkson bodies as continuous magnitudes, the three dimensions asprinciples of geometry, the purpose of geometry, and lastly'*on optics, with its subdivisions, optics proper, Catoptriea andoypa0i/c?j, scene-painting (a sort of perspective), with somefundamental principles of optics, e.g. that all light travelsalong straight lines (which are broken in the cases of reflectionand refraction), and the division between optics and naturalphilosophy (the theory of light), it being the province of thelatter to investigate (whatis a matter of indifference to optics)whether visual (1) rays issue from the eye, (2) images proceedfrom the object and impinge on the eye, or (3) the interveningair is aligned or compacted with the beam-like breath oremanation from the eye.Nos. 80-6 again in the same collection give the Peripateticexplanation of the name mathematics, adding that the term

GEMINUS 225was applied by the early Pythagoreans more particularlyto geometry and arithmetic, sciences which deal with the pure,the eternal and the unchangeable, but was extended by laterwriters to cover what we call ' mixed ' or applied mathematics,which, though theoretical, has to do with sensible objects, e.g.astronomy and optics. Other extracts from Geminus are foundin extant manuscripts in connexion with Damianus's treatiseon optics (published by R. Schone, Berlin, 1897). The defini-and geometry also appear, but with decidedtions of logisticdifferences, in the scholia to Plato's (-harmtdes 165 K.Lastly,isolated extracts appear in Eutocius, (1) a remark reproducedin the commentary on Archimedes's Plane Equilibriums tothe effect that Archimedes in that work gave the name ofpostulates to what are really axioms, (2) the statement thatbefore Apollonius's time the conies were produced by cuttingdifferent cones (right-angled, acute-angled, and obtuse-angled)by sections perpendicular in each case to a generator. 1The object of Geminus's work was evidently the examinationof the first principles, the logical building up of mathematicson the basis of those admitted principles,and thedefence of the whole structure against the criticisms ofthe enemies of the science?, the Epicureans and Sceptics, someof whom questioned the unproved principles, and others thelogical validity of the deductions from them. Thus ingeometry Geminus jlealt first with the principles or hypotheses(dpxai, viro0t(ri$) and then with the logical deductions, thetheorems and problems (ra fiera ray dp\d$). The distinctionis between the things which must be taken for granted butarc incapable of proof and the things which must not beassumed but are matter for demonstration. The principlesconsisting of definitions, postulates, and axioms, Geminussubjected them severally to a critical examination from thispoint of view, distinguishing carefully between postulates andaxioms, and discussing the legitimacy or otherwise of thoseformulated by Euclid in each class. In his notes on the definitionsGeminus treated them historically, giving .the variousalternative definitions which had been suggested for eachfundamental concept such as ' line ','surface ', figure '/body','angle ', &c., and frequently adding instructive classifications*1Eutocius, COMM. on ApoUoniuis Conies, ad itnf.

228 SUCCESSORS OF THE GREAT GEOMETERSof the different species of the thingdefined. Thus in thecase of 'lines' (which include curves) he distinguishes, first,the composite (e.g.a broken line forming an angle) and theincomposite. The incomposite are subdivided into those*forming a 'figure (crx^^ôTrotova-ai) or determinate (e.g.circle, ellipse, cissoid) and those not forming a figure, indeterminateand extending without limit (e.g. straight lino,parabola, hyperbola, conchoid). In a second classificationincomposite lines are divided into '(1) simple namely the circle',and straight line, the one ' making a figure the other extendingwithout ' '',limit, and (2)mixed '. Mixed 'lines again are'divided into (a)lines in planes ', one kind being a line meetingitself (e.g. the cissoid) and another a line extendingwithout limit, and (6)'lines on solids', subdivided into linesformed by sections (e.g. conic sections, spiric curves) and'lines round solids'(e.g.a helix round a cylinder, sphere, orcone, the first of which is uniform, homoeomeric, alike in allits parts, while the others are non-uniform). Geminus gavea corresponding division of surfaces into simple and mixed,the former being plane surfaces and spheres, while examplesof the latter are the tore or anchor-ring (though formed byabout an axis) and the conicoids ofthe revolution of a circlerevolution (the right-angled conoid, the obtuse-angled conoid,and the two spheroids, formed by the revolution of a parabola,a hyperbola, and an ellipse respectively about theiraxes). He observes that, while there are three homoeomericor uniform 'lines' (the straight line, the circle, and thecylindrical helix), there are only two homoeomeric surfaces,the plane and the sphere. Other classifications are those of'angles ' (according to the nature of the two lines or curveswhich form them) and of figures and plane figures.When Proclus gives definitions, &c., by Posidonius,it isevident that he obtained them from Gerninus's work. Suchare Posidonius's definitions of ''figure and ' parallels and his',division of quadrilaterals into seven kinds. We may assumefurther that, even where Geminus did not mention the nameof Posidonius, he was, at all events so far as the philosophy ofmathematics was concerned, expressingmainly those of his master.views which were

GEMINUS 227Attempt to prove theParallel-Postulate.Geminus devoted much attention to the distinction betweenpostulates and axioms, giving the views of earlier philosophersand mathematicians (Aristotle, Archimedes, Euclid,Apollonius, the Stoics) on the subject as well as his own. Itwas important in view of the attacks of the Epicureans andSceptics on mathematics, for (as Geminus says) it is as futileto attempt to prove the indemonstrable (as Apollonius didwhen he tried to prove the axioms) as it is incorrect to assume'what really requires proof, as Euclid did in the fourth postulate[that all right angles are equal] and in the fifth postulate[the parallel-postulate] VThe fifth postulate was the special stumbling-block.Geminus observed that the converse is actually proved byEuclid in I. 17; also that it is conclusively proved that anangle equal to a right angle is not necessarily itself a rightangle (e.g. the ' angle ' between the, circumferences of two semicircleson two equal straight lines with a common extremityand at right angles to one another) we cannot therefore admit;that the converses are incapable of demonstration. 2And'we have learned from the very pioneers of this science not tohave, regard to mere plausible imaginings when it is a questionof the reasonings to be included inour geometricaldoctrine. As Aristotle says, it is as justifiable toîsk scientificproofs from a rhetorician as to accept mere plausibilitiesfrom a geometer. So in this case . . (that of the parallelpostulate)the fact that, when the right angles are lessened, thestraight lines converge is true and necessary but the statementthat, since they converge more and more as they are;produced, they will sometime meet is plausible but not necessary,in the absence of some argument showing that this istrue in the case of straight lines. For the fact that some linesexist which approach indefinitely but yet remain non-secant(acrtf/zTTTooro*), although it seems improbable and paradoxical,is nevertheless true and fully ascertained with reference toother species of lines [the hyperbola and itsasymptote andthe conchoid and its asymptote, as Geminus says elsewhere].May not then the same thing be possible in the case of1Proclus on Eucl. I, pp. 178-82. 4> 183. 14-184. 10.2/&., pp. 183. 26-184. 5.Q 2

228 SUCCESSORS OF THE GREAT GEOMETERSstraight lines which happens in the case of the lines referredto ? Indeed, until the statement in the postulateis clinchedby proof, the facts shown in the case of the other lines maydirect our imagination the opposite way. And, though thecontroversialarguments against the meeting of the straightlines should contain much that is issurprising, there not allthe more reason why we should expel from our body ofdoctrine this merely plausible and unreasoned (hypothesis) ?It is clear from this' that we must seek a proof of the presenttheorem, and that it is alien to the special character ofpostulates/ lMuch of this might have been written by a moderngeometer. Geminus's attempted remedy was to substitutea definition of parallels like that of Posidonius, based on thenotion of equidistanee. An-Naiiizi gives the definition asfollows: 'Parallel straight lines are straight lines situated inthe same plane and such that the distance between them, ifthey are produced without limit in both directions at the sametime, is everywhere the same', to which Geminus adds thestatement that the said distance is the shortest straight linethat can be drawn between them. Starting from this,Geminus proved to his own satisfaction the propositions ofEuclid regarding parallels and finally the parallel-postulate.He firstgave the propositions (1) that the 'distance ' betweenthe two lines as Defined is perpendicular to both, and (2) that,if a straight line is perpendicular to each of two straight linesand meets both, the two straight lines are parallel, and the'distance' is the intercept on the perpendicular (proved byreductio ad absurdum). Next comp (3) Euclid's propositionsI. 27, 28 that, if two lines are parallel, the alternate anglesmade by any transversal are equal, &c. (easily proved bydrawing the two equal 'distances' through the points ofintersection with the transversal), and (4) Eucl. I. 29, the converseof I. 28, which is proved by reductio (id absurdum, bymeans of (2) and (3). Geminus still needs Eucl. I. 30, 31(about parallels) and I. 33, 34 (the first two propositionsrelating to parallelograms) for his final proof of the postulate,which is to the followingeffect.Let AS, CD be two straight lines met by the straight line1Proclus on Eucl. I, pp. 192. 5-193. 3.

GEMINUS 229EF, and let the interior angles BEF, EFD be together lessthan two right angles.Take any point 11 on FD and draw HK parallel to ABThen, if we bisect EF at L, LFnt, M, MFmeeting EF in K.at X, and so on, we shall at last have a length, as FX, lessUthan FK. Draw FG, XOP parallel to AB. Produce FO to Q,Tand let /S f be the middle point of FQ and Ji the middle point ofFti. Draw through /f, #, Q respectively the* straight lines7UW, jSTZ7, QK parallelto EF. Join J/7?, Z6 f and producethem to 7 T ,K. Produce FG to T.Then, in the triangles JWiY, jffOP, two angles are equalrespectively, the vertically opposite angles FOX, HOP andtlu> alternate angles XFO, PRO and ^ = OR T therefore; ;7w = j#y.And AW, 7^G in the parallelogram FXPG are equal ;there-KG = 2 A T .Y = / T J/ (whence J//i is parallel to FG or AB)foreSimilarly we prove that SU=>2FM = FL, and L8 isparallelto FG or 4U.Lastly, by thetriangles FL8, QVti, in which the sides FM,ti(J are equal and two angles are respectively equal, Q V =Since then EL, QV are equal and parallel, so are EQ tand (says Gominus)it follows that AB passes through Q.LV,

230 SUCCESSORS OF THE GREAT GEOMETERSWhat follows is actually that both EQ and AB (or EB)are parallel to LV, and Geminus assumes that EQ, ABare coincident (in other words, that through a given pointonly one parallel can be drawn to a given straight line, anassumption known as Playfair's Axiom, though it is actuallystated in Proclus on Eucl. I. 31).The proof therefore, apparently ingenious as it is, breaksdown. Indeed the method is unsound from the beginning,since (as Saccheri pointed out), before even the definition ofparallels by Geminus can be used, it has to be proved thatthe geometrical locus of points equidistant from a straight'line is a straight line ',and this cannot be proved without apostulate. But the attempt is interesting as the first whichhas come down to us, although there must have been manyothers by geometers earlier than Geminus.Coming now to the things which follow from the principles(ra perk ray />X^y)> we gather from Proclus that Geminuscarefully discussed such generalities as the nature of elements,the different views which had been' held of the distinctionbetween theorems and problems, the nature and significanceof StopiarfjioL (conditions and limits of possibility), the meaningof * porism ' in the sense in which Euclid used the word in hisPorisms as distinct from its other meaning of c corollary ', thedifferent sorts of problems and theorems, the two varieties ofconverses (complete and partial), topical or focus-theorems,with the classification of loci. He discussed also philosophicalquestions, e.g. the question whether a line is made up ofindivisible parts (e a/ze/D

GEMINUS 231and make equal angles with it, the straight lines are equal.1As Apollonius wrote on the cylindrical helix and proved thefact of its uniformity, we may fairly assume that Geminuswas here drawing upon Apollonius.Enough has been said to show how invaluable a source ofinformation Geminus's work furnished to Proclus and allwriters on the history of mathematics who had access to it.In astronomy we know that Geminus wrote an7777/0-* 9 ofPosidonius's work, the Meteorologica or ?rep2 /ireo>pa>j>.Thisisthe source of the famous extract made from Geminus byAlexander Aphrodisiensis, and reproduced by Simplicius inhis commentary on the Physics of Aristotle, 2 on which Schiaparellirelied in his attempt to show that it was Heraclides ofPontus, not Aristarchus of Sanaos, who first put forward theheliocentric hypothesis. The extract is on the distinctionbetween physical and astronomical inquiry as applied to theheavens. It is the business of the physicist to consider thesubstance of the heaven and stars, their force and quality,their coming into being and decay, and lie is in a position toprove the facts about their size, shape, and arrangement;astronomy, on the other hand, ignores the physical side,proving the arrangement of the heavenly bodies by considerationsbased on the view that the heaven is a real 007*09, and,when it tells us of the shapes, sizes and distances of the earth,sun and moon, of eclipses and conjunctions, and of the qualityand extent of the movements of the heavenly bodies, it isconnected with the mathematical investigation of quantity,size, form, or shape, and uses arithmetic and geometry toprove its conclusions. Astronomy deals, jiot with causes, butwith facts; hence it often proceeds by hypotheses, statingcertain expedients by which the phenomena may be saved.For example, why do the sun, the moon and the planetsappear to move irregularly ? To explain the observed factswe may assume, for instance, that the orbits are eccentriccircles or that the stars describe epicycles on a carryingcircle; and then we have to go farther and examine otherways in which it is possible for the phenomena to be brought'about. Hence we actually find a certain person [Heraclides1Proclus on Eucl. I, pp. 112. 22-113. 3, p. 251. 3-11.2Simpl. in Phy*., pp. 291-2, ed. Diels.

282 SUCCESSOES OF THE GREAT GEOMETERSof Pontus] coming forward and saying that, even on theassumption that the earth moves in a certain way, whilethe sun is in a certain way at red, the apparent irregularitywith reference to the sun may be saved! Philological considerationsas well as the other notices which we possessabout Heraclides make it practically certain that ' Heraclidesof Pontus' is an interpolation and that Geminus said msimply, 'a certain person', without any name, though hedoubtless meant Aristarchus of Samos. 1Simplicius says that Alexander quoted this extract fromthe epitome of thegrjyrjy Geminus,which has come down to us under the title Tepivov e/crayooyr;e/s ra $aiv6p.tva? What, if any,is the relation of this workto the exposition of Posidonius's Meteorologies or the epitomeof it just mentioned ? One view is that the original Isayoyeof Geminus and the 17777079 of Posidonius were one and thesame work, though the Isagoye as we have it is not byGeminus, but by an unknown compiler. The objections tothis are, first, that it does not contain the extract given bySimplicius, which would have come in usefully at the beginningof an Introduction to Astronomy, nor the other extractgiven by Alexander from Geminus and relating to the rainbowwhich seems likewise to have come from the1Of. Aristarchus of Samos, pp. 275-83.2 Edited by Manitius (Teubner, 1898).3 Alex. Aphr. on Aristotle's Meteorologica, iii. 4, 9 (Ideler, ii, p. 128;p. 152. 10, Hayduck).

GEMINUS 233secondly, that it docs not anywhere mention the name ofPosidonius (not, perhaps, an insuperable objection) and,;thirdly, that there are views expressedin it which are notthose held by Posidonius but contrary to them. Again, thewriter knows how to give a sound judgement as betweendivergent views, writes in good style on the whole, and canhardly have been the mere compiler of extracts from Posidoniuswhich the view in question assumes him to be. Itseems in any case safer to assume that the Isagoge and thewere grjyrj(ri$ separate works. At the same time, the Isagoge,as we have it, contains errors which we cannot attribute toGemirms. The choice, therefore, seems to lie between twoalternatives : either the book isby Geminus in the main, buthas in the course of centuries suffered deterioration by interpolations,mistakes of copyists, and so on, or it is a compilationof extracts from an original Isagoge by Geminus with foreignand inferior elements introduced either by the compiler himselfor by other prentice hands.The result is a tolerable elementarytreatise ilhitable for teaching purposes and containingthe most important doctrines of Greek astronomy representedfrom the standpoint of 1Hipparchus. Chapter treats of thezodiac, the solar year, the irregularity of the sun's motion,which is explained by the eccentric position* of the sun's orbitrelatively to tho zodiac, the order and the periods of revolutionof the planets and the moon. In 23 we are told that allthe fixed stars do not lie on one spherical surface, but someare farther away than others a doctrine due to the Stoics.Chapter 2, again, treats of the twelve signs of the zodiac,chapter 3 of the constellations, chapter 4 of the axis ofthe universe and the poles, chapter 5 of the circles on thesphere (the equator and the parallel circles, arctic, summertropical,winter-tropical, antarctic, the colure-circles, the zodiacor ecliptic, the horizon, the meridian, and the Milky Way),chapter 6 of Day and Night, their relative lengths in differentlatitudes, their lengthening and shortening, chapter 7 ofthe times which the twelve signs take to rise.Chapter 8is a clear, interesting and valuable chapter on the calendar,the length of months and years and the various cycles, theoctaeteris, the 16-years and 160-years cycles, the 19-yearscycle of Euctemon (and Meton), and the cycle of Callippus

234 SUCCESSORS OF THE GREAT GEOMETERS(76 years). Chapter 9 deals with the moon's phases, chapters10, 11 with eclipses of the sun and moon, chapter12 with theproblem of accounting for the motions of the sun, moon andplanets, chapter 13 with Risings and Settings and the varioustechnical terms connected therewith, chapter 14 with thecircles described by the fixed stars, chapters 15 and 16 withmathematical and physical geography, the zones, &c. (Geminusfollows Eratosthenes's evaluation of the circumference of theearth, not that of Posidonius). Chapter 17, on weather indications,denies the popular theory that changes of atmosphericconditions depend on the rising and setting of certain stars,and states that the predictions of weather (eTncrrj/jaoYa/.)incalendars (rrapaTrTyy/zara) are only derived from experienceand observation, and have no scientific value. Chapter 18 ison the eAiy//6y, the shortest period which contains an integralnumber of synodic months, of days, and of anomalistic revolutionsof the moon this ;periodis three times the Chaldaeanperiod of 223 lunations used for predicting eclipses.The endof the chapter deals with the maximum, mean, and minimummotion of the moon. The as a whole does notdailychaptercorrespond to the rest of the book it deals with more difficult;matters, and isthought by Manitius to be a fragment only ofa discussion to which the compiler did not feel himself equal.At the end of the work is a calendar (Parupeyma) giving thenumber of days taken by the sun to traverse each sign ofthe zodiac, the risings and settings of various stars and theweather indications noted by various astronomers, Democritus,Eudoxus, Dositheus, Euctemoii, Meton, Callippus this calendar;is unconnected with the rest of the book and the contentsare in several respects inconsistent with it, especially thedivision of the year into quarters which follows Callippusrather than Hipparchus. Hence it has been, since Boeckh'stime, generally considered not to be the work of Geminus.Tittel, however, suggests that it is not impossible that Geminusmay have reproduced an older Parapegma of Callippus.

XVISOME HANDBOOKSTHE descriptionof the handbook on the elements ofastronomy entitled the Introduction to the Phaenomenct andattributed to Geminus might properly have been reservedfor this chapter. It was, however, convenient to deal withGeminus in close connexion with Posidonius; for Geminuswrote an exposition of Posidonius's Meteorologica related to theoriginal work in such a way that Simplicius, in quoting a longpassage from an epitome of this work, could attribute thepassage to either Geminus or ' Posidonius in Geminus ' and it;is evident that, in other subjects too, Geminus drew from, andwas influenced by, Posidonius.The small work De motu circular i corpomm caelestium byCLEOMEDES (KXco/uôyy KVK\iKtj Ot&pia) in two Books is theproduction of a much less competent person, but is much morelargely based on Posidonius. This is proved by several referencesto Posidonius by name, but it is specially true of thevery long first chapter of Book II (nearly half of the Book)which seems for the most part to be copied bodily fromPosidonius, in accordance with the author's remark at theend of Book I that, in giving the refutation of the Epicureanassertion that the sun is just as large as it looks, namely onefoot in diameter, he will give so much as suffices for such anintroduction of the particular arguments used by 'certainauthors who have written whole treatises on this one topic(i.e. the size of the sun), among whom is Posidonius'. Theinterest of the book then lies mainly in what is quoted fromPosidonius ;its mathematical interest is almost nil.The date of Cleomedes is not certainly ascertained, but, ashe mentions no author later than Posidonius, it is permissibleto suppose, with Hultsch, that he wrote about the middle of

236 SOME HANDBOOKSthe first century B. c. As he seems to know nothingworks of Ptolemy, he can hardly, in any case, have livedof thelater than the beginning of the second centuryA. D.Book I begins with a chapter the object of which is toprove that' the universe, which has the shape of a sphere,is limited and surrounded by void extending without limit inall directions, and to refute objections to this view. Thenfollow chapters on the five parallel circles in the heaven andthe zones, habitable and uninhabitable (chap. 2) ;on themotion of the fixed stars and the independent (rrpoaiptTiKaL)movements of the planets including the sun and moon(chap. 3); on the zodiac and the effect of the sun's motion init (chap. 4) on the inclination of the axis of the universe and;its effects on the lengths of days and nights at different places(chap. 5); on the inequality in the rate of increase in thelengths of the days and nights according to the time of year,the different lengths of the seasons due to the motion of thesun in an eccentric circle, the difference between a day-andnightand an exact revolution of the universe owing to theseparate motion of the sun (chap. 6) on the habitable;regionsof the globe including Britain and the ' island of Thulo ',saidto have been visited by Pytheas, where, when the sun is inCancer and visible, the day is a month long and so on ; (chap. 7).Chap. 8 purports to prove that the universe is a sphere byproving first that the earth is a sphere, and then that the airabout it, and the ether about that, must necessarily make uplarger spheres. The earth isproved to be a vsphere by themethod of exclusion ;it is assumed that the only possibilitiesare that it is (a)flat and plane, or (6) hollow and deep, or(c) square, or (d) pyramidal, or (e) spherical, and, the first fourhypotheses being successively disposed of, only the fifthremains. Chap. 9 maintains that the earth is in the centre ofthe universe ; chap. 10, on the size of the earth, contains theinteresting reproduction of the details of the measurements ofthe earth by Posidonius and Eratosthenes respectively whichhave been given above in their proper places (p. 220, pp. 1 06-7) ;chap. 1 1 argues that the earth is in the relation of a point to,i. e. is negligible in size in comparison with, the universe oreven the sun's circle, but not the moon's circle (cf. p. 3 above).Book II, chap. 1, is evidently the piece de resistance, con-

CLEOMEDES 237sisting of an elaborate refutation of Epicurus and his followers,who held that the sun is just as large as it looks, and furtherasserted (according to Cleomedes) that the stars are lit up asthey rise and extinguished as they set. The chapter seems tobo almost wholly taken from Posidonius; it ends* with somepages of merely vulgar abuse, comparing Epicurus with Thersites,with more of the same sort. The value of the chapterlies in certain historical traditions mentioned in it, and in theaccount of Posidonius's speculation as to the size and distanceof the sun, which does, as a matter of fact, give results muchnearer the truth than those obtained by Aristarchus, Hipparchus,and Ptolemy. Cleomedes observes (1) that by means ofwater-clocks it is found that the apparent diameter of the sunisl/750tli of the sun's circle, and that this method ofmeasuring it is said to have been first invented by theEgyptians; (2) that Hipparchusis said to have found thatthe sun is 1,050 times the size of the earth, though, as regardsthis, we have the better authority of Adrastus (in Theon ofSmyrna) and of Chalcidius, according to whom Hipparchusmade the sun nearly 1,880 times the size of the earth (bothfigures refer of course to the solid content). We have alreadydescribed Posidonius's method of arriving at the size anddistance of the sun (pp. 220-1). After he has given this, Cleomedes,apparently deserting his guide, adds a calculation ofhis own relating to the sizes and distances of the moon andthe sun which shows how littlehe was capable of any scientificinquiry. 1 Chap. 2 purports to provethat the sun is1He says (pp. 146. 17-148. 27) that in an eclipse the breadth of theearth's shadow is stated to be two moon-breadths ; hence, he says, itseems credible (mQavov) that the earth is twice the size of the moon (thispractically assumes that the breadth of the earth's shadow is equal tothe diameter of the earth, or that the cone of the earth's shadow isa cylinder!). Since then the circumference of the earth, according toKratostherios, is 250,000 stades, and its diameter therefore ' more than80,000 ' (he evidently takes = TT 8), the diameter of the moon will be40,000 stades. Now, the moon's circle being 750 times the moon'sdiameter, the radius of the moon's circle, i.e. the distance of the moonfrom the earth, will be Jtli of this (i.e. TT = 3) or 125 moon-diameters;therefore the moon's distance is 5,000,000 stades (which is much toogreat). Again, since the moon traverses its orbit 13 times to the sun'sonce, he assumes that the sun's orbit is 13 times as large as the moon's,and consequently that the diameter of the sun is 13 times that of themoon, or 520,000 stades and its distance 13 times 5,000,000 or 65,000,000stades !

238 SOME HANDBOOKSlarger than the earth ;the size of the moon and the starsand the remaining chapters deal with(chap. 3),the illuminatipnof the moon by the sun (chap. 4), the phases of the moon andits conjunctions with the sun (chap. 5), the eclipsesof themoon (chap. 6),the maximum deviation in latitude of the fivefor Marsplanets (given as 5 for Venus, 4 for Mercury, 2|and Jupiter, 1 for Saturn), the maximum elongations ofMercury and Venus from the sun (20 and 50 respectively),and the synodic periods of the planets (Mercury 116 days,Venus 584 days, Mars 780 days, Jupiter 398 days, Saturn378 days) (chap. 7).There is only one other item of sufficient interest to bementioned here. In Book II, chap. 6, Cleomedes mentionsthat there were stories of extraordinary eclipses which ' themore ancient of the mathematicians had vainly tried to explain';the supposed c paradoxical' case was that in which,while the sun seems to be still above the horizon, the eclipsedmoon rises in the east. The passage has been cited above(vol. i, pp. 6-7), where I have also shown that Cleomedes himselfgives the true explanation of .the phenomenon, namelythat it is due to atmospheric refraction.The first and second centurfes of the Christian era sawa continuation of the work of writing manuals or introductionsto the different mathematical subjects. About A. D. 100came NICOMACHUS, who wrote an Introduction to Arithmeticand an Introduction toHarmony] if we may judge by aremark of his own, 1 he would appear to have written an introductionto geometry also. The Arithmetical Introduction hasbeen sufficiently described above (vol. i, pp. 97-112).There is yet another handbook which needs to be mentionedseparately, although we have had occasion to quote from itseveral times already* This is the book by THEON OF SMYRNAwhich goes by the title Expositio rerum mathematicarum adlegendum Platonem utilium. There are two main divisionsof this work, contained in two Venice manuscripts respectively.The first was edited by Bullialdus (Paris, 1644), thesecond by .T. H. Martin (Paris, 1849); the whole has been1Nicom. Arith. ii. 6. 1.

THEON OP SMYRNA 239edited by E. Killer (Teubner, 1878) and finally,with a Frenchtranslation, by J. Dupuis (Paris, 1892).Theon's date is approximately fixed by two considerations.He is clearly the person whom Theon of Alexandria called'the old Theoii', rov -rraXaiov Ion/a, 1 and there is no reasonto doubt that he is the ' Theoii the mathematician ' (6 fiaOrjfiaTtKos)who is credited by Ptolemy witfi four observationsof the planets Mercury and Venus made in A.D. 127, 129, 130and 132. 2 The latest writers whom Theoii himself mentionsare Thrasyllus, who lived in the reign of Tiberius, andAdrastus the Peripatetic, who belongs to the middle of thesecond century A.D. Thecm's work itself is a curious medley,valuable, not intrinsically, but for the numerous historicalnotices which it contains. The title, which claims that thebook contains things useful for the study of Plato, must notbe taken too seriously. It was no doubt an elementaryintroduction or vade-mecum for students of philosophy, butthere is little in it which has special reference to the mathematicalquestions raised in Plato. The connexion consistsmostly in the long proem quoting the views of Plato on theparamount importance of mathematics in the training ofthe philosopher, and the mutual relation of the five differentbranches, arithmetic, geometry, stereometry, astronomy andmusic. The want of care shown by Theon in the quotationsfrom particular dialogues of Plato prepares us for the patchworkcharacter of the whole book.In the first chapter he promises to give the mathematicaltheorems most necessary for the student of Plato to know,in arithmetic, music, and geometry, with its application tostereometry and astronomy/* But the promise is by no meanskept as regards geometry and stereometry indeed, in a:later passage Theoii seems to excuse himself from includingtheoretical geometry in his plan, on the ground that all thosewho are likely to read his work or the writings of Plato maybe assumed to have gone through an elementary course oftheoretical geometry. 4 But he writes at length on figured1Theon of Alexandria, Comm. on Ptolemy's Syntaxis, Basel edition,pp. 390, 395, 396.2Ptolemy, Syntaxis, ix. 9, x. 1, 2.8Theon of Smyrna, ed. Hiller, p. 1. 10-17.

240 SOME HANDBOOKSnumbers, plane and solid, which are of course analogous tothe corresponding geometrical figures,and he may have consideredthat he was in this way sufficiently fulfilling hispromise with regard to geometry and stereometry. Certaingeometrical definitions, of point, line, straight line, the threedimensions, rectilinear plane and solid figures, especiallyparallelograms and parallelepipedal figures including cubes,pi intitules (square bricks) and SoKities (beams), and scalenein thefigures with sides unequal every way ( = P&pfoKoiclassification of solid numbers), are dragged in later (chaps.53-5 of the section on music) 1 in the middle of the discussionof proportions and means; if this passageis not an interpolation,it confirms the supposition that Theon included inhis work only this limited amount of geometry and stereometry.Section I is on Arithmetic in the same sense as Nicomachus'sIntroduction. At the beginning Theon observes that arithmeticwill be followed by music. Of music in its threeaspects, music in instruments (*v opydvois), music in numbers,i.e. musical intervals expressed in numbers or pure theoreticalmusic, and the music or harmony in the universe, the firstkind (instrumental music) is not exactly essential, but the othertwo must be discussed 2immediately after arithmetic. The contentsof the arithmetical section have been sufficiently indicatedin the chapter on Pythagorean arithmetic (vol. i, pp. 112-13) ;it deals with the classification of numbers, odd, even, andtheir subdivisions, prime numbers, composite numbers withequal or unequal factors, plane numbers subdivided intosquare, oblong, triangular and polygonal numbers, with theirrespective '.gnomons' and their properties as the sum ofsuccessive terms of arithmetical progressions beginning with1 as the first term, circular and spherical numbers, solid numberswith three factors, pyramidal numbers and truncatedpyramidal numbers, perfect numbers with their correlatives,the over-perfect and the deficient; this is practically whatwe find in Nicomachus. But the special value of Theoii'sexposition lies in the fact that it contains an account of thefamous ' side- ' and c diameter- ' numbers of the Pythagoreans. 31Theon of Smyrna, ed. Hiller, pp. 111-13. 2/&., pp. 16. 24-17. 11.3/&., pp. 42. 10-45. 9. Of. vol. i, pp. 91-3.

THEON OF SMYRNA 241In the Section on Music Theon says he will first speak ofthe two kinds of music, the audible or instrumental, and theintelligible or theoretical subsisting in numbers, after whichhe promises to deal lastly with ratio as predicable of mathematicalentities in general and the ratio constituting theharmony in the universe, 'not scrupling to set out once againthe things discovered by our predecessors, just as we havegiven the things handed down in former times by the Pythagoreans,with a view to making them better known, withoutourselves claiming to have discovered any of them'. 1 Thenfollows a discussion of audible music, the intervals whichgive harmonies, &c., including substantial quotations fromThrasyllus and Adrastus, and references to views of Aristoxenus,Hippasus, Archytas, Eudoxus and Plato. Withchap. 17 (p. 72) begins the account of the 'harmony innumbers', which turns into a general discussion of ratios,proportions and means, with more quotations from Plato,Eratosthenes and Thrasyllus, followed by Thrasyllus's divisiocanonis, chaps. 35, 36 (pp. 87-93). After a promise to applythe latter division to the sphere of the universe, Theonpurports to return to the subject of proportion and means.This, however, does not occur tillchap. 50 (p. 106), theintervening chapters being taken up with a discussion ofthe 5e*ay and rerpaKri/y (with eleven applications of thelatter) and the mystic or curious properties of the numbersfrom 2 to 1;here we have a part of the theoloyumeiia ofarithmetic. The discussion of proportions and the differentkinds of means after Eratosthenes and Adrastus is againinterrupted by the insertion of the geometrical definitionsalready referred to (chaps. 53-5, pp. 111-13), after whichTheon resumes the question of means for * 'more precisetreatment.The Section on Astronomy begins on p.120 of Killer'sedition. Here again Theon ismainly dependent uonAdrastus, from whom he makes long quotations. Thus, onthe sphericity of the earth, he says that for the necessaryconspectus of the argumentsit will be sufficient torefer to the grounds stated summarily by Adrastus. Inexplaining (p. 124) that the unevennesses in the surface of1Theon of Smyrna, ed. Hillcr, pp. 46. 20-47. 14.1621.2 R

242 SOME HANDBOOKSthe earth, represented e.g. by mou'ntains, are negligible incomparison with the size of the whole, he quotes Eratosthenesand Dicaearchus as claiming to have discovered that theperpendicular height of the highest mountain above the normallevel of the land is no more than 10 stades ;and to obtain thediameter of the earth he uses Eratosthenes's figure of approximately252,000 stades for the circumference of the earth,which, with the Archimedean value of ^- for TT, gives adiameter of about 80,182 stades. The principal astronomicalcircles in the heaven are next described (chaps. 5-12, pp.129-35) ;then (chap. 12) the assumed maximum deviations inlatitude are given, that of the sun being put at 1, that of themoon and Venus at 12, and those of the planets Mercury,Mars, Jupiter and Saturn at 8, 5, 5 and 3 respectively; theobliquity of the ecliptic is given as the side of a regular polygonof 15 sides described in a circle, i.e. as 24 (chap. 23, p. 151).Next the order of the orbits of the sun, moon and planets is explained(the system is of course geocentric) ; we are told (p. 13 8)that ' some of the Pythagoreans ' made the order (reckoningoutwards from the earth) to be moon, Mercury, Venus, sun,Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put thesun next to the moon, and the mathematicians, agreeing withEratosthenes in this, differed only in the order in which theyplaced Venus and Mercury after the sun, some putting Mercurynext and some Venus (p. 143).The order adopted by 4 someof the Pythagoreans ' is the Chaldacan order, which was notfollowed by any Greek before Diogenes of Babylon (secondcentury B.C.); 'some of the Pythagoreans' are therefore thelater Pythagoreans (of whom Nicomachus was one) the other;order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, wasthat of Plato and the early Pythagoreans. In chap. 15with the sun and moon and(p. 138sq.) Theon quotes verses of Alexander 'the Aetoliun'(not really the ' Aetoliaii ',but Alexander of Ephesus, a contemporaryof Cicero, or possibly Alexander of Miletus, asChalcidius calls him) assigning to each of the planets (includingthe earth, though stationary)the sphere of the fixed stars one note, the intervals betweenthe notes being so arranged as to bring the nine into anoctave, whereas with Eratosthenes and Plato the earth wasexcluded, and the eight notes of the octachord were assigned

THEON OF SMYRNA 243to the seven heavenly bodies and the sphere of the fixed stars.The whole of this passage (chaps. 15 to 16, pp. 138-47)is nodoubt intended as the 'promised account of the harmony inthe universe ', although at the veryimplies that this has still to be explainedend of the work Theonon the basis ofThrasyllus's exposition combined with what he has alreadygiven himself.The next chapters deal with the forward movements, thestationary points, and the retrogradations, as they respectivelyappear to us, of the five planets, and the 'saving of the pheno-of eccentric circles andmena ' by the alternative hypothesesepicycles (chaps. 17-30, pp. 147-78). These hypotheses areexplained, and the identity of the motion produced by thetwo is shown by Adrastus in the case of the sun (chaps. 26, 27,pp. 166-72). The proofis introduced with the interestingremark that * Hipparchus says it is worthy of investigationby mathematicians why, on two hypotheses so different fromone another, that of eccentric circles and that of concentriccircles with epicycles, the same results appear to follow '. Itis not to be supposed that the proof of the identity could beother than easy to a mathematician like Hipparchus; theremark perhaps merely suggests that the two hypotheses werediscovered quite independently, and it was not till later thatthe effect was discovered to be the same, when of course thefact would seem to be curious and a mathematical proof wouldimmediately be sought. Another passage (p. 188) says thatHipparchus preferred the hypothesis of the epicycle, as beinghis own. If this means that Hipparchus claimed to havediscovered the epicycle-hypothesis,it must be a misapprehension;for Apollonius already understood the theory of epicyclesin all its generality. According to Theon, the epicyclehypothesisis more ' according to nature ' but it was; presumablypreferred because it was applicable to all the planets,whereas the eccentric-hypothesis, when originally suggested,applied only to the three superior planets ;in order to makeit apply to the inferior planets it is necessary to suppose thecircle described by the centre of the eccentric to be greaterthan the eccentric circle itself, which extension of the hypothesis,though known to Hipparchus, does not seem to haveoccurred to Apollonius.R 2

244 SOME HANDBOOKSWe next have (chap. 31, p. 178) an allusion to the systemsof Eudoxus, Callippus and Aristotle, and a description(p. 180 sq.) of a system in which the 'carrying' spheres(called ' 'hollow ')have between them solid spheres which bytheir own motion will roll (dve\igov moves) about the centre of the universe, andthat Anaximenes said that the moon has its light from thesun and (explained) how its eclipses come about' (Anaximenesis here apparently a mistake for Anaxagoras).1Chalcidius, Comm. on Timaeus, c. 110. Cf. Aristarcltus of Samos,pp. 256-8.

XVIITRIGONOMETRY: HIPPARCHUS, MENELAUS,PTOLEMYWE have seen that S^haeric, the geometry of the sphere,was very early studied, because it was required so soon asastronomy became mathematical with the ;Pythagoreans theword fyrfiaeric, applied to one of the subjects of the quadrivium,actually meant astronomy. The subject was so far advancedbefore Euclid's time that there was in existence a regulartextbook containing the principal propositions about greatand small circles on the sphere, from which both Autolycusand Euclid quoted the propositions as generally known.These propositions, with others of purely astronomical interest,were collected afterwards in a work entitled tiirtiaerica,in three Books, by THEODOSIUS.Suidas has a notice, 8. v. 0eo86

246 TRIGONOMETRYdosius was of Bithynia and not later in date than Vitruvius(say 20 B.C.); but the order in which Strabo gives thenames makes it not unlikely that he was contemporary withHipparchus, while the character of his Sphaerica suggests adate even earlier rather than later.Works by Theodosius.Two other works of Theodosius besides the Sphaerica,namely On habitations and On Days and Nights, seem tohave been included in the ' Little Astronomy ' (fiiKpos dcrrpovopovfjisvoS)sc. TOKOS). These two treatises need not detain uslong. They are extant in Greek (in the great MS. VaticanusGraecus 204 and others), but the Greek text has not apparentlyyet been published. In the first, Oti habitations, in 12propositions, Theodosius explains the different phenomena dueto the daily rotation of the earth, and the particular portionsof the whole system which are visible to inhabitants of thedifferent zones.In the second, On Days and Nights, containing13 and 19 propositions in the two Books respectively,Theodosius considers the arc of the ecliptic described by thesun each day, with a view to determining the conditions to besatisfied in order that the solstice may occur in the meridianat a given place, and in order that the day and the night mayreally be equal at the equinoxes; he shows also that thevariations in the day and night must recur exactly afteris commen-a certain time, if the length of the solar yearsurable with that of the day, while on the contrary assumptionthey will not recur so exactly.In addition to the works bearing on astronomy, Theodosiusis said l to have written a commentary, now lost, on the tyoSiovor Method of Archimedes (see above, pp. 27-34).Contents of the Sphaerica.We come now to the Sphaerica, which deserves a shortdescription from the point of view of this chapter. A textbookon the geometry o the sphere was wanted as a supplementto the Elements of Euclid. In the Elements themselves1Suidas, loc. cit.

THEODOSIUS'S SPHAERICA 247(Books XII and XIII) Euclid included no general propertiesof the sphere except the theorem proved in XII. 16-18, thatthe volumes of two spheres are in the triplicate ratio of theirdiameters ; apart from this, the sphere is only introduced inthe propositions about the regular solids, where it isprovedthat they are severally inscribable in a sphere, and it was doubtlesswith a view to his proofs of this property in each case thathe gave a new definition of a sphere as the figure described bythe revolution of a semicircle about its diameter, instead ofthe more usual definition (after the manner of the definitionof a circle) as the locus of all points (in spaceinstead of ina plane) which are equidistant from a fixed point (the centre).No doubt the exclusion of the geometry of the sphere fromthe Elements was due to the fact that it was regarded asbelonging to astronomy rather than pure geometry.Theodosius defines the sphere as ' a solid figure containedby one surface such that all the straight lines falling upon itfrom one point among those lying within the figure are equalto one another ',which is exactly Euclid's definition of a circlewith * solid ' inserted before * figure ' and ' surface ' substitutedfor c line '. The early part of the work is then generallydeveloped on the lines of Euclid's Book III on the circle.Any plane section of a sphere is a circle (Prop. 1). Thestraight line from the centre of the sphere to the centre ofa circular section is perpendicular to the plane of that section(1, For. 2 ;cf. 7, 23); thus a plane section serves for findingthe centre of the sphere just as a chord does for finding thatof a circle (Prop. 2). The propositions about tangent planes(3-5) and the relation between the sizes of circular sectionsand their distances from the centre (5, 6) correspond toEuclid III. 16-19 and 15; as the small circle corresponds toany chord, the great Circle (' greatest circle ' in Greek) correspondsto the diameter. The polesof a circular sectioncorrespond to the extremities of the diameter bisectinga chord of a circle at right angles (Props. 8-10). Greatcircles bisecting one another (Props. 11-12) correspond tochords which bisect one another (diameters), and great circlesbisecting small circles at right angles and passing throughtheir poles (Props. 13-15) correspond to diameters bisectingchords at right angles. The distance of any point of a great

248 TRIGONOMETRYcircle from its pole is equal to the side of a square inscribedin the great circle and conversely (Props. 16, 17).certain problems To find a :straight line equalNext cometo the diameterof any circular section or of the sphere itself (Props. 18, 19) ;to draw the great circle through any two given points onthe surface (Prop. 20); to find the pole of any given circularsection (Prop. 21). Prop. 22 applies Eucl. III. 3 to thesphere.Book II begins with a definition of circles on a spherewhich touch one another ; this happens * when the commonsection of the planes (of the circles) touches both circles '.Another series of propositions follows, corresponding againto propositions in Eucl., Book III, for the circle. Parallelcircularsections have the same poles, and conversely (Props.1, 2). Props. 3-5 relate to circles on the sphere touchingone another and therefore having their poles on a greatcircle which also passes through the point of contact (cf.Eucl. III. 11, [12] about circles touching one another). Ifa great circle touches a small circle, it also touches anothersmall circle equal and parallel to it (Props. 6, 7),and if agreat circle be obliquely inclined to another circular section,it touches each of two equal circles parallel to that section(Prop. If8).two circles on a sphere cut one another, thegreat circle drawn through their poles bisects the interceptedsegments of the circles (Prop. 9). If there are any number ofparallel circles on a sphere, and any number of great circlesdrawn through their poles, the arcs of the parallel circlesintercepted between any two of the great circles are similar,and the arcs of the great circles intercepted between any twoof the parallel circles are -equal (Prop. 10).The last proposition forms a sort of transition to the portionof the treatise (II.11-23 and Book III) which contains propositionsof purely astronomical interest, though expressed aspropositions in pure geometry without any specific referenceto the various circles in the heavenly sphere. The propositionsare long and complicated, and it would neither be easjnor worth while to attempt an enumeration. They deal withcircles or parts of circles (arcs intercepted on one circle byseries of other circles and the like).We have no difficulty irrecognizing particular circles which come into many proposi

THEODOSIUS'S SPHAERICA 249tions. A particular small circle is the circle which is thelimit of the stars which do not set, as seen by an observer ata particular place on the earth's surface ;the pole of thiscircle is the pole in the heaven.A great circle which touchesthis circle and is obliquely inclined to the ' parallel circles ' is thecircle of the horizon ;the parallel circles of course represent-the apparent motion of the fixed stars in the diurnal rotation,and have the pole of the heaven as pole. A second greatcircle obliquely inclined to the parallel circles is of course thecircle of the zodiac or ecliptic.The greatest of the ' parallelcircles ' is naturally the equator. All that need be said of thevarious propositions (except two which will be mentionedseparately) is that the sort of result provedis like that of12 and 13 of Euclid's Phaenomena to the effect that inProps.the half of the zodiac circle beginning with Cancer (or Capricornus)equal arcs set (or rise) in unequal times those which;are nearer the tropic circle take a longer time, those furtherfrom it a shorter; those which take the shortest time arethose adjacent to the equinoctial points those which are;equidistantfrom the equator rise and set in equal times. In likemanner Theodosius (III. 8) in effect takes equal and contiguousarcs of the ecliptic all on one side of the equator,draws through their extremities great circles touching thecircumpolar cparallel'circle, and proves that the correspondingarcs of the equator intercepted between the latter greatcircles are unequal and that, of the said arcs, that correspondingto the arc of the eclipticwhich is nearer the tropic circleis the greater. The successive great circles touching thecireumpolar circle are of course successive positions of thehorizon as the earth revolves about its axis, that is to say,the same length of arc on the ecliptic takes a longer or shortertime to rise accordingas it is nearer to or farther from thetropic, in other words, farther from or nearer to the equinoctialpoints.'It is, however, obvious that investigations of this kind,which only prove that certain arcs 'are greater than others,and do not give the actual numerical ratios between them, areuseless for any practical purpose such as that of telling thehour of the night by the stars, which was one of the fundamentaluroblems in Greek astronomv; and in order to find

250 TRIGONOMETRYthe required numerical ratios a new method had to be invented,namely trigonometry.It isNo actual trigonometry in Theodosius.perhaps hardly correct to say that spherical trianglesare nowhere referred to in Theodosius, for in III. 3 the congruence-theoremfor spherical triangles corresponding to Eucl.I. 4 is practically proved; but there is nothing in the bookthat can be called trigonometrical. The nearest approach isin III. 11, 12, where ratios between certain straight lines arecompared with ratios between arcs. ACc (Prop. 11) is a greatcircle through the poles A, A' \ CDc, C'D are two other greatcircles, both of which are at right angles to the plane of ACfc,but CDc is perpendicular to AA', while C'D is inclined to it atan acute angle. Let any other great circle AB'BA' throughAA' cut CD inLet the 'parallelany point B between C and /), and C'D in B'.'C'c' be the diameter of the 'parallelcircle EB'e be drawn through B','and letcircle touching the greatcircle C'D. Let L, K be the centres of the ' parallel circles,and let R, p be the radii of the ' 'parallelcircles CDc, Cfcfrespectively. It is required to prove that2R :2p > (arc CB):(arc C'B').Let (7'0, Ee meet in JV, and join NB'.Then B'N, being the intersection of two planes perpendicu-is perpendicular to that plane andlar to the plane of AC'CA ',therefore to both Ee and C'U.'

THEODOSIUS'S SPHAERICA 251Now, the triangle NLO being right-angled* at L, NO > NL.Measure NT along NO equal to NL, and join TB'.Then in the triangles B'NT, B f NL two sides B'N, NT areequal to two sides B'N, NL and the includedy angles (bothbeing right) are equal therefore the ;triangles are equal in allrespects, and LNLB'= LNTB'.Now 2R:2 = OC':C'K= ON:NT[= tan jmj': tan .V> LCOB-./.NOB'> (arc 6'): (are fi'C' ').If a', //, c' are the sides of the spherical triangle AB'C', thisresult is equivalent (since the angle COB subtended bythe arcGB is equal to A) to 1 : sin b' = tan A :> a :a',tan ofwhere a = BG, the side oppositeA in the triangle ABC.The proofis based on the fact (proved in Euclid'sand assumed as known by Aristarchus of Samoa and Archimedes)that, if a, )8 are angles such that n > Oi > /?,tan OL/ tan ft > a//3.While, therefore, Theodosius proves the equivalent of theformula, applicable in the solution of a spherical triangleright-angled at (7, that tana = nin 6 tan A, he is unable, forwant of trigonometry, to find the actual value of a/a 7 ,andcan only find a limit for it. He is exactly in the same positionas Aristarchus, who can only approximate to the values of thetrigonometrical ratios which he needs, e.g.sin 1, cos 1, sin 3,by bringing them within upper and lower limits with the aidof the inequalitiestana a > fl.tan J8 sin^S'

252 TRIGONOMETRYWe may contrast with thispropositionof Theodosius thecorresponding proposition in Menelaus's Sphaerica (III. 15)dealing with the more general case in which (?', instead ofbeing the tropical point on the ecliptic, is, like B', any pointbetween the tropical point and D. If R, p have the samemeaning as above and r lt r. z are the radii of the parallel circlesthrough J3' and the new C", Menelaus proves thatsin asin a 7which, of course, with the aid of Tables, gives the meansof finding the actual values of a or a' when the other elementsare given.The proposition III.12 of Theodosius proves a result similarto that of III. 11 for the case where the great circles AB'B,AC'C, instead of being great circles through the poles, arcgreat circles touching ' the circle of the always- visible stars ',i. e. different positions of the horizon, and the points C', B' areany points on the arc of the oblique circle between the tropicaland the equinoctial points ;in this case, with the same notation,4 R 2 :p > (arc:BG) (arc 5'C")-It is evident that Theodosius was simply a laborious compiler,and that there was practically nothing original in hiswork. It has been proved, by .means of propositions quotedverbatim or assumed as known by Autolycus in his MovingSphere and by Euclid in his Phaenomendt, that the followingpropositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11,12, 13, 15, 20 ;II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22 ;III. Ib, 2, 3, 7, 8, those shown in thick type being quotedword for word.But this is not all.The beginnings of trigonometry.In Menelaus's Spliaerica,, III. 15, thereis a reference to the proposition (III. 11) of Theodosius provedabove, and in Gherard of Cremona's translation from theArabic, as well as in Halley's translation from the Hebrewof Jacob b. Machir, there is an addition to the effect that thisproposition was used by Apolloniuswhich is givenin a book the title ofin the two translations in the alternative

BEGINNINGS OF TRIGONOMETRY 253forms ' liber aygregativus ' and ' liber de principiis univorsalibus'.Each of these expressions may well mean the workof Apollonius which Marinus refers to as the 'GeneralTreatise* (fiKaOoXov Tr/oay/iarem). There is no apparentreason to doubt that the remark in question was reallycontained in Menelaus's original work ; and, even if it is anArabian interpolation, it is not likely to have been madewithout some definite authority. If then Apollonius was thediscoverer of the proposition, the fact affords some ground forthinking that the beginnings of trigonometry go as far back,at least, as 1Apollonius. Tannery indeed suggested that notonly Apollonius but Archimedes before him may have compileda ' table of chords ',or at least shown the way to sucha compilation, Archimedes in the work of which we possessonly a fragment in the Measurement of a Circle, and Apolloniusin the VKVTOKIOV, where he gave an approximation to the valueof TT closer than that obtained by Archimedes; Tannerycompares the Indian Table of Sines in the Su**ya-Siddhanta,where the angles go by 24ths of a right angle (l/24th=3 45',2/24ths=7 30', &c.), as possibly showing Greek influence.This is, however, in the region of conjecture ;the first personto make systematic use of trigonometry is, so far as we know,Hipparchus.HIPPAKCHUS, the greatest astronomer of antiquity, wasborn at Nicaea in Bithynia. The period of his activity isindicated by references in Ptolemyhim the limits of which are from 161 B.C. to 126 B.C.to observations made byPtolemyfurther says that from Hipparchus's time to the beginning ofthe reign of Antoninus Pius (A.D. 138) was 265 years.2Thebest and most important observations made by Hipparchuswere made at Rhodes, though an observation of the vernalequinox at Alexandria on March 24, 146 B.C., recorded by himmay have been his own. His main contributions to theoreticaland practical astronomy can here only be indicated in thebriefest manner.1*2Tannery, Recherches sur Fhist. de Vastronomie ancienne, p. 64.Ptolemy, Syntaxis^ vii. 2 (vol. ii, p. 15).

254 TRIGONOMETRYThe work of Hipparchus.Discovery of precession.1. The greatest is perhaps his discovery of the precessionof the equinoxes. Hipparchus found that the bright starSpica was, at the time of his observation of it, 6 distantfrom the autumnal equinoctial point, whereas he deduced fromobservations recorded by Timocharis that Timocharis hadmade the distance 8. Consequently the motion had amountedto 2 in the period between Timocharis's observations, made in283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or166 years; this gives about 46-8" or 43-4" a year, as comparedwith the true value of 50 -3 75 7".Calculation of mean lunar month.2. The same discovery is presupposed in his work On thelength of the Year, in which, by comparing an observationof the summer solstice by Aristarchus in 281/0B.C. with hisown in 136/5 B.C., he found that after 145 years (the intervalbetween the two dates) the summer solstice occurred halfa day-and-night earlier than it should on the assumption ofexactly 365 days to the year ;hence he concluded that thetropical year contained about ^th of a day-and-night lessthan 365 J days. This agrees very nearly with Censorinus'sstatement that Hipparchus's cycle was 304 years, four timesthe 76 years of Callippus, but with 111,035 days in itinstead of 111,036 (= 27,759 x 4). Counting in the 304 years12x304 + 112 (intercalary) months, or 3,760 months in all,Hipparchus made the mean lunar month 29 days 12" hrs.44 min. 2^ sec., which is less than a second out in comparisonwith the present accepted .figure of 29-53059 days!3.Hipparchus attempted a new determination of the sun'smotion by means of exact equinoctial and solstitial observations;he reckoned the eccentricity of the sun's courseand fixed the apogee at the point 5 30' of Gemini. Moreremarkable still was hisinvestigation of the moon'scourse. He determined the eccentricity and the inclinationof the orbit to the ecliptic, and by means of records ofobservations of eclipses determined the moon's period withextraordinary accuracy (as remarked above). We now learn

HIPPARCHUS 255that the lengths of the mean synodic, the sidereal, theanomalistic and the draconitic month obtained by Hipparchusagree exactly with Babylonian cuneiform tables of date notlater than Hipparchus, and it is clear that Hipparchus wasin full possessionastronomy.of all theIm]yroved estimates of sizesand moo/6.results established by Babylonianawl distances of sun4.Hipparchus improved on Aristarchus's calculations of thesizes and distances of the sun and moon, determining theapparent diameters more exactly and noting the changes inthem ;he made the mean distance of the sun 1,245D, the meandistance of the moon 33 7), the diameters of the sun andmoon 12$ D and J D respectively, where D is the meandiameter of the earth.Epicyclesand eccentrics.5.Hipparchus, in investigating the motions of the sun, moonand planets, proceeded on the alternative hypotheses of epicyclesand eccentrics he did not invent these; hypotheses,which were already fully understood and discussed byWhile the motions of the sun and moon couldApollonius.with difficulty be accounted for by the simple epicycle andeccentric hypotheses, Hipparchus found that for the planets itwas necessary to combine the two, i.e. to superadd epicycles tomotion in eccentric circles.dat

356 TRIGONOMETRYImproved Instruments.7. He made great improvements in the instruments used forobservations. Among those which he used were an improveddioptra, a ' meridian-instrument ' designed for observations inthe meridian only, and a universal instrument (aorpoAajSoi/opyavov) for more general use. He also made a globe onwhich he showed the positions of the fixed stars as determinedby him it ; appears that he showed a larger number of starson his globe than in his catalogue.Geography.In geography Hipparchus wrote a criticism of Eratosthenes,in great part unfair. He checked Eratosthenes's data bymeans of a sort of triangulation he insisted on the;necessityof applying astronomy to geography, of fixing the position ofplaces by latitude and longitude, and of determining longitudesby observations of lunar eclipses.Outside the domain of astronomy and geography, Hipparchuswrote a book On, things borne down ly their weight fromwhich Siinplicius (on Aristotle's De caelo, p. 264 sq.) quotestwo propositions. It is possible, however, that even in thiswork Hipparchus may have applied his doctrine to the case ofthe heavenly bodies.In pure mathematics he is said to have considered a problemin permutations and combinations, the problem of finding thenumber of different possible combinations of 10 axioms orassumptions, which he made to be 103,049 (v. I. 101,049)or 310,952 according as the axioms were affirmed or denied 1 :Whenit seems impossible to make anything of these figures.the Fihrist attributes to him works On c the art of algebra,known by the title of the Rules ' and On ' the division of numbers',we have no confirmation : Suter suspects some confusion,in view of the fact that the article immediately following inthe Fihrist is on Diophantus, who also ' wrote on the art ofalgebra 1 .1Plutarch, Quaest. Conviv. viii. 9. 3, 732 r, De Stoicorum repugn. 29.1047 D.

HIPPARCHUS 257Firstsystematic use of Trigonometry.We come now to what is the most important from thepoint of view of this work, Hipparchus's share in the developmentof trigonometry. Even if he did not invent it,Hipparchus is the first person of whose systematic use oftrigonometry we have documentary evidence. (1) Theonof 'Alexandria says on the Syntaxis of Ptolemy, & propos ofPtolemy's Table of Chords in a circle (equivalent to sines),that Hipparchus, too, wrote a treatise in twelve books onstraight lines (i.e. chords) in a circle, while another in sixbooks was written by Menelaus. 1 In the tiyntaxis I. 10Ptolemy gives the necessary explanations as to the notationused in his Table. The circumference of the circle is dividedinto 360 parts or degrees; the diameter is also divided into120 parts, and one of such parts is the unit of length in termsof which the length of each chord is expressed; each part,whether of the circumference or diameter, is divided into 60parts, each of these again into 60, and so on, according to thesystem of sexagesimal fractions. Ptolemy then sets out theminimum number of propositions in plane geometry uponwhich the calculation of the chords in the Table is based (Siarrjy K rail/ ypa/i/io>i> /ieflo&Kfjs1 OLVT&V (ruorao'eooy). The propositionsare famous, and it cannot be doubted that Hipparchusused a set of propositions of the same kind, though hisexposition probably ran to much greater length. As Ptolemydefinitely set himself to give the necessary propositions in theshortest form possible, it will be better to give them underPtolemy rather than here. (2) Pappus, in speaking of Euclid'spropositions about the inequality of the times which equal arcsof the zodiac take to rise, observes that 'Hipparchus in his bookOn the rising of the twelve signs of the zodiac shows by meansof numerical calculations (Si api6p.S>v} that equal arcs of thesemicircle beginning with Cancer which set in times havinga certain relation to one another do not everywhere show thesame relation between the times in which they rise ',2 and soon. We have seen that Euclid, Autolycus, and even Theodosiuscould only prove that the said times are greater or less1Theon, Comm. on Syntax?s, p. 110, ed. Halma.2Pappus, vi, p. 600. 9-13.

258 TRIGONOMETRYin relation to one another ; they could not calculate the actualtimes. As Hipparchus proved corresponding propositions bymeans of numbers, we can only conclude that he used propositionsin spherical trigonometry, calculating arcs from otherswhich are given, by means of tables. (3) In the only workof his which survives, the Commentary on the Phaenomenaof Eudoxus and Aratus (an early work anterior to thediscovery of the precession of the equinoxes), Hipparchusstates that (presumably in the latitude of Rhodes) a star whichlies 27^ north of the equator describes above the horizon anarc containing 3 minutes less than 15/24ths of the wholecircle 1 ;then, after some more inferences, he says, 'For eachof the aforesaid facts is proved by means of lines (Sia rS>vypafifji&y) in the general treatises on these matters compiledby me In other '. places 2 of the Commentary he alludes toa work On simultaneous risings (ra IT* pi T>V (rvvava,To\S>v),and in II. 4. 2 he says he will state summarily, about each ofthe fixed stars, along with what sign of the zodiac it rises andsets and from which degree to which degree of each sign itrises or sets in the regions about Greece or wherever thelongest day is1 4| equinoctial hours, adding that lie has givenspecial proofs in another work designed so that it is possiblein practically every place in the inhabited earth to follow5the differences between the concurrent risings and settings/'Where Hipparchus speaks of proofs by means of lines ',hedoes not mean a merely graphical method, by constructiononly, but theoretical determination by geometry, followed bycalculation, just as Ptolemy uses the /cexpression rS>v ypapp.S>vof his calculation of chords and the expressions aipiKalfleets and ypa/ifJiiKal Seêiy of the fundamental propositionin spherical trigonometry (Menelaus's theorem applied to thesphere) and its various applications to particularcases. Itis significant that in the Syntaxis VIII. 5, where Ptolemyapplies the proposition to the very problem of finding thetimes of concurrent rising, culmination and setting of thefixed stars, he says that the times can be obtained 'by linesbe certainonly ' (8ia p.6va>v r&v ypa/jLfjLOH/)*Hence we maythat, in the other books of his own to which Hipparchus refers1Ed. Manitius, pp. 148-50. 2 16., pp. 128. 5, 148. 20.3 4lb., pp. 182. 19-184. 5.Syntaxis, vol. ii, p. 193.

HIPPARCHUS 259in his Comment

260 TRIGONOMETRYby ^^ from $$$ or 5 7 T , Heron's figure). There is little doubtthat it is to Hipparchus's work that Heron refers, though theauthor is not mentioned.While for our knowledge of Hipparchus's trigonometry wehave to rely for the most part upon what we can infer fromPtolemy, we fortunately possess an original source of informationabout Greek trigonometry in its highest developmentin the Hphaerica of Menelaus.The. date of MENELAUS of Alexandria isroughly indicatedby the fact that Ptolemy quotes an observation ofhis made in the first year of Trajan's reign (A.D. 98). Hewas therefore a contemporary of Plutarch, who in factrepresents him as being present at the dialogue De facie inorbe lunae, where (chap. 17) Lucius apologizes to Menelaus 'themathematician' for questioning the fundamental propositionin optics that the angles of incidence and reflection are equal.He wrote a variety of treatises other than the Sphaerica.We have seen that Theon mentions his work on Chords in aCircle in six Books. Pappus saysthat he wrote a treatise(IT paypare fa)on the setting (or perhaps only rising) ofdifferent arcs of the zodiac. 1 Proclus quotes an alternativeproof by him of Eucl. I. 25, which is direct instead of byreductio ad absurdum? and he would seem to have avoidedthe latter kind of proof throughout. Again, Pappus, speakingof the many complicated curves 'discovered by Demetrius ofAlexandria (in his "Linear considerations") and by Philonof Tyana as the result of interweaving plectoids and othersurfaces of all kinds ', says that one curve in particular wasinvestigated by Menelaus and called by him c paradoxical '(7ra/>a5ooy) 3 the;nature of this curve can only be conjectured(see below).But Arabian tradition refers to other works by Menelaus,(1) Elements of Geometry, edited by Thabit b. Qurra, in threeBooks, (2)a Book on triangles, and (3) a work the title ofwhich is translated by Wenrich de cognitione quantitatisdiscretae corporum perrnixtorum. Lightis thrown on thislast titleby one al-Chazim who (about A,D. 1121) wrote a1Pappus, vi, pp. 600-2.2 Proclus ori Eucl. I,8 pp. 345. 14-346. 11.Pappus, iv, p. 270. 25.

MENELAUS OF ALEXANDRIA 261treatise about the hydrostatic balance, i.e. about the determinationof the specific gravity of homogeneous or mixedthe course of which he mentions Archimedes andbodies, inthe treatise (3)Menelaus (among others) as authorities on the subject hence;must have been a book on hydrostatics discussingsuch problems as that of the crown solved by Archimedes.The alternative proof of Eucl. I. 25 quoted byProclus might have come either from the Elements of Geometryor the Book on triangles. With regard to the geometry, the*liber trium fratrum '(written by three sons of Musa b. Shakirin the ninth century) says that it contained a solution of theduplication of the cube, which is none other than that ofArchytas. The solution of Archytas having employed theintersection of a tore and a cylinder (with a cone as well),there would, on the assumption that Menelaus reproduced thesolution, be a certain appropriateness in the suggestion ofTannery 1 that the curve which Menelaus called the napdSogosypa/*/z?7 was in reality the curve of double curvature, knownby the name of Viviani, which is the intersection of a spherewith a cylinder touchingit internally and having for itsdiameter the radius of the sphere. This curve is a particularcase of Eudoxus's hippopede, and it has the property that theportion left outside the curve of the surface of the hemisphereon which it lies is equal to the square on the diameter of thesphere; the fact of the said area being squareable wouldjustify the application of the word napdSogos to the curve,and the quadratureitself would not probably be beyond thepowers of the Greek mathematicians, as witness Pappus'sdetermination of the area cut off between a complete turn ofa certain spiral on a sphere and the great circle touching2the origin.The Sphaerica of Menelaus.it atThis treatise in three Books is fortunately preserved inthe Arabic, and although the extant versions differ considerablyin form, the substance isbeyond doubt genuine;the original translator was apparently Ishaq b. Hunain(died A.D. 910). There have been two editions, (1) a Latin1Tannery, MJmoires scientifiqueS) ii, p. 17.2Pappus, iv, pp. 264-8.

262 TRIGONOMETRYtranslation by Maurolycus (Messina, 1558) and (2) Halley'sedition (Oxford, 1758). The former is unserviceable becauseMaurolycus's manuscript was very imperfect, and, besidestrying to correct and restore the propositions, he addedseveral of his own. Halley seems to have made a freetranslation of the Hebrew version of the work by Jacob b.Machir (about 1273), although he consulted Arabic manuscriptsto some extent, following them, e.g., in dividing the work intothree Books instead of two. But an earlier version directfrom the Arabic is available in manuscripts of the thirteenthto fifteenth centuries at Paris and elsewhere ;this version iswithout doubt that made by the famous translator Gherardof Cremona (1114-87). With the help of Halley's edition,Gherard's translation, and a Ley den manuscript (930) ofthe redaction of the work by Abu-Nasr-Mansur made inA.D. 1007-8, Bjornbo has succeeded in presenting an adequatereproduction of the contents of the SphaericaÎn this Book for the firstBook I.time we have the conception anddefinition of a spherical triangle. Menelaus does not troubleto give the usual definitions of points and circles related tothe sphere, e.g. pole, great circle, small circle, but begins withthat of a spherical triangle as ' the area included by arcs ofgreat circles on the surface of a sphere ', subject to the restriction(Def .2) that each of the sides or legs of the triangle is anarc less than a semicircle. The angles of the triangle are theangles contained by the arcs of great circles on the sphere(Def. and 3), one such angle is equal to or greater than anotheraccording as the planes containing the arcs forming the firstangle are inclined at the same angle as, or a greater anglethan, the planes of the arcs forming the other (Dcfs. 4, 5).The angleis a right angleif the planes of the arcs are at iîghtangles (Def. 6). Pappus tells us that Menelaus in his Sphaericacalls the figure in question (the spherical triangle) a ' threeside' (rp/TrAeirpoj/) 2 ;the word triangle (rpiytovov) was of course1BjOrnbo, Studien fiber Menelaos* Sphttrik (Abhandlungen zur Gesch. d.math. Wissenschaften, Heft xiv. 1902).2Pappus, vi, p. 476. 16.

MENELAUS'S SPHAERICA 263already appropriated for the plane triangle. We should gatherfrom this, as well as from the restriction of the definitions tothe spherical triangle and itsparts, that the discussion of thespherical triangle as such was probably new and if the; prefacein the Arabic version addressed to a prince and beginningwith the words, O ' prince! I have discovered an excellentmethod of 'proof . is . . genuine, we have confirmatory evidencein the writer's own claim.Mcnelaus's object, so far as Book I is concerned, seems tohave been to give the main propositions about sphericaltriangles corresponding to Euclid's propositions about planetriangles.At the same time he does not restrict himself toEuclid's methods of proof even where they could be adaptedto the case of the sphere; he avoids the form of proof byreductio ad absurdum, but, subject to this, he prefers theeasiest proofs. In some respects his treatment is more completethan Euclid's treatment of the analogous plane cases.In the congruence-theorems, for example, we have I. 4 acorresponding to Eucl. I. 4, I. 4b to Eucl. I. 8, I. 14, 16 toEucl. I. 26 a, b; but Menelaus includes (I. 13) what we knowas the ' ambiguous case ', which is enunciated on the lines ofEucl. VI. 7. I. 12 is a particular case of I. 16. Menelausincludes also the further case which has no analogue in planetriangles, that in which the three angles of one triangle areseverally equal to the three angles of the other (1.17). Hemakes, moreover, no distinction between the congruent andthe symmetrical, regarding both as covered by congruent. 1. 1is a, problem, to construct a spherical angle equal to a givenspherical angle, introduced only as a lemma because requiredin later propositions.I. 2, 3 are the propositions aboutisosceles triangles corresponding to Eucl. I. 5, 6 ;Eucl. 1. 18, 19(greater side opposite greater angle and vice versa) have theiranalogues in I. 7, 9, and Eucl. I.24, 25 (two sides respectivelyequal and included angle, or third side, in one triangle greaterthan included angle, or third side, in the other) in I. 8. I. 5(two sides of a triangle together greater than the third) correspondsto Eucl. I. 20. There is yet a further group of propositionscomparing parts of spherical triangles, I. 6, 18, 19, whereI. 6 (corresponding to Eucl. I. 21) is deduced from I. 5, just asthe first part of Eucl. I. 21 is deduced from Eucl. I. 20.

264 TRIGONOMETRYEucl. I. 16, 32 are not true of spherical triangles, andMenelaus has therefore the corresponding but differentpropositions.L 10 proves that, with the usual notation a, 6, c,A,B9 (7, for the sides and opposite angles of a sphericaltriangle, the exterior angle at C, or 180 (7, < = or >Aaccording as c + a > = or < 180, and vice versa. The proofof this and the next proposition shall be given as specimens.In the triangle ABC suppose that c + a > = or < 180 ;letD be the pole opposite to A.Then, according as c + a > = or(since AD = 180),< 180, BC> = or < BDand therefore LD > = or < /.BCD (= 180-G'), [I. 9]i.e.(since LD = /.A) 180-(7< = or >A.Menelaus takes the converse for granted.As a consequence of this, I. 11 provesthat A + B + G> 180.Take the same triangle ABC, with the pole D oppositeto A, and from B draw the great circle BE such thatLDBE = LBDE.Then CE+EB = CD < 180, so that, by the precedingproposition, the exterior angle ACB to the triangle BCE isgreater than LCBE,i.e.C > LCBE.Add A or D (= LEBD) to the unequals ;thereforeC + A > Z.CBD,whence A + B + C > LCBD + B or 180.After two lemmas I. 21, 22 we have some propositions introducingM, N, P the middle points of a, 6, c respectively.I. 23proves, e.g., that the arc MJf of a great circle >c, and I. 20that AM < = or >|a according as A > = or < (B + C). .Thelast group of propositions, 26-35, relate to the figure formed

MENELAUS'S SPHAERICA 265by the triangle ABC witt great circlesdrawn through B tomeet AC (between A and 0) in D, E respectively, and thecase where D and E coincide, and they prove different resultsarising from different relations between a and c (a>c), combinedwith the equality of AD and EC (or DC), of the anglesABD and EEC (or DBG), or of a + c and BD + BE (or 2BD)respectively, according as a + c< = or > 180.Book II has practically no interest for us.The object of itis to establish certain propositions, of astronomical interestonly, which are nothing more than generalizations or extensionsof propositions in Theodosius's Mphaerica, Book III.Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10,Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11is an extension of Theodosius III. 13. The proofs are quitedifferent from those of Theodosius, which are generally verylong-winded.Book III.It will have been noticedTrigonometry.gives the geometry of the spherical triangle,that, while Book I of Menelausneither Book Inor Book II contains any trigonometry. This is reserved forBook III. As I shall throughout express the various resultsobtained in terms of the trigonometrical ratios, sine, cosine,tangent, it is necessary to explain once for all that the Greeksdid not use this terminology, but, instead of sines, they usedthe chords subtended by arcs of acircle. In the accompanying figurelet the arc AD of a circle subtend anangle a at the centre 0. Draw AMperpendicular to OD andy produceitto meet the circle again in A'. Thensin a = AM/AO, and AM is\AA'or half the chord subtended by anangle 2 a at the centre, which mayshortly be denoted by |(crd. 2 a).Since Ptolemy expresses the chords as so many 120th parts ofthe diameter of the circle, while AM / AO = AA'/2AO,itfollows that sin a and (crd. 2 a) are equivalent. Cos a isof course sin (90 a) and is therefore equivalent to crd.(180-2a).

266 TRIGONOMETRY(a) ' Menelaus's theorem' for the sphere.The first 'proposition of Book III is the famous Menelaus'stheorem 'with reference to a spherical triangle and any transversal(great circle) cutting the sides of a triangle, producedif necessary. Menelaus does not, however, use a sphericaltriangle in his enunciation, but enunciates the proposition in'terms of intersecting great circles. Between two arcs ADB,AEG of great circles are two other arcs of great circles DFCand BFE which intersect them and also intersect each otherin F. All the arcs are less than a semicircle. It is requiredto prove thatsin EA ~" sin FD sin BAIt appears that Menelaus gave three or four cases, sufficientto prove the theorem completely. The proof depends on twosimple propositions which Menelaus assumes without proof;the proof of them is given by Ptolemy.(1) In the figure on the last page, if OD be a radius cuttinga chord AB in 0, thenFor draw AM, BN perpendicular to OD.Then= (crd. 2AD): l(

MENELAUS'S SPHAERTCA 267For, if AM, jBJVare perpendicular to 0(7, we have, as before,.2BC)Now letthe arcs of great circles ADB, AEC be cut by thearcs of great circles DFC, BFK which themselves meet in F.Let G be the centre of the sphere and join GB, GF, GE, AD.Then the straight lines AD, GB, being in one plane, areeither parallel or not parallel.If they are not parallel, theywill meet either in the direction of D, B or of A, G.Let AD, GB meet in T.Draw the straight lines ARC, DLC meeting GE, GFin K, Lrespectively.Then K, L, T must lie on a straight line, namely the straightline which is the section of the planes determined by the arcEFB and by the triangle ACD. 1Thus we have two straight lines AC, AT cut bystraight lines (77), TK which themselves intersect in L.1the twoTherefore, by Menelaus's proposition in plane geometry,CK _ CL DTKA~ LD'TASo Ptolemy. In other words, since the straight lines GB, GE, GF,which are in one plane, respectively intersect the straightlines AD, AC,CD which are also in one plane, the points of intersection T, K, L are inboth planes, and therefore lie on the straight line in which the planesintersect.

268 TRIGONOMETRYBut, by the propositions proved above,CK sin GE CL sin OF 1)T sin PRtherefore, by substitution, we havesin GE _ sinCF sin DB ^sin EA ~~ sin jPD * sin BA *Menelaus apparently also gave the prooffor the cases inwhich AD, GB meet towards A, G, and in which AD, GB areparallel respectively, and also proved that in like manner, inthe above figure,sin OA _sin CD sin FBsin AE~~ s5TS^*siirgA There CFE instead of(the triangle cut by the transversal beingADC). Ptolemy 1 gives the proof of the above case only, anddismisses the last-mentioned result with a 'similarly '.()8)Deductions fromMenelaus's Theorem.III. 2 proves, by means of I. 14, 10 and III. 1, that, if ABC,A'B'C' be two spherical triangles in which A = A', and C, C/are either equal or supplementary, sin c/sin a = siii-c'/sin a'and conversely. The particular case in which C, C' are rightangles gives what was afterwards known as the ' regulaquattuor quantitatum' and was fundamental in Arabiantrigonometry. 2 A similar association attaches to the result ofIII. 3, which is the so-called c tangent ' or c shadow-rule ' of theArabs.If ABC, A'B'C' be triangles right-angled at A, A', and(7, C' are equal and both either > or < 90, and if P, P' bethe poles of AC, A'C', thensinAB _ sinJ/jysin BPsin AC ~ sin A'C' ' sin B'P' "Apply the triangles so that (7 falls on C 9C'B' on GB as GE,then the result follows directly fromand G A' on GA as CD ;III. 1. Since sin BP = cos AB, and sin B'P' = cos A'B', theresult becomessin CMtan.A.Bwhich is the c tangent-rule ' of the Arabs. 31Ptolemy, Syntaxis, i. 13, vol. i, p. 76.2 See Braunmiihl, Gesch. der Trig, i, pp. 17, 47, 58-60, 127-9,8 Cf. Braunmtihl. 00. cit. i, DD. 17-18. 58. 67-9. &c.

MENELAUS'S SPH'AERIGA 269It follows at once (Prop. 4) that, if AM, A'M' are greatcircles drawn perpendicular to the bases BO, B'C' of twospherical triangles ABC, A'B'W in which B = K,C=C',sin BM sin MC /. . . . . ,tan AM \sin K'M'~~III. 5 proves that, if* ir^f (since both are equal to - J/T/"/ rsin M^'C' \tan AM /there are two spherical triangles ABC,/p''C 1D(A')A'KC' right-angledat A, A' and such that (7=C", while 6and b' are less than 90,sin (a + b) sin (a' -f //)from which we may deduce 1sin (a //)sin (a' b')the formulasin (a -f b) _1 -f cos C~~sin (a1 />)cos C 'which is equivalent to tan b = tan a cos C.(y) Anharmonic property of four great circles throughotiepoint.But more important than the above result isthe proof assumes as known the anharmonicproperty of four great circlesdrawn from a point on a sphere in relationto any great circle intersecting themall,viz. that, if ABCD, A'R'Wl)' be twotransversals,sin AD sin BC sin __A'D'" "i "\/^ A Tl """"" "sin DO sin AB sin D C sin A B1f\t/^t*A / li/Braunmiihl, op. cit. i, p. 18; BjOrnbo, p. 96.the fact that

270 TRIGONOMETRYIt follows that this proposition was known before Menelaus'stime. It is most easily proved by means of ' Menelaus'sTheorem', III. 1, or alternatively itmay be deduced for thesphere from the corresponding proposition in plane geometry,just as Menelaus's theorem is transferred by him from theplane to the sphere in III. We 1.may therefore fairly concludethat both the* anharmonic property and Menelaus'stheorem with reference to the sphere were already includedin some earlier text-book ; and, as Ptolemy, who built so muchupon Hipparchus, deduces many of the trigonometricalformulae which he uses from the one theorem (III. 1) ofMenelaus, it seems probable enough that both theorems wereknown to Hipparchus. The corresponding plane theoremsappear in Pappus among his lemmas to Euclid's Porisms? andthere is therefore every probability that they were assumedby Euclid as known.(S) Propositions analogousto End. VI. 3.Two theorems following, III. 6, 8, have their analogy inEucl. VI. 3. In III. 6 the vertical angle A of a sphericaltriangle is bisected by an arc of a great circle meeting BG inD, and it isproved that sin BD/sin DC = sin JiA/sin AC\in III. 8 we have the vertical angle bisected both internallyand externally by arcs of great circles BC in D andE9meetingand the proposition proves the harmonic propertysin BE _sin 7)smEC ~~ sin DCIII. 7 is to the effect that, if arcs of great circles be drawnthrough B to meet the opposite side AC of a spherical trianglein D, E so that ZABD = / EBC, thensin EA .sin AD sin 2 AB'As this is analogous to plane propositions given by Pappus aslemmas to different works included in the Treasury ofAnalysis,it is clear that these works were familiar toMenelaus.1Pappus, vii, pp. 870-2, 874.

MENELAUS'S SPHAERICA 271III. 9 and III. 10 show, for a spherical triangle, that (1) thegreat circles bisecting the three angles, (2) the great circlesthrough the angular points meeting the opposite sides atright angles meet in a point.The remaining propositions, III. 11-15, return to the samesort of astronomical problem as those dealt with in Euclid'sPhaeiiomena, Theodosius's tiphaeriea and Book II of Menelaus'sown work. Props. 11-14 amount to theorems inspherical trigonometry such as the following.Given arcs a lf 2, or.., 4 , ftl9 2 , 3 , /?4such that,90and also a x> @ 19a 2> a,3> 3I 4> /3 4 ,then(1) If sin a t : sin a 2 : sin 3 : sin a 4 = sin )3j : sin/J 2: sin /3 3 : sin 4 ,.=^>i=,.If sin( I +/g1 ) _ sin ( 2 + /32 )_ sMa:) L + j83sin(a l -/3 1 ) sin (a 2 -/? 2 ) sin(a3 -â,1then(S\Ifsin fai-ot,) sn(-/ 2 )sin(a3 4 )sin (/3.,- /34 )Again, giventhree series of three arcs such that!> 2> a 3, t> 2> |8 3I90 > y l> y2> y3,nd sin (o^ ya ) : sin (a2 y2 ) : sin (a3 y :j)= sin (j8 1- y t ) : sin (0 2 - y2) : sin (/33 - y3)= siny^sinyg-.sinyg

272TRIGONOMETRY(1)K ,>/,theni- 2,> o o5an(i(2) If ft l< !thenIII. 15, the last proposition,is in four parts.The first partis the proposition corresponding to Theodosius III. 11 abovealluded to. Let BA, BC be two quadrants of great circles(in which we easily recognize the equator and the ecliptic),P the pole of the former, PA 19 PA% quadrants of great circlesmeeting the other quadrants in A 19 A% and O l9 0% respectively.Let R be the radius of the sphere, r, r,,r:jthe radii of the'parallel circles' (with pole P) through 0, C 19C:J respectively.Then shallsnsnRrIn the triangles PCG^ BA^C.A the angles at C,and the angles at (7 3 equal ; therefore (III. 2)snsin PGsin PC*are right,

MENELAUS'S SPHAERICA 273But, by III. 1 applied to the triangle BC^ cut by thetransversalsn sn C sinsinBA% sin #C 3 sin PC\sin AÂS _ sin PJ. 1sin BA.} __ sin P^ Tsin (7, 6 Y .,""sin PC.sin J5(7o" "sin PC*sin PCsin PC,I t> I O 1 .)from above,Part 2 of the proposition proves that, if PC ZA 2be drawnsuch that sin 2 P6 f 2 = sin PA 2 . sin PC, or ?' 22= jRr (where r 2 isthe radius of the parallel circle through (7 2),BC 2BA 2is amaximum, while Parts 3, 4 discuss the limits to the value ofthe ratio between the arcs AÂ^ and C\C^Nothing is known of the life of CLAUDIUS PTOLEMY exceptthat he was of Alexandria, made observations between theyears A.D. 125 and 141 or perhaps 151, and therefore presumablywrote his great work about the middle of the reign ofAntoninus Pius (A.I). 138-61). A tradition handed down bythe Byzantine scholar Theodoras Meliteniota (about 1361)states that he was born, not at Alexandria, but at Ptolemaisfl 'EpfjLtiov.Arabian traditions, going back probably toHunain b. Ishaq, say that he lived to the age of 78, and givea number of personal details to which too much weight mustnot be attached.The MadrjfjLartKr) crvvragi* (Arab. AlmcKjesf).Ptolemy's great work, the definitiveachievement of GreekSwrdgecos fttftXfa iy,astronomy, bore the title MadijfjiaTiKfjsthe Mathematical Collection in thirteen Books. By the timeof the commentators who distinguished the lesser treatises onastronomy forming an introduction to Ptolemy's work asfUKpb? acrrpoi>o//oi$/

274 TRIGONOMETRYthe superlativepfyurros, made up a word Al-majisti, whichbecame Almagest ;and it has been known by this name eversince. The complicated character of the system expoundedby Ptolemy is no doubt responsible for the fact that itspeedily became the subject of elaborate commentaries.Commentaries on the Syntaxis.Pappus 1 cites a passage from his own commentary on*Book I of the Mathematics, which evidently means Ptolemy'swork. Part of Pappus's commentary on Book V, as well ashis commentary on Book VI, are actually extant in theoriginal. Theon of Alexandria, who wrote a commentary onthe Syntaxis in eleven Books, incorporated as much as wasavailable of Pappus's commentary on Book V with fullacknowledgement, though not in Pappus's exact words. Inhis commentary on Book VI Theon made much more partialquotations from Pappus indeed the ;greater part of the commentaryon this Book is Theon's own or taken from othersources. Pappus's commentaries are called scholia, Theon'svTTOfjLvriijiaTa. Passages in Pappus's commentary on Book Vallude to ' the scholia preceding this one ' (in the plural), andin particular to the scholium on Book IV. It is therefore allbut certain that lie wrote on all the Books from I to VI atleast. The text of the eleven Books of Theon's commentarywas published at Basel by Joachim Camerarius in 1538, butit is rare and, owing to the way in which it is printed, withinsufficient punctuation marks, gaps in places, and any numberof misprints, almost unusable ;accordingly little attention hasso far been paid to it except as regards the first two Books,which were included, in a more readable form and with a Latintranslation, by Halma in his edition of Ptolemy.Translations and editions.The ttyniaxis was translated into Arabic, first (we are told)by translators unnamed at the instance of Yahya b. Khalid b.Barmak, then by.al-Hajjaj, the translator of Euclid (about786-835), and again by the famous translator Lshaq b. Hunain(d. 910), whose translation, as improved by Thabit b. Qurra1Pappus, viii, p. 1106. 13.

PTOLEMY'S S7NTAXI8 275(died 901), is extant in part, as well as the version by Nasiraddinat-Tusi (1201-74).The first edition to be published was the Latin translationmade by Gherard of Cremona from the Arabic, which wasfinished in 1175 but was not publishedtill 1515, when it wasbrought out, without the author's name, by Peter Liechtensteinat Venice. A translation from the Greek had been madeabout 1160 by an unknown writer for a certain HenricusAristippus, Archdeacon of Catania, who, having been sent byWilliam I, King of Sicily,on a mission to the ByzantineEmperor Manuel I. Comnenus in 1158, brought back withhim a Greek manuscript of the Syntaxis as a present; thistranslation, however, exists only in manuscripts in the Vaticanand at Florence. The first Latin translation from the Greek'to be published was that made by Georgius of Trebizond 'forPope Nicolas V in 1451 ; this was revised and published byLucas Gauricus at Venice in 1528. The editio princeps of theGreek text was brought out by Grynaeus at Basel in 1538.The next complete edition was that of Halma published1813-16, which is now rare. All the more welcome, therefore,is the definitive Greek text of the astronomical worksof Ptolemy edited by Heiberg (1899-1907), to which is nowadded, so far as the Syntaxis is concerned, a most valuablesupplement in the German translation (with notes) by Manitius(Teubner, 1912-13).Summaryof Contents.The Syntaxisis most valuable for the reason that it containsvery full particulars of observations and investigationsby Hipparchus, as well as of the earlier observations recordedby him, e.g. that of a lunar eclipse in 721 B.C.Ptolemybased himself very largely upon Hipparchus, e.g. in thepreparation of a Table of Chords (equivalent to sines), thetheory of eccentrics and epicycles, &c. and it is ; questionablewhether he himself contributed anything of great value excepta definite theory of the motion of the five planets, for whichHipparchus had only collected material in the shape of observationsmade byhis predecessors and himself. A very shortindication of the subjects of the different Books is all that can

276 TRIGONOMETRYbo given here. Book I: Indispensable preliminaries to thestudy of the Ptolemaic system, general explanations ofthe different motions of the heavenly bodies in relation tothe earth as centre, propositions required for the preparationof Tables of Chords, the Table itself, some propositions inspherical geometry leading to trigonometrical calculations ofthe relations of arcs of the equator, ecliptic, horizon andmeridian, a * Table of Obliquity ',for calculating declinationsfor each degree-point on the ecliptic, and finally a method offinding the right ascensions for arcs of the ecliptic equal toone-third of a sign or 10. Book II: The same subject continued,i.e. problems on the sphere, with special reference tothe differences between various latitudes, the length of thelongest day at any degree of latitude, and the like. Book III :On the length of the year and the motion of the sun on theeccentric and epicycle hypotheses. Book IV : The length of themonths and the theory of the moon. Book V : The constructionof the astrolabe, and the theory of the moon continued,the diameters of the sun, the moon and the earth's shadow,the distance of the sun and the dimensions of the sun, moonand earth. Book VI :Conjunctions and oppositions of sunand moon, solar and lunar eclipses and their periods. BooksVII and VIII are about the fixed stars and the precession ofthe equinoxes, and Books IX-XIII are devoted to the movementsof the planets.Trigonometry inPtolemy.What interests the historian of mathematics isthe trigonometryin Ptolemy. It is evident that no part of the trigonometry,or of the matter preliminary to it, in Ptolemy was new.What he did was to abstract from earlier treatises, and tocondense into the smallest possible space, the minimum ofpropositions necessary to establish the methods and formulaeused. Thus at the beginning of the preliminaries to theTable of Chords in Book I he says:'We will first show how we can establish a systematic andspeedy method of obtaining the lengths of the chords based onthe uniform use of the^ smallest possible number of propositions,so that we may not only have the lengths of the chords

PTOLEMY'S NYNTAXia 277set out correctly, but may be in possession of a ready proof ofour method of obtaining them based on geometrical considerations/lHe explains that he will use the division (1) of the circle into360 equal parts or degrees and (2) of the diameter into 120equal parts, and will express fractions of these parts on thesexagesimal system. Then come the geometrical propositions,as follows.(a) Lemma for finding sin 18 and sin 36.To find the side of a pentagon and decagon inscribed ina circle or, in other words, the chords subtending arcs of 72and 36respectively.Let AH be the diameter of a circle, O the centre, OC theradius perpendicular to A B.Bisect OB at JJ, join DC, and measureI)E along DA equal to DC. Join EC.Then shall OK be the side of the inscribedregular decagon, and EC the sideof the inscribed regular pentagon.For, since OB is bisected at />,)*= DE*Therefore BE. EO = 0(72= OS 2 ,and BE is divided in extreme und mean ratio.But (Eucl. XIII. 9) the sides of the regular hexagon and theregular decagon inscribed in a circle when placed in a straightline with one another form a straight line divided in extremeand mean ratio at the point of division.Therefore, BO being the side of the hexagon, EO i^thesideof the decagon.Also (by Eucl. XIII. 10)(side of pentagon) 2 = (side of hexagon) 2 + (side of decagon)2therefore EC is the side of the regular pentagon inscribedin the circle.1Ptolemy, Syutaxi*) i. 10, pp. 31 2.

278 TRIGONOMETRYThe construction in fact easily leads to the resultswhere a is the radius of the circle.Ptolemy does not however use these radicals, but calculatesthe lengths in terms of ' parts ' of the diameter thus.DO = 30, and DO = 2 900 ;00 = 60 and 00* = 3600 ;therefore DE* = DO = 2 4500, and DE = 67* 4' 55" nearly ;therefore side of decagon or = (cixl.36) /J#-jDO = 37''4'55"Again 0# = 2 (37/'4' 55")*= 1375 .4' 15", and 0(7 2 =3600;therefore CE* = 4975 . 4' 15", and CE = 70'' 32' 3" nearly,i.e. side of pentagon or (crd. 72)= 70'' 32' 3".The method of extracting the square root is explained byTheon in connexion with the first of these cases, \/4500 (seeabove, vol. i, pp. 61-3).The chords which are the sides of other regular inscribedfigures, the hexagon, the square and the equilateral triangle,are next given, namely,crd. 60= 60'',crd. 90 = \/(2 .60-) = >/(7200) = 84'' 51' 10",crd. 120 = V(3. 60 2 ) = \/( 10800) = 103'' 55' 23".(/3) Equivalent of sin 2 -f cos- = 1.It isnext observed that, if x be any arc,(crd. a) 2 + {crd. (180 -a;)}2= (diam.)2= 120 2 ,a formula which is of course equivalent to sin 2 Q + cos 2 6 = 1.We c&i therefore, from crd. 72, derive crd. 108, fromcrd. 36, crd. 144, and so on.(y) 'Ptolemy's theorem', giving the equivalent ofsin (6 (f>)= sin cos cos 6 sin 0.,The next step is to find a formula which will give uscrd. (a j8)when crd. a and crd. ft are given. (This forinstance enables us to find crd. 12 from crd. 72 and crd. 60.)

PTOLEMY'S SYNTAX1S 279The proposition giving the required formula depends upona lemma, which is the famous 'Ptolemy'stheorem '.Given a quadrilateral ABCD inscribed in a circle, thediagonals being AC, BD, to prove thatAC.BD =. DC+ AD .The proofis well known.is equal to the angle DBC, and let BEmeet AC in E.Then the triangles ABE, DBC areequiangular, and thereforeAB:AE=BD:DC,or AB.DC=AE.BD.(1)Again, to each of the equal anglesABE, DBC add the angle EBD;EC.Draw BE so that the angle ABEI I \ X ^*>then the angle ABD is equal to the angle EBC, and thetriangles ABD, EBC are equiangular;therefore BC :orBy adding (1) and (2),AB . DC+ADCE = BD :DA,AD.BC=CE.BD..we obtain(2)BC = AC. BD.Now let AB, AC be two arcs terminating at A, the extremityof the diameter A I) of a circle, and letAC (= a) be greater than AB (= /3,.Suppose that (crd. AC) and (crd. AB)are given:it isrequired to find(crd. BC).Join BD, CD.Then, by the above theorem,Now AB, AC are given; therefore BD = crd. (180 -AB)and CD = crd. (180 ^4(7) are known. And AD is known.Hence the remaining chord BG (crd. BC)is known.

280 TRIGONOMETRYThe equation in fact gives the formula,{crd. (a- )} . (crd. 180) = (crd. a) . {crd. (180-/3)}-(crd. /3).{crd.(180-a)},which is, of course, equivalent tosin (0 0)= sin 6 cos cos 6 sin 0, where a = 20, ft= 20.By means of this formula Ptolemy obtained= 12'' 32' 36".crd. 12 = crd. (72- 60)(5) Equivalent of siii^d = (1 cos 0).But, in order to get the chords of smaller angles still, wewant a formula for finding the chord of half an arc when thechord of the arc is given. This is the subject of Ptolemy'snext proposition.Let BG be an arc of a circle with diameter AC, and let thearc BC be bisected at D. Given (crd. EG),it is required tofind (crd. DC).Draw DF perpendicular to AC,and join AB, AD, BD, DC. MeasureAE along AC equal to A Li, and joinDK.Then shall FC be equal to EF, orFO shall be half the difference betweenAC and AB.For the triangles ABD, AED areequal in allrespects, since two sidesof the one are equal to two sides of the other and the includedangles BAD, BAD, standing on equal arcs, are equal.ThereforeED = BD = DC,and the right-angled triangles DEF, DCF are equal in allrespects, whence EF = FC, or CF = %(AC-AB).NowAC. CF = CD*,whence (crd. CD)* = \AG (AC-AB)= 4 (crd. 180).{(crd.l80 )-~(crd.l80 ~â)j.This is,of course, equivalent to the formula= (l-cos 0).

PTOLEMY'S tiYNTAXIS 281By successively applying this formula, Ptolemy obtained(crd. 6), (crd. 3) and finally (crd. 1|) = 12' 34' 15" and(crd. |) = OP 47' 8". But we want a table going by halfdegrees,and hence two more things are necessary ;we have toget a value for (crd. 1) lying between (crd. l) and (crd. |),and we have to obtain an addition formula enabling us when(crd. a) is given to find {crd. (a + ^)}, and so on.(e) Equivalent of cos (6 + 0)= cos 6 cos sin sin.To find the addition formula. Suppose AD is the diameterof a circle, and AB, BC two arcs. Given (crd. AB) and(crd. BC), to find (crd. AC), Draw the diameter BOE, andjoin CE, CD, DE, BD.Now, (crd, AB) being known,(crd. BD) is known, and thereforealso (crd. DE), which is equal to(crd. AB)] and, (crd. BC) beingknown, (crd. CE) is known.And, by Ptolemy's theorem,BD . CE = BC . DE + BE .The diameter BE and allCD.the chords in this equation exceptCD being given, we can find CD or crd. (180 AC). We havein fact(crd. 180) .[crd.= !crd.(180-4#)].{c^thus crd. (180AC) and therefore (crd. AC)is known.If AB = 20, BC =20, the result is equivalent tocos (6 + 0)= cos cos -sin Q sin 0.(0 Method of interpolation based on formulasin a /sin ft < a//3 (where % > TT a > /9).Lastly we have to find (crd. 1), having given (crd. 1|) and(crd. |).Ptolemy uses an ingenious method of interpolation based ona proposition already assumed as known by Aristarchus.If AB, BC be unequal chords in a circle, BC being the

282 TRIGONOMETRYgreater, then shall the ratio of CB to BA be less than theratio of the arc CB to the arc BA.Let BD bisect the angle ABC, meeting AC in E andin//.Now FE: EA = A FED :Componeiido, FA :the circumference in D. The arcsAD, DC are then equal, and so arethe chords AD, DC. Also CE>EA(since CB:BA = CE:EA).Draw DF perpendicular to AC;then AD>DE>DF, so that thecircle with centre D and radius DEwill meet DA in G and DF produced&AED< (sector IIED):(sector QED)< LFDE-.LEDA.AE < L FDA : L ADE.Doubling the antecedents, we haveand, separando, CE :CA : AE < L CDA : L A DE,EA < L CDE: L EDA;therefore(since CB-.BA = CE-.EA)CB-.BA < LODE: LEVA< (arc CB):(arc BA),i. e. (crd. CB) : (crd. BA) < (arc CB): (arc BA ).[This is of course equivalent to sin a : sin /8 < a :ft,whereIt follows (1) that (crd. 1 ) : (crd. |) < 1 :| ,and (2) that (crd. l):(crd. 1) (crd. 1 )> f .(crd. l).But (crd. |) = OP 47' 8", so that |(crd. |)nearly (actually V 2' 50%");= \v 2' 50"and (crd. l) = V 34' 15", so that |(crd. l) = 1?' 2' 60".Since, then, (crd. 1) is both less and greater than a lengthwhich only differs inappreciably from IP 2' 50", we may saythat (crd. 1) = 1?'2' 50" as nearly as possible.

PTOLEMY'S SYNTAXIS 283(rj)Table of Chord*.From thisPtolemy deduces that (crd.f ) is very nearlyO/' 3V 25", and by the aid of the above propositions he is ina position to complete his Table of Chords for arcs subtendingangles increasing from to 180 by steps of ;in otherwords, a Table of Sines for angles from J to 90 by stepsof i.r(0) Further use of pro portioiialincrease.Ptolemy carries further the principle of proportional increaseas a method of finding approximately the chords ofarcs containing an odd number of minutes between 0' and 20'.Opposite each chord in the Table he enters in a third column-3ô th of the excess of that chord over the one before, i.e. thechord of the arc containing 30' less than the chord in question.For example (crd. 2)is stated in the second column of theTable as W 37' 4".in theThe excess of (crd. 2) over (crd.2)Table is 0'' 31' 24": ^th of this is 0*' l' 2" 48'", which istherefore the amount entered in the third column opposite(crd. 2J). Accordingly, if we want (crd. 2 25'), we take(crd. 2) or 2P 5' 40" and add 25 times OP l'2"48'"; or wetake (crd. 2) or 2/' 37' 4" and subtract 5 times 0'' l' 2" 48"'.Ptolemy adds that if, by using the approximation for 1 and^, we gradually accumulate an error, we can check the calculationby comparing the chord with that of other related arcs,e.g. this double, or the supplement (the difference between thearc and the semicircle).Some particular results obtained from the Table may bementioned. Since (crd. 1) = 1 P 2' 50", the whole circumference= 360 (1^2' 50"), nearly, and, the length of the diameterbeing 1201', the value of TT is 3 (1 +A+ *Sro)= 3 + *Tr+ rahy>which is the value used later by Ptolemy and is equivalent to3-14166... Again, > 55' 23"), we have ^3 = ^ (103 + ff + ?ffs)43 55 23which is correct to 6 places of decimals.Speaking generally,

284 TRIGONOMETRYthe sines obtained from Ptolemy's Table are correct to 5places.(i) Plane trigonometry iu effect used.There are other cases in Ptolemy in which plane trigonometryis in effect used, e.g. in the determination of theeccentricity of the sun's orbit. 1 Supposethe eccentric circle with centre 0,and AB, CD are chords at rightangles through E, the centre of theearth. To find OE. The arc BUis known (= a, say) as also the arcGA (=0).If BF be the chordparallel to CD, and CG the chordparallel to AB, and if JV, P be themiddle points of the arcs BF, GC,Ptolemy finds (1) the arc BFthat ACBD is= a(+ 0-180 ),then the chord BF,crd. (a +0-180), then the half of it, (2) the arc GO= arc (a + 02/8) or arc (a 0), then the .chord 76', andon the half-lastly half of He it. then adds the squareschords, i.e. he obtainsthat is, OJP/v* = cos2 J (a + j8) + sin 2 i (a - 0).He proceeds to obtain the angle OEC from its sineOR/UE,which he expresses as a chord of double the angle in thecircle on OE as diameter in relation to that diameter.Spherical trigonometry:formulae in solution ofspherical triangles.In spherical trigonometry, as already stated, Ptolemyobtains everything that he wants by using the one fundamentalproposition known as 'Meiielaus's theorem' appliedto the sphere (Menelaus III. 1), of which he gives a prooffollowing that given by Menelaus of the first case taken inhis proposition.Where Ptolemy has occasion for other propositionsof Menelaus's Sphaerica, e.g. III. 2 and 3, he does1Ptolemy, Syntaxia, iii. 4, vol. i, pp. 234-7.

PTOLEMY'S SYNTAX IS 285not quote those propositions, as he might have done, but provesthem afresh by means of Menelaus's theorem. 1 The applicationof the theorem in other cases gives in effect thefollowing different formulae belonginga spherical triangle ABC right-angled at G',sin a = sine sin -4,tan a = sin b tan A,cos c = cos a cos 6,tan b '= tan r cos A.to the solution ofviz.One illustration of Ptolemy's procedure will be sufficient. 2Let 11Air be the horizon, PEZH the meridian circle, EE'the equator,'//// the ecliptic,F anequinoctial point. Let EK f %//,cut the horizon in A, B. Let P bethe pole, and let the great circlethrough P, B cut the equator at (!.Now let it be required to find thetime which the arc FB of the ecliptictakes to rise; this time will bemeasured by the arc FA of theequator. (Ptolemy has previously found the length of thearcs EC, the declination, and FC. the right ascension, of B,I. 14, 16.)By Menelaus's theorem applied to the arcs AE', E'P cut bythe arcs All' ,PC which also intersect one another in 7?.crd.27 J 7/' crd. __ ~ 2P# crd.26M'crd. 2 U'K' crd.TTTC' crd. 2 AE'*sinP//' sinP7i sin GAtnat> is. -. yTf r ^. .~ r7-

286 TRIGONOMETRYThus AC is found, and therefore FC-AC or FA.The lengths of BO, FC are found in I. 14, 16 by the samemethod, the four intersecting great circles used in the figurebeing in that case the equator EE', the eclipticZZ' 9the greatcirclePBCP' through the poles,and the great circle PKLP'passing through the poles of both the ecliptic and the equator.In this case the two arcs PL, AE' are cut by the intersectinggreat circles PC, FK, and Menelaus's theorem gives (1)sin PL _sin CP sinUPsin KL "" siiTBO 'sin FKBut sinPZ=l, sinKL = smBFC, sin OP =1, *inFK = l,and it follows thatsin BO = sin #Psin BFC,corresponding to the formula for a triangle right-angled at C,and(2)We havesin a = sin c sin A.sin PK _sinPjSsin OPsin KL ~~ sin sin PA 'sin PK = cos A'7, = cos JMY7, sin P# = cos BG, sin FL=l,"so thatcorrespondingtan jBC = sin C'Ptan BFT!,to the formulatan a = sin b tan A.While, therefore, Ptolemy's method implicitly gives theformulae for the solution of right-angled triangles abovequoted, he does not speak of right-angled triangles at all, butonly of arcs of intersecting great circles. The advantagefrom his point of view is that he works in sines and cosinesonly, avoiding tangents as such, and therefore he requirestables of only one trigonometrical ratio, namely the sine (or,as he has it, the chord of the double arc).The Ana/cmma.Two other works of Ptolemy should be mentioned here.The first is the Analemma. The object of this is to explaina method of representing on one plane the different points

THE ANALEMMA OF PTOLEMY 287and arcs of the heavenly sphere by means of orthogonalprojection upon three planes mutually at right angles, themeridian, the horizon, and the ' prime vertical '. The definiteproblem attacked is that of showing the position of the sun atany given time of the day, and the use of the method andof the instruments described in the book by Ptolemy wasconnected with the construction of sundials, as we learn fromVitruvius. 1 There was another dvaXrîfjia besides that ofPtolemy the author of it was Diodorus of Alexandria, a contemporaryof Caesar and Cicero (' Diodorus, famed among the;makers of gnomons, tell me the time '! says the Anthology 2 ),and Pappus wrote a commentary upon it in which, as h3 tellsus, 3 he used the conchoid in order to trisect an angle, a problemevidently required in the Aiialemma in order to divide anyarc of a circle into six equal parts (hours). The wordar/aXiy/z/za evidently means'taking up ' ('Aufiiahme ')in thesense of 'making a graphic representation 'of something, inthis case the representation on a plane of parts of the heavenlysphere. Only a few fragments remain of the Greek text ofthe A nalemma of Ptolemy; these are contained in a palimpsest(Ambros. Gr. L. 99 sup., now 491) attributed to the seventhcentury but probably earlier. Besides this, we have a translationby William of Moerbeke from an Arabic version.This Latin translation was edited with a valuable commentaryby the indefatigable Commandinus (Rome, 1562); but it isnow available in William of Moerbeke's own words, Heiberghaving edited it from Cod. Vaticanus Ottobon. lat. 1 850 of thethirteenth century (written in William's own hand), and includedit with the Greek fragments (so tar as they exist) inparallel columns in vol. ii of Ptolemy's works (Teubner, 1907).The figure is referred to three fixed planes (1) the meridian,(2) the horizon, the (3) prime vertical; these planes are theplanes of the three circles APZBy ACB, ZQO respectivelyshown in the diagram below. Three other great circles areused, one of which, the equator with pole P, is fixed; theother two are movable and were called by special names;the first is the circle represented by any position of the circleof the horizon as it revolves round COG' as diameter (GSM in1Vitruvius, De architect, ix. 4.8Pappus, iv, p. 246. 1.2 Anil. Palat. xiv. 139.

288 TRIGONOMETRYthe diagram is one position of it, coinciding with the equator),and it was called tKrrjpopos KVK\O$ ('the circle in six parts ')because the highest point of it above the horizon correspondsto the lapse of six hours ; the second, called the hour-circle, isthe circle represented by any position, as BtiQA, of the circleof the horizon as it revolves round BA as axis.The problem is, as above stated, to find the position of thesun at a given hour of the day. In order to illustratethe method, it is sufficient, with A. v. Braunmuhl, 1 to take thesimplest case where the sun is on the i.e.equator, at one ofthe equinoctial points, so that the hectemoron circle coincideswith the equator.Let S be the position of the sun, lying on the equator M8C,P the pole, MZA the meridian, BOA the horizon, JitiQAhour-circle, and let the vertical greatthecircle ZtiV be drawnthrough S, and the vertical great circle ZQC through Z thezenith and C the east-point.We are given the arc =90 S6/ f, where / is the hofcr-where is the elevation ofangle, and the arc MB = 90 0,the pole ;and we have to find the arcs >S'F (the sun's altitude).VC, the 'ascensional difference', SQ and QO. Ptolemy, infact, practically determines the positionof S in terms ofcertain spherical coordinates.Draw the perpendiculars, SF to the plane of the meridian,8H to that of the horizon, and SE to the plane of the prime1Braunmuhl, Gescli. der Trigonomrtrie, i, pp. 12, 13,

THE ANALEMMA OF PTOLEMY 289vertical ;and draw FG perpendicular to BA, and ET to OZ.Join 7/G, and we have FG = 8U, GH = J%' = #21We now represent tiF in a separate figure (for clearness'sake, as Ptolemy uses only one figure), where B'Z'A' correspondsto BZA, P' to P and Q'M' to OM. Set off the arcP'S' equal to OS (= 90 -i), and draw S'F' perpendicularto O'M'. Then tf'J/'= 8M, and B'F'= 8F; it is as if in theoriginalfigure we had turned the quadrant MSC round MOtill it coincided with the meridian circle.In the two figures draw IFK, 1'F'K' parallel to BA, B'A\and LFG, L'F'G' parallel to OZ. O'//.because if weThen (1) arc XI = areas' = arc (90 -NF),turn the quadrant %8V about %0 till it coincides with themeridian, S falls on /, and V on It. It follows that therequired arc #J r = arc /?'/' in the secondWfigure.(2) To find the arc VC, set off G'X (in the second figure)along equal to Fti or JW, and draw O'X through tomeet the circle in X'. Then arc #',Y' = arc IT/; for it is as ifwe had turned the quadrant HVC about BO till it coincidedwith the meridian, when (since G'X = Ftf = (r7/) // wouldcoincide with A" and K with X'.to JX'.Therefore BFis also equal(3) To find QC or ^, set off along T'F* in the second figureT'Y equal to F'N', and draw O'F through to F' on the circle.Then arc -B'F'ss arc Qf/;for it is as if we turned the primevertical ZQC about ZO till it coincided with the meridian,when (since !TF=W= TE) E would fall on F, the radiusOEQ on 0'FF Xand Q on F'.(4) Lastly, arc BS = arc BL = arc B'L', because S, L areIS93.9U

290 TRIGONOMETRYboth in the plane L8HG at right angles to the meridian;therefore arc 8Q = arc L'Z'.Hence all four arcs 8V9 VC, QC, QS are represented in theauxiliary figure in one plane.So far the procedure amounts to a method of graphicallyconstructing the arcs required as parts of an auxiliary circlein one plane. But Ptolemy makes it clear that practical1calculation followed on the basis of the figure. The linesused in the construction are SF= sin t(where the radius =1),FT=OFsiu(f>, FG = OF8in(9Q- is knownfore the arc 1*8' [= 90-*],[in fact O'F' /D- {crd. (I80-2.It follows from these two results thatLastly, the arc #V( = h) being equal to /f /', the angle h isequal to the angle O'T'T' in the triangleI'O'T'. And in thistriangle O'l', the radius, is known, while O'T' has been found ;and we have thereforeO'T' crd.(2/t) crd. (180- 2t) -- crd. (180-*2 A)p,ym,==,fromabove.[In other words, sin h = cos t cos ; or, if u = 8C = 90 /,sin h = sin u cos 0, the formula for finding sin h in the rightangledspherical triangle 8V CJ]For the azimuth o> (arc BV = arc B'X'), the figure gives_ _ fl^' _ S itan _o> - - -Qr(y jfj, Q, ;fj ff/^ - tan .^,^1See Zeuthen in BiNiotheoa mathemaiica, i :5 , 1900, pp. 23-7.

THE ANALEMMA OF PTOLEMY 291or tan VG tanSCcoaSCV in the right-angled sphericaltriangle SVG.Thirdly,ci/ ET/ ci/ rv /v tv iJ^ ./^ \J J? A0\r 02 cos= 7TTo> TT/T/T/= tan t .that is, ^-^rrrr =9^"rrTr 7tan&lf am/Mfwhich is Menelaus, Sphcterica, ÎII. 3, applied to the right-angled spherical triangles ZBQ,MBS with the angle B common.Zeuthen points out that later in the same treatise Ptolemyfinds the arc 2oc described above the horizon by a star ofgiven declination 8', by a procedure equivalent to the formulacos a = tan 5' tan 0,and this is the same formula which, as we have seen,Hipparchus must in effect have used in his Commentary onthe Phaenomena of Eudoxus ami Aratus.Lastly, with regard to the calculations of the height h andthe azimuth CD in the general case where the sun's declinationis 8', Zeuthen has shown that they may be expressed by theformulaesin h = (cos 5' cos t sin #' tan )cos 0,;, ,cos 8' sin tand tan co= .-~,~~ >'+ (cos 8' cos t sin 8* tan

292 TRIGONOMETRYseeing that Diodorus wrote his Analemmtt in the next century.The other alternative source for Hipparchus's sphericaltrigonometry is the Menelaus-theorem applied to the sphere,on which alone Ptolemy, as we have seen, relies in hisSyntaxis. In any case the Table of Chords or Sines was infull use in Hipparchus's works, for it is presupposed by eithermethod.The Planisphaerium.With the Analemma of Ptolemyis associated anotherwork of somewhat similar content, the Planisphaerium.This again has only survived in a Latin translation from anArabic version made by one Maslama b. Ahmad al-Majiiti,Cordova (born probably at Madrid, died 1007/8) ;ofthe translationis now found to be, not by Rudolph of Bruges, but by'Hermannus Secundus', whose pupil Rudolph was; it wasfirst published at Basel in 1536, and again edited, with commentary,by Commandinus (Venice, 1558). It has 'beenre-edited from the manuscripts by Heiberg in vol. ii. of histext of Ptolemy. The book is an explanation of the systemof projection known as stereoyraphic, by which points on theheavenly sphere are represented on the plane of the equatorby projection from one point, a pole Ptolemy naturally takes;the south pole as centre of projection, as it is the northernhemisphere which he is concerned to represent on a plane.Ptolemy is aware that the projections of all circles on thesphere (great circles other than those through the poleswhich project into straight lines and small circles eitherparallel or not parallel to the equator) are likewise circles.It is curious, however, that he does not give any generalproof of the fact, but is content to prove it of particularcircles, such as the ecliptic, the horizon, &c. This is remarkable,because it is easy to show that, if a cone be describedwith the pole as vertex and passing through any circle on thesphere, i.e. a circular cone, in general oblique, with that circleas base, the section of the cone by the plane of the equatorsatisfies the criterion found for the ' subcontrary sections ' byApollonius at the beginning of his Conies, and is therefore acircle. The fact that the method of stereographic projection isso easily connected with the property of subcontrary sections

THE PLAN1SPHAERIUM OF PTOLEMY 293of oblique circular cones has led to the conjecture that Apolloniuswas the discoverer of the method. But Ptolemy makes nomention of Apollonius, and all that we know is that Syuesiusof Gyrene (a pupil of Hypatia, and born about A.D. 365-370)attributes the discovery of the method and its application tcHipparchusit is curious that he does not mention ;Ptolemy'streatise on the subject, but speaks of himself alone as havingperfected the theory. While Ptolemy is fully aware thatcircles on the sphere become circles in the projection, he saysnothing about the other characteristic of this method of projection,namely that the angles on the sphere are representedby equal angles 011 the projection.We must content ourselves with the shortest allusion toother works of Ptolemy. There are, in the first place, otherminor astronomical works as follows :(1) $acrei? a7rAai/a>*> aorrlpaw of which only Book II survives,(2) 'TnoOtaeis rS>v TrAayoo/iej/eoi/ in two Books, the firstof which is extant in Greek, the second in Arabic only, (3) theinscription in Canobus, (4) IIpo\*ip

294 TRIGONOMETRYon the mirror where the reflection takes place'; Ptolemy usesthe principle to solve various special cases of the followingproblem (depending in general on a biquadratic equation andnow known as the problem of Alhazen), *Given a reflectingsurface, the position of a luminous point, and the positionof a point through which the reflected ray is required to pass,to find the point on the mirror where the reflection will takeplace/ Book V is the most interesting, because it seems tobe the first attempt at a theory of refraction. It containsmany details of experiments with different media, air, glass,and water, and gives tables of angles of refraction (r) correspondingto different angles of incidence (i) these are calculatedon the supposition that r and i are connected by;anequation of the following form,r = aibi*,where a, 6 are constants, which is worth noting as the firstrecorded attempt to state a law of refraction.The discovery of Ptolemy's Optics in the Arabic at oncemade itclear that the work De S'peculis formerly attributedto Ptolemy is not his, and itis now practically certain that itis, at least in substance, by Heron. This is established partlyby internal evidence, e.g. the style and certain expressionsrecalling others which are found in the same author's Automataand Dloptra, and partly by a quotation by Damianus(0/6 hypotheses in Optics, chap. 14) of a proposition proved by'the mechanician Heron in his own Catoptrica ', which appearsin the work in question, but is not found in Ptolemy's Optics,or in Euclid's. The proposition in questionis to the effectthat of all broken straight lines from the eye to the mirrorand from that again to the object, that particular broken lineis shortest in which the two parts make equal angles with thesurface of the mirror; the inference is that, as nature doesnothing in vain, we must assume that, in reflection from amirror, the ray takes the shortest course, i.e. the angles ofincidence and reflection are equal. Except for the notice inDamianus and a fragment in Olympiodorus *containing theproof of the proposition, nothing remains of the Greek text ;1Olympiodorus on Aristotle, Meteor, iii. 2, ed. Ideler, ii, p. 96, ed.Stiive,pp.212. 5-213. 20.

THE OPTICS OF FTOLEMY 295but the translation into Latin (now included in the Teubneredition of Heron, ii, 1900, pp. 316-64), which was made byWilliam of Moerbeke in 1269, was evidently made from theGreek and not from the Arabic, as is shown by Graecisms inthe translation.A mechanical work, lie pi fon&v.There are allusions in 1Simplicius and elsewhere to a bookby Ptolemy of mechanical content, nepl pojrSiv, on balancingsor turnings of the scale, in which Ptolemy maintained asagainst Aristotle that air or water (e.g.) in their own 'placehave no weight, and, when they are in their own * place either',remain at rest or rotate simply, the tendency to go up or tofall down being due to the desire of things which are not intheir own places to move to them. Ptolemy went so far as tomaintain that a bottle full of air was not only not heavierthan the same bottle empty (as Aristotle held), but actuallylighter when inflated than when empty. The same work isapparently meant by the book on ' the elements ' mentionedby Simplicius. 2 Suidas attributes to Ptolemy three Books ofMechauica.Simplicius 3 also mentions a single book, Trepi &aoracrea>y,'On divieiisioii', i.e. dimensions, in which Ptolemy tried toshow that the possible number of dimensions is limited tothree.'Attempt to provethe Parallel-Postulate.Nor should we omit to notice Ptolemy's attempt to provethe Parallel-Postulate. Ptolemy devoted a tract to this4subject, and Proclus has given us the essentials of the argumentused. Ptolemy gives, first, a proof of Eucl. I. 28, andthen an attempted proof of I. 29, from which he deducesPostulate 5.Simplicius 011 Arist. De caelo, p. 710. 14, Heib. (Ptolemy, ed. Heib.,1vol. ii, p. 263).2Ib., p. 20. 10 sq.3/&., p. 9. 21 sq., .(Ptolemy, ed. Heib., vol. ii, p. 265).4Proclus on Eucl. I,vol. ii, pp. 266-70).pp. 362. Usq., 365. 7-367. 27 (Ptolemy, ed. Heib.,

296 TRIGONOMETRYI. To proveI. 28, Ptolemy takes two straight lines AB, CD,and a transversal EFGH. We have to prove that, if the sumC /G DHof the angles BFG, FGD is equal to two right angles, thestraight lines AB, CD are parallel, i.e. non-secant.Since AFG is the supplement of BFG, and FGC of FGD, itfollows that the sum of the angles AFG, FGC is also equal totwo right angles.Now suppose, if possible, that FB, GD, making the sum ofthe angles FG, FGD equal to two right angles, meet at K;then similarly FA, GG making the sum ofFGC equal to two right angles must also meet, saythe angles AFG,at L.[Ptolemy would have done better to point out that notonly are the two sums equal but the angles themselves areequal in pairs, i.e. AFG to FGD and FGC to BFG, and we cantherefore take the triangle KFG and applyit to FG on the otherside so that the sides FK, GK may lie along G(J, FA respectively,in which case GG, FA will meet at the point whereK falls.]Consequently the straight lines LABK, LCDK enclose aspace :which is impossible.It follows that AB, CD cannot meet in either direction ;they are therefore parallel.II. To prove I. 29, Ptolemy takes two parallel lines AB,CD and the transversal FG, and argues thus. It is requiredto prove thatZ AFG + Z GGF = two right angles.For, if the sum is not equal to two right angles, it must beeither (1) greater or (2)less.(1) If it is greater, the sum of the angles on the other side,BFG, FGD, which are the supplements of the first pair ofangles, must be less than two right angles.But AF, GG are no more parallel than FB, GD, so that,FG makes one pair of angles AFG, FGC togeilier greater thanif

PTOLEMY ON THE PARALLEL-POSTULATE 297two rigid angles,it must also make the other pair BFG, FGDtogether greater than two right angles.But the latter pair of angles were provedless than tworight angles which is :impossible.Therefore the sum of the angles AFG, FGC cannot begreater than two right angles.(2) Similarly we can show that the sum of the two anglesAFGy FGC cannot be less than two right angles.Therefore L AFG + Z CGF - two right angles.[The fallacy here lies in the inference which I have markedby italics. When Ptolemy says that Ab\ CG are no moreparallel than FB, GD, he is in effect assuming that throughany o)ie point only one pandlel can bedrawn toagiven straightline, which is an equivalent for the very Postulate he isendeavouring to prove. The alternative Postulate is knownas ' Playfair's axiom ',but it is of ancient origin, since it isdistinctly enunciated in Proclus's note on Eucl. I. 31.]III. Post. 5 is now deduced, thus.Suppose that the straight lines making with a transversalangles the sum of which is less than two right angles do notmeet on the side on which those angles are.Then, a fortiori, they will not meet on the other side onwhich are the angles the sum of whicli is greater than tworight angles. [This is enforced by a supplementary propositionshowing that, if the lines met on that side, Eucl. I. 16would be contradicted.]Hemse the straight lines cannot meet in either direction :they are therefore parallel.But in that case the angles made with the transversal arewhich contradicts the assumption.eyual to two right angles :Therefore the straight lines will meet.

XVIIIMENSURATION:HERON OF ALEXANDRIAControversies as toHeron's date.THE vexed question of Heron's date lias perhaps calledforth as much discussion as any doubtful point in the historyof mathematics. In the early stages of the controversy muchwas made of the supposed relation of Heron to Ctesibius.The Belopoewa of Heron has, in the best manuscript, theheading "JH/oau/oy Krrjo-ifiiov JBeAoTrouVca, and from this, coupledwith an expression used by an anonymous Byzantine writerof the tenth century, 6 'Ao-Kprjvbs KTrjo-ifiios 6 rov X\gav8p

CONTROVERSIES AS TO HERON'S DATE 299above quoted ;the title, however, in itself need not implymore than that Heron's work was a new edition of a similarwork by Ctesibius,and the Krrj(nftiov may even have been ad'dedby some well-read editor who knew both works and desired toindicate that the greater part of the contents of Heron's workwas due to Ctesibius. One manuscript has^Hpan/os *A\*av-5pcv kv KVK\O> tvQti&v). Now, althoughthe first beginnings of trigonometry may go back as far asApollonius, we know of no work giving an actual Table ofChords earlier than that of Hipparchus. We get, therefore,at once the date 150 B.C. or thereabouts as the terminus postquein. A terminus aide quernis furnished by the date of thecomposition of Pappus's Collect ion] for Pappus alludes to, anddraws upon, the works of Heron. As Pappus was writing inthe reign of Diocletian (A.D. 284-305),it follows that Heroncould not be much later than, say, A.D. 250. In speaking ofthe solutions by * the old geometers ' (ol naXcuol yeoyecrpa*) ofthe problem of finding the two mean proportionals, Pappus mayseem at first sight to include Heron along with Eratosthenes,Nicomedes and Philon in that designation, and it has beenargued, on this basis, that Heron lived long before Pappus.But a close examination of the passage1shows that this isby no means necessary. The relevant words are as follows :*The ancient geometers were not able to solve the problemof the two straight lines [the problem of finding two meanproportionals to them] by ordinary geometrical methods, sincethe problem is by nature '* solid " but . . .by attackingit withmechanical means they managed, in a wonderful way, toreduce the question to a practical and convenient construction,as may be seen in the Mesolaboii of Eratosthenes and in themechanics of Philon and Heron . . . Nicomedes also solved itby means of the cochloid curve, with which he also trisectedan angle.'1Pappus, iii, pp. 54-6.

300 HERON OF ALEXANDRIAPappus goes on to say that he will give four solutions, oneof which is his own ;the first, second, and third he describesas those of Eratosthenes, Nicomedes and Heron. But in theearlier sentence he mentions Philon along with Heron, and weknow from Eutocius that Heron's solution is practically thesame as Philon's. Hence we may conclude that by the thirdsolution Pappus really meant Philon's, and that he only mentionedHeron's Mechanics because it was a convenient place inwhich to find the same solution.Another argument has been based on the fact that theextracts from Heron's Mechanics given at the end of Pappus'sBook VIII, as we have it, are introduced by the author witha complaint that the copies of Heron's works in which hefound them were in many respects corrupt, having lost bothbeginning and end. 1 But the extracts appear to have beenadded, not by Pappus, but by some later writer, and theargument accordingly falls to the ground.The limits of date being then, say, 160 B.C. to A. D.250, ouronly course is to try to define, as well as possible, the relationin time between Heron and the other mathematicians whocome, roughly, within the same limits. This method has ledone of the most recent writers on the subject (Tittel-) toplace Heron jiot much later .than 100 B.C., while another/*relying almost entirely on a comparison between passages inPtolemy and Heron, arrives at the very different conclusionthat Heron was later than Ptolemy ^ncl belonged in fact tothe second centuryA. D.In view of the difference between these results, it will beconvenient to summarize the evidence relied on toestablishthe earlier date, and to consider how far it is or is not conclusiveagainst the later. We begin with the relation ofHeron to Philon. Philon is supposed to come not more thana generation later than Ctesibius, because it would appear thatmachines for throwing projectiles constructed by Ctesibiusand Philon respectively were both available at one time forinspection by experts on the subject 4 ;it is inferred thatPappus, viii, p. 1116. 4-7.12 Art. * Heron von Alexandreia ' in Pauly-Wissowa's JKeal-JKncyclop&dieder class. Altertumswisnenschaft, vol. 8. 1, 1912.8 I. Hammer-Jensen in Hermes, vol. 48, 1913, pp. 224-35.4Philon, Meek. Synt. iv, pp. 68. 1, 72. 86,

CONTROVERSIES AS TO HERON'S DATE3Q1Philon's date cannot be later than the end of the secondcentury B.C. (If Ctesibius flourished before 247 B.C. the argumentwould apparently suggest rather the beginning than theend of the second century.) Next, Heron is supposed to havebeen a younger contemporary of Philon, the grounds beingthe following. (1) Heron mentions a ' stationary-automaton 'representation by Philon of the Nauplius-story, 1 and this isidentified by Tittel with a representation of the same story bysome contemporary of Heron's (ol KaO 9 ^/xay 2 ).But a carefulperusal of the whole passage seems to me rather to suggestthat the latter representation was not Philon's, and thatPhilon was included by Heron among the 'ancient* automaton-makers,and not amonghis contemporaries. 3(2) Anotherargument adduced to show that Philon was contemporary1Heron, Autom., pp. 404. 11-408. 9. -/&., p. 412. 13.3The relevant remarks of Heron are as follows. (1) He says that hohas found no arrangements of * stationary automata 1 better or moreinstructive than those described by Philon of Byzantium (p. 404. 11).As an instance he mentions Philon's setting of the inNauplius-story,which he found everything good except two things (a) the mechanismfor the appearance of Athene, which was too difficult ^pyadc'o-rcpor),and(/>)the absence of an incident promised by Philon in his description,namely the falling of a thunderbolt on Ajax with a sound of thunderaccompanying it (pp. 404. 15-408. 9). This latter incident Heron couldnot find anywhere in Philon, though he had consulted a great numberof copies of his work. He continues (p. 408. 9-13) that we are not tosuppose that he isrunning down Philon or charging him with not beingcapable of carrying out what he promised. On the contrary, the omissionwas probably due to a slip of memory, for it is easy enough to makestage-thunder (he proceeds to show how to do it). But the rest ofPhilon's arrangements seemed to him satisfactory, and this, he says, is'why he has not ignored Philon's work : for I think that my readers willget the most benefit if they are shown, first what has been well said bythe ancients and then, separately from this, what the ancients overlookedor what in their work needed improvement ' (pp. 408. 22-410. 6). (2) Thenext chapter (pp. 410. 7-412. 2) explains generally the sort of thing theautomaton-picture has to show, and Heron says he will give one examplewhich he regards as the best. Then (3), after drawing a contrast between*the simpler pictures made by the ancients ',which involved three differentmovements only, and the contemporary (nl *#' fjp.as) representations ofinteresting stories by means of more numerous and varied movements(p. 412. 3-15), he proceeds to describe a setting of the Nauplius-story.This is the representation which Tittel identifies with Philon's. But itis to be observed that the description includes that of the episode of thethunderbolt striking Ajax (c.30, pp. 448. 1-452. 7) which Heron expresslysays that Philon omitted. Further, the mechanism for the appearanceof Athene described in c. 29 is clearly not Philon's 'more difficult'arrangement, but the simpler device described (pp. 404. 18-408. 5) aspossible and preferable to Philon's (cf. Heron, vol. i, ed. Schmidt, pp.Ixviii-lxix).

302 HERON OF ALEXANDRIAwith Heron is the fact that Philon has some criticisms ofdetails of construction of projectile-throwers which are foundin Heron, whence it is inferred that Philon had Heron's workspecifically in view. But if Heron's BeXorrouKa was based onthe work of Ctesibius, it is equally possible that Philon maybe referring to Ctesibius.A difficulty in the wayof the earlier date is the relation inwhich Heron stands to Posidonius. In Heron's Mechanics,i. 24, there is a 'definition of ' 'centre of gravity which isattributed by Heron to 'Posidonius a Stoic'. But this canhardly be Posidonius of Apamea, Cicero's teacher, because thenext sentence in Heron, stating a distinction drawn by Archimedesin connexion with this definition, seems to imply thatthe Posidonius referred to lived before Archimedes. But theDefinitions of Heron do contain definitions of geometricalProclus to Posidonius ofnotions which are put down byApamea or Rhodes, and, in particular, definitions of ' 'figureand of ' parallels'.Now Posidonius lived from 135 to 51 B.C.,and the supporters of the earlier date for Heron can onlysuggest that either Posidonius was not the first to give thesedefinitions, or alternatively, if he was, and ifthey wereincluded in Heron's Definitions by Heron himself and not bysome later editor, all that this obliges us to admit is thatHeron cannot have lived before the first centuryB. o.Again, if Heron lived at the beginning of the first centuryB.C., it is remarkable that he is nowhere mentioned byVitruvius. The De architecture* was apparently brought outin 14 B.C. and in the preface to Book VII Vitruvius givesa list of authorities on machinationes from whom he madeextracts. The list contains twelve names and has everyappearance of being scrupulously complete but, while it;includes Archytas (second), Archimedes (third), Ctesibius(fourth), and Philon of Byzantium (sixth), it does not mentionHeron. Nor is it possible to establish interdependencebetween Heron and Vitruvius ;the differences seem, on thewhole, to be more numerous than the resemblances. A few ofthe differences may be mentioned. Vitruvius uses 3 as thevalue of TT, whereas Heron always uses the Archimedean value3^. Both writers make extracts from the AristotelianMT]%aviKa TrpojSATj/xara,but their selections are different. The

CONTROVERSIES AS TO HERON'S DATE 303machines used by the two for the same purpose frequentlydiffer in details e. ; g. in Vitru vius's hodometer a pebble dropsinto a box at the end of each Roman 1mile, while in Heron'sthe distance completed is marked by a pointer. 2It is indeedpointed out that the water-organ of Heron is in many respectsmore primitive than that of Vitruvius; but, as the instrumentsare altogether different, this can scarcely be said toprove anything.On the other hand, there are points of contact betweencertain propositions of Heron and of the Roman agrimensores.Columella, about A.D. 62, gave certain measurements ofplane figures which agree with the formulae used by Heron,notably those for the equilateral triangle, the regular hexagon(in this case not only the formula but the actual figures agreewith Heron's) and {-he segment of a circle which is less thana semicircle, the formula in the last case beingwhere s is the chord and h the height of the segment.Herethere might seem to be dependence, one way or the other ;but the possibilityis not excluded that the two writers maymerely have drawn from a common source for; Heron, ingiving the formula for the area of the segment of a circle,states that it was the formula used by 'the more accurateinvestigators' (oi a/cpi/Seorepoj/ e^jrij/corer). 3We have, lastly, to consider the relation between Ptolemyand Heron. If Heron lived about 100 B.C., he was 200 yearsearlier than Ptolemy (A.D. 100178). The argument used toprove that Ptolemy came some time after Heron is based ona passage of Proclus where Ptolemyis said to have remarkedon the untrustworthiness of the method in vogue among the"more ancient 'writers of measuring the apparent diameter ofthe sun by means of water-clocks.4 Hipparchus, says Proclus,used his dioptra for the purpose, and Ptolemy followedhim. Proclus proceeds:'Let us then set out here not only the observations ofthe ancients but also the construction of the dioptra of134Vitruvius, x. 14. 2Heron, Dioptra, c. 34.Heron, Metrica, i. 31, p. 74. 21.120. 9-15, 124, 7-26.Proclus, Hypohtpo^ pp.

304 HEKON OF ALEXANDRIAAnd first we will show how we can measure anHipparchus.interval of time by means of the regular efflux of water,a procedure which was explained by Heron the mechanicianin his treatise on water-clocks.'Theon of Alexandria has a passage to a similar effect. 1firstsays that the most ancient mathematicians contriveda vessel which would let water flow out uniformly through asmall aperture at the bottom, and then adds at the end, almostin the same words as Proclus uses, that Heron showed howthis ismanaged in the first book of his work on wjiterclocks.Theon's account is from Pappus's Commentary onthe Syntaods, and this is also Proclus's source, as is shown bythe fact that Proclus gives a drawing of the water-clockwhich appears to have been lost in Theon Js transcription fromPappus, but which Pappus must have reproduced from thework of Heron. Tittel infers that Heron must have rankedas one of the 'more ancient' writers as compared withPtolemy. But this again does not seem to be a necessaryinference. No doubt Heron's work was a convenient place torefer to for a description of a water-clock, but it does notnecessarily follow that Ptolemy was referringHeto Heron'sclock rather than some earlier form of the same instrument.An entirely different conclusion from that of Tittel isreached in the article c Ptolemaios and Heron ' already alludedto. 2 The arguments are shortly these. (1) Ptolemy says inhis Geography (c. 3) that his predecessors had only been ableto measure the distance between two places (as an arc of agreat circle on the earth's circumference) in the case wherethe two places are on the same meridian. He claims that hehimself invented a way of doing this even in the case wherethe two places are neither on the same meridian nor on thesame parallel circle, provided that the heights of the pole at"the two places respectively, and the angle between the greatcircle passing through both and the meridian circle throughone of the places, are known. -Now Heron in his Dioptradeffls with the problem of measuring the distance betweentwo places by means of the dioptra, and takes as an example1Theon, Comm. on the Syntaxin, Basel, 1538, pp. 261 sq. (quoted inProclus, Hypotyposis, ed. Manitius, pp. 309-11).2Hammer-Jensen, op. dt.

CONTROVERSIES AS TO HERON'S DATE 305the distance between Rome and Alexandria. 1Unfortunatelythe text is in places corrupt and deficient, so that the methodcannot be reconstructed in detail. But it involved the observationof the same lunar eclipse at Rome and Alexandriarespectively and the drawing of the aiuilemma for Rome'.That is to say, the mathematical method which Ptolemyclaimsto have invented isspoken of by Heron as a thinggenerally known to experts and not more remarkable thanother technical matters dealt with in the same book. ConsequentlyHeron must have been later than Ptolemy. (It isright to add that some hold that the chapter of the Dioptrain question is not germane to the subject of the treatise, andwas probably not written by Heron but interpolated by somelater editor ;if this is so, the argument based uponthe ground.) (2) The dioptra described in Heron's work is ait falls tofine and accurate instrument, very much better than anythingPtolemy had at his disposal. If Ptolemy had been aware ofits existence, it is highly unlikely that ho would have taken''the trouble to make his separate and imperfect parallacticinstrument, since it could easily have been grafted on toHeron's dioptra. Not only, therefore, must Heron have beenlater than Ptolemy but, seeing that the technique of instrument-makinghad made such strides in the interval, he musthave been considerablylater.Ptolemy, as we have seen, disputedair has weight even when surrounded by(3) In his work ntpl poTrtov 2the view of Aristotle thatair. Aristotlesatisfied himself experimentally that a vessel full of air isheavier than the same vessel empty ; Ptolemy, also by experiment,convinced himself that the former is actually thelighter. Ptolemy then extended his argument to water, andheld that water with water round it has no weight, and thatthe diver, however deep he dives, does not feel the weight of:>>the water above him. Heron asserts that water has noappreciable weight and has no appreciable power of compressingthe ail' in a vessel inverted and forced clown intothe water. In confirmation of this he cites the case of thedriver, who is not prevented from breathing when far below1Heron, Dioptra, c. 35 (vol. iii, pp. 302-6).2Simplicius on De caelo, p. 710. 14, Heib. (Ptolemy, vol. ii, p. 263).3Heron, Pneumatica, i. Pref. (vol. i, p. 22. 14 sq.).W3.2 X

306 HERON OF ALEXANDRIAthe surfo.ee. He then inquires what is the reason why thediver is not oppressed ^ though he has an unlimited weight ofwater on his back. He accepts, therefore, the view of Ptolemyas to the fact, however strange this may seem. But he is notsatisfied with the explanation given:''Some say',he goes on,it is because water in itself is uniformly heavy (/cro/Sapcy avrb'KaO* ai)ro) this seems to be equivalent to Ptolemy's dictum*that water in water has no weight but they give no explanationwhatever why divers . . .' He himself attempts anexplanation based on Archimedes. It is suggested, therefore,that Heron's criticism is directed specifically against Ptolemyand no one else. (4) It is suggested that the Dionysius to whomHeron dedicated his Definitionsis a certain Dionysius whowas praefectus urbi at Rome in A.D. 301. The grounds arethese (a) Heron addresses Dionysius as AIOVVCTM Aa/nrporare,where Aa/nr/>6raroy obviously corresponds to the Latin clarissimus,a title which in the third century and under Diocletianwas not yet in common use. Further, 'this Dionysius wascuratoi* aquarnm and curator operum publicorwni,so that hewas the sort of person who would have to do with theengineers, architects and craftsmen for whom Heron wrote.Lastly, he is mentioned in an inscription commemorating animprovement of water supply and dedicated ' to Tiberinus,father of all waters, and to the ancient inventors of marvellousconstructions 1(repertoribus admirabilium fabricawimprecis viris), an expression which is not found in any otherinscription, but which recalls the sort of tributq that Heronfrequently pays to his predecessors. This identification of thetwo persons named Dionysiusis an ingenious conjecture, butthe evidence is not such as to make itanything more. 1The result of the whole investigation just summarized is toplace Heron in the third century A.D., and perhaps little, ifanything, earlier than Pappus. Heiberg accepts this conclusion,2 which may therefore, I sup'pose, be said to hold the fieldfo* the present.1Dionysius was of course a very common name. Diophantus dedicatedhis Arithmetica to a person of this name VTifud>TaT /ioi Atoi/ucm), whom hepraised for his ambition to learn the solutions of arithmetical problems.This Dionysius must have lived in the second half of the third centuryA.D., and if Heron also belonged to this time, is it not possible that.Heron's Dioriysius was the same person ?2Heron, vol. v fp. ix.

CONTROVERSIES AS TO HERON'S DATE 307Heron was known as 6 'AXtgavSpevs (e.g. by Pappus) or6 priyaviKos (mechanicus), to distinguish him from otherpersons of the same name Proclus and Damianus use the;latter title, while Pappus also speaks of ol nepl TOV "HpcovaCharacter of works.Heron was an almost encyclopaedic writer on mathematicaland physical subjects. Practical utility rather than theoreticalcompleteness was the object aimed at ;his environment inEgypt no doubt accounts largely for this. His Metrica beginswith the old legend of the traditional origin of geometry inEgypt, and in the Dioptra we find one of the very problemswhich geometry was intended to solve, namely that of reestablishingIxmndaries of lands when the flooding of theNile had destroyed the land-marks: When ' the boundariesof an area have become obliterated to such an extent thatonly two or three marks remain, in addition to a plan of thearea, to supply afresh the remaining marks/ 1 Heron makeslittle or no claim to originality ;he often quotes authorities,but, in accordance with Greek practice, he more frequentlyomits to do so, evidently without any idea of misleading anyone; only when he has made what is in his opinion anyslight improvement on the methods of his predecessors doeshe trouble to mention the fact, a habit which clearly indicatesthat, except in these cases, he issimply giving the besttraditional methods in the form which seemed to him easiestof comprehension and application.The Melrica seems to berichest in definite references to the discoveries of predecessors;the names mentioned are Archimedes, Dionysodorus,Eudoxus, Plato ; in the Dioptra Eratosthenes is quoted, andin the introduction to the (fatoptricu Plato and Aristotle arementioned.The practical utility of Heron's manuals being so great, itwas natural that they should have great vogue, and equallynatural that the most popular of them at any rate should bere-edited, altered and added to by later writers; this wasinevitable with books which, like the Elements of Euclid,were in regular use in Greek, Byzantine, Roman, and ArabianJHeron, Dioptra, c. 25, p. 268. 17-19.X 'i

808 HERON OF ALEXANDRIAeducation for centuries. The geometrical or mensurationalbooks in particular gave scope for expansion by multiplicationof examples, so that it is difficult to disentangle the genuineHeron from the rest of the collections which have come downto us under his name. Hultsch's considered criterion is asfollows: 'The Heron texts which have come down to ourtime are authentic in so far as they bear the author's nameand have kept the original design and form of Heron's works,but are unauthentic in so far as, being constantly in use forpractical purposes, they were repeatedly re-edited and, in thecourse of re-editing, were rewritten with a view to theparticularneeds of the time.'Listof Treatises.Such of the works of Heron as have survived have reachedus in very different ways. Those which have come down inthe Greek are :I. The Metrica, first discovered in 1896 in a manuscriptof the eleventh (or twelfth) century at Constantinople byR. Scheme and edited by his son, H. Schone (Heronis Opera, iii,Teubner, 1903).II. On the Dioptra, edited in an Italian version by Venturiin 1814 ;the Greek text was first brought out by A. J. H.Vincent 1 in 1858, and the critical edition of itby H. Scheme isincluded in the Teubner vol. iii just mentioned.III. The Pneumatica, in two Books, which appeared first ina Latin translation by Commandinus, published after hisdeath in 1575; the Greek text was first edited by Th^venotin Veterum mathematicorum opera Graece et Latitte cdita(Paris, 1693), and is now available in Iferonis Opera,i (Teubner,1899), by W. Schmidt.IV. On the art of constructing automata (irpt avroâro"TTOtijrjjcijy), or The automaton-theatre, first edited in an Italiantranslation by B. Baldi in 1589 the Greek text was; includedin Th^venot's Vet. math., and now forms part of HeroiiisOpera, vol. i,by W. Schmidt.V. Belopoewa (on the construction of engines of war), edited1Notices et extraits des manuscrits de la Bibliothtque impMale, xix, Dt. 2.pp. 157-337.P

LIST OF TREATISES 309by B. Baldi(Augsburg, 1616), TWvenot (Vet. math.), Kochlyand Rustow (1853) and by Wescher (Poliorc&tique dcs Grecs,1867, the first critical edition).VI. The Cheirobalistra ("Hpawo? x LPâ^crT P a ^ wravKevt}Kal a-vfifjiTpia edited (?)), by V. Prou, Notices et extraits> xxvi. 2(Paris, 1877).VII. The geometrical works, Defmitiones, Geometria, Geodaesia,Stereometrica I and 11, Meumrae, Liber Geeponicu8>edited by Hultsch with Variae collectioues (Hero MB Alexandrini(jeometriourum et stereometricortim relicfuiae, 1864).This edition will now be replaced by that of Heiberg in theTeubner collection (vols. iv, v),which contains much additionalmatter from the Constantinople manuscript referred to,but omits the Liber Geeponicus (except a few extracts) and theGeodaeîa (which contains only a few extracts from theGeometry of Heron).Only fragments survive of the Greek text of the Mechanicsin three Books, which, however, is extant in the Arabic (nowedited, with German translation, in lleroius Opera, vol. ii,by L. Nix and W. Schmidt, Teubner, 1901).A smaller separate mechanical treatise, the Bapov\Ko$, isquoted by Pappus. 1 'The object of it was to move a givenweight by means of a given force ',and the machine consistedof an arrangement of interacting toothed wheels with differentdiameters.At the end of the Dioptrais a description of a hodometer formeasuring distances traversed by a wheeled vehicle, a kind oftaxameter, likewise made of a combination of toothed wheels.A work on Water-clock* (nept vSpicw copocr/eoTmW) is mentionedin the Pneiimatica as having contained four Books,and is also alluded to by Pappus. 2 Fragments are preservedin Proclus (Hypotyposix, chap. 4) and in Pappus's commentaryon Book V of Ptolemy's tiyntaxis reproduced by Theon.Of Heron's Commentary on Euclid's Elements only verymeagre fragments survive in Greek (Proclus), but a largenumber of extracts are fortunately preserved in the Arabiccommentary of un-Nairizi, edited (1) in the Latin version ofGherard of Cremona by Curtze (Teubner, 1899), and (2) by1Pappus, viii, p. 1060. 5* 3/&., p. 1026. 1.

310 HERON OF ALEXANDRIABesthorn and Heiberg (Codex Leidensis 399. 1, five parts ofwhich had appeared up to 1910). The commentary extendedas far as Elem. VIII. 27 at least.The Catoptrica, as above remarked under Ptolemy, exists ina Latin translation from the Greek, presumed to be by Williamof Moerbeke, and is included in vol. ii of Heronis Opera,edited, with introduction, by W. Schmidt.Nothing is known of the Camarica (' on vaultings ')mentionedby Eutocius (on Archimedes, Sphere and Cylinder), theZygia (balancings) associated by Pappus with the Automata, 1or of a work on the use of the astrolabe mentioned in theFihrist.We are in this work concerned with the treatises of mathematicalcontent, and therefore can leave out of account suchworks as the Pneumatica, the Automata, and the Belopoeica.The Pneumatica and Automata have, however, an interest tothe historian of physics in so far as they employ the force ofcompressed air, water, or steam. In the P neumatica the'reader will find such things as siphons, Heron's fountain ','penny -in-the-slot 'machines, a fire-engine, a water-organ, andmany arrangements employing the force of steam.Geometry.(a) Commentary OR Euclid's Elements.In giving an account of the geometryand mensuration(or geodesy) of Heron it will be well, I think, to beginwith what relates to the elements, and first the Commentaryon Euclid's Elements, of which we possess a numberof extracts in an-Nairizi and Proclus, enabling us to forma general idea of the character of the work. Speakinggenerally, Heron's comments do not appear to have containedmuch that can be called important. They may be classifiedas follows :(1) A few general notes, e.g. that Heron would not admitmore than three axioms.(2) Distinctions of a number of particularcoxes of Euclid'spropositions according as the figure is drawn in one wayor another.1Pappus, viii, p. 1024. 28.

GEOMETRY 311Of this class are the different cases of I. 35, 36, III. 7, 8(where the chords to be compared are drawn on different sidesof the diameter instead of on the same side), III. 12 (which isnot Euclid's at all but Heron's own, addingexternal to that of internal contact in III.the case of11*, VI. 19 (wherethe triangle in which an additional line is drawn is taken tobe the smaller of the two), VII. 19 (where the particular caseis given of three numbers in continued proportionfour proportionals).instead of(3) Alternative proofs.It appears to be Heron who first introduced the easy butuninstructive semi-algebraical method of proving the propositionsII. 2-10 which is now so popular. On this method thepropositions are proved without ''figures as consequences ofII. 1corresponding to the algebraical formulaa (b + c + d + = . ..)ab + oc + ad + .Heron explains that it is not possible to prove II. 1 withoutdrawing a number of lines (i.e. without actually drawing thercctanglcH), but that the following propositions up to II. 10can be proved by merely drawing one line. He distinguishestwo varieties of the method, one by dissolutio, the other bycomposltio, by which he seems to mean splittiiig-u-p of rectanglesand squares and combination, of them into others.But in his proofs he sometimes combines the two varieties.Alternative proofs are given () of some propositions ofBook III, namely III. 25 (placed after III. 30 and startingfrom the arc instead of the chord), III. 10 (proved by meansof III. 9), III. 13 (a proof preceded by a lemma to the effectthat a straight line cannot meet a circle in more than twopoints).A class of alternative proof is (6) that which is intended tomeet a particular objection (eixrrao-is) which had been or mightbe raised to Euclid's constructions. Thus in certain casesHeron avoids product ny a certain straight line, where Euclidproduces it, the object being to meet the objection of one whoshould deny our right to assume that there is any spaceavailable. Of this class are his proofs of I. 11, 20 and hisnote on 1. 16.Similarly in I. 48 lie supposes the right-angled. .

312 HERON OF ALEXANDRIAtriangle which is constructed to be constructed on the sameside of the common side as the given triangleis.A third class (c)is that which avoids reciuct'io ad ab&ardum,e.g. a direct proof of I. 19 (for which he requires and givesa preliminary lemma) and of I. 25.(4) Heron supplies certain converses of Euclid's propositionse.g. of II. 12, 13 and VIII. 27.(5)A few additions to, and extensions of, Euclid's propositionsare also found. Some are unimportant, e. g, the constructionof isosceles and scalene trianglesin a note on 1. 1 and theconstruction of two tangents in III. 17. The most importantextension is that of III. 20 to the case where the angle at thecircumference is greater than a right angle, which gives aneasy way of proving the theorem of III. 22. Interesting alsoare the notes on I. 37 (on I. 24 in Proclus), where Heronproves that two triangles with two sides of the one equalto two sides of the other and with the included angles swpplementcM'yare equal in area, and compares the areas where thesum of the included angles (one being supposed greater thanthe other) is less or greater than two right angles, and on I. 47,where there is a proof (depending on preliminary lemmas) ofthe fact that, in the figure of Euclid's proposition ^see nextpage), the straight lines AL, BG, CE meet in a point. Thislast proof is worth giving. First come the lemmas.(1) If in a triangle ABG a straight line I)E be drawnparallel to the base BG cutting the sides AB, AC or thosesides produced in D, E, and if F be the8 L F M Cmiddle point of BCythen the straight lineAF (produced if necessary) will also bisectDE.(UK is drawn through A parallel toDE, and HDL, KEM through J), E parallelto AF meeting the base in L, M respec-tively. Then the triangles ABF, AFGbetween the same parallels are equal. So are the trianglesDBF, EFC. Therefore the differences, the triangles ADF,are equal and so therefore are the parallelograms HF,AEFtKF. Therefore LF'= or FM9 DG = GE.)(2) is the converse of Eucl. 1. 43. If a parallelogram is

GEOMETRY 313cut into four others ADGE, DF, FGCB, CE, so that DF, CEare equal, the common vertex G will lie on the diagonal AB.Heron produces AG to meet OF in H, and then proves thatAHB is a straight line.Since DF, CE are equal, so arethe triangles DGF, ECG. A eldingthe triangle GCF, we have thetriangles EOF, DCF equal, andDE, OF are parallel.But (byI. 34, 29, 26)the trianglesAKE, GKJ) are congruent,HO that EK=KD\ and by lemma (1)Now, init follows that CH=HF.the triangles F11B, CHG, two sides (BF, FH andGO, Gil) and the included angles are equal;therefore thetriangles are congruent, and the angles BllF, G11C are equal.Add to each the angle GHF, andZ B1IF+ L FUG = Z CHG + Z GHF = two right angles.To prove his substantive proposition Heron draws AKLperpendicular to J3G, and joins JKO meeting AK in M. Thenwe have only to prove that liMG is a straight line.Complete the parallelogram FAllO* and draw the diagonalsOA, FH meeting in F. Through M draw PQ, UR parallelrespectively to BA, AG.

314 HERON OF ALEXANDRIANow the triangles FAH, BAG are equal in all respects ;LEFA == LAEG= L CAK (since AK is at right angles to BC).thereforeBut, the diagonals of the rectangle FH cutting one anotherin F, we have FY = YA and LHFA = ZO^l^;therefore LOAF LCAK, and OJ. is in a straight linewith AKL.Therefore, OM being the diagonal of SQ ywe add AM to each, FM = J/fT.Also, since J

THE DEFINITIONS 3154concave' and 'convex', lune, garland (these last two arecomposite of homogeneous parts) and axe (mXtKvs), bounded byfour circular arcs, two concave and two convex, Defs. 27-38.Rectilineal figures follow, the various kinds of triangles andof quadrilaterals, the gnomon in a parallelogram, and thegnomon in the more general sense of the figure which addedto a given figure makes the whole into a similar figure,polygons, the parts of figures (side, diagonal, height of awhich willtriangle), perpendicular, parallels, the three figuresfillup the space round a point, Defs. 39-73. Solid figures arenext classified according to the surfaces bounding them, andlines on surfaces are divided into (1) simple and circular,(2) mixed, like the conic and spiric curves, Defs. 74, 75. Thesphere is then defined, with its parts, and stated to bethe figure which, of all figures having the same surface, is thegreatest in content, Defs. 76-82. Next the cone, its differentspecies and its parts are taken up, with the distinctionbetween the three conies, the section of the acute-angled cone(' by some also called ellipse ')and the sections of the rightangledand obtuse-angled cones (also called parabola andhyperbola), Defs. 83-94; the cylinder, a section in general,the spire or tore in its three varieties, open, continuous (orjust closed) and ' crossing-itself which ', respectively have''sections possessing special properties, square rings whichare cut out of cylinders (i.e. presumably rings the cross-sectionof which through the centre is two squares), and various otherfigures cut out of spheres or mixed surfaces, Defs. 95-7 ;rectilineal solid figures, pyramids, the five regular solids, thesemi-regular solids of Archimedes two of which (each withfourteen faces) were known to Plato, Defs. 98-104; prismsof different kinds, parallelepipeds, with the special varieties,the cube, the beam, SOKO? (length longer than breadth anddepth, which may be equal), the brick, Tr\iv6i$ (length lessthan breadth and depth), the (Woy or jSoo/zicr/coywithlength, breadth and depth unequal,Defs. 105-14.Lastly come definitions of relations, equality of lines, sur-cfaces, and solids respectively, similarity of figures, reciprocalfigures', Defs. 115-18; indefinite increase in magnitude,parts (which must be, homogeneous with the wholes, so thate. g. the horn-like angleis not a part or submultiple of a right

316 HERON OF ALEXANDRIAor any angle), multiples, Defs. 119-21 ; proportion in magnitudes,what magnitudes can have a ratio to one another,magnitudes in the same ratio or magnitudes in proportion,definition of greater ratio, Defs. 122-5; transformation ofratios (coniponendo, separando, convertendo, alternando, invertendoand ex aequali), Defs. 126-7; commensurable andincommensurable magnitudes and straight lines, Defs. 128,129. There follow two tables of measures, Defs, 130-2.The Definitions are very valuable from the point of view ofthe historian of mathematics, for they give the different alternativedefinitions of the fundamental conceptions; thus wefind the Archimedean 'definition' of a straight line, otherApollonius,others from Posidonius, and so on. No doubt thesome editor or editorsdefinitions which we know from Proclus to be due tocollection may have been recast byafter Heron's time, but it seems, at least in substance, to goback to Heron or earlier still. So far as it contains originaldefinitions of Posidonius, it cannot have been compiled earlierthan the first century B.C.; but its content seems to belong inthe main to the period before the Christian era. Heibergadds to his edition of the Definitions extracts from Heron'sGeometry, postulates and axioms from Euclid, extracts fromGeminus on the classification of mathematics, the principlesof geometry, &c., extracts from Proclus or some early collectionof scholia on Euclid, and extracts from Anatolius andTheon of Smyrna, which followed the actual definitions in themanuscripts. These various additions were apparently collectedby some Byzantine editor, perhaps of the eleventh century.Mensuration.The Metrica, Geometrica, Stereometrua, Geodaesia,Mensurae.We now come to the mensuration of Heron. Of thedifferent works under this head the Metr'uu is the mostimportant from our point of view because it seems, more thanany of the others, to have preserved its original form. It isalso more fundamental in that it gives the theoretical basis ofthe formulae used, and is not a mere application of rules toparticular examples.It is also more akin to theory in that it

MENSURATION 317does not use concrete measures, but simple numbers or unitswhich may then in particular cases be taken to be i eet, cubits,or any other unit of measurement. Up to 1896, when amanuscript of it was discovered by R. Schone at Constantinople,it was only known by an allusion to it in Eutocius(on Archimedes's Measurement of a Circle), who statesthe way to obtain an approximation to the squarethatroot ofa non-square number is shown by Heron in his Metrica, aswell as by Pappus, Theon, and others who had commented onthe Syntaxis of Ptolemy. 1 Tannery 2 had already in 1894discovered a fragment of Heron's Metrica giving the particularrule in a Paris manuscript of the thirteenth century containingProlegomena to the Syntaxis compiled presumably fromthe commentaries of Pappus and Theon. Another interestingdifference between the Metrica and the other works is that inthe former the Greek way of writing fractions (whichis ourmethod) largely preponderates, the Egyptian form (whichexpresses a fraction as the sum of diminishing submultiples)being used comparatively rarely, whereas the reverse is thecase in the other works.In view of the greater authority of the Metrica, we shalltake it as the basis of our account of the mensuration, whilekeeping the other works in view. It is desirable at theoutset to compare broadly the contents of the various collections.Book I of the Metrica contains the mensuration ofsquares, rectangles and triangles (chaps. 1-9), parallel-trapezia,rhombi, rhomboids and quadrilaterals with one angle right(10-16), regular polygons from the equilateral triangle to theregular dodecagon (17-25), a ring between two concentriccircles (26), segments of circles (27-33), an ellipse (34), a parabolicsegment (35), the surfaces of a cylinder (36), an isoscelescone (37), a sphere (38)and a segment of a sphere (39).Book II gives the mensuration of certain solids, the solidcontent of a cone (chap. 1),a cylinder (2), rectilinear solidfigures, a parallelepiped, a prism, a pyramid and a frustum,&c. (3-8), a frustum of a cone (9, 10), a sphere and a segmentof a sphere (11, 12), a vpire or tore (13), the section of acylinder measured in Archimedes's Method (14),and the solid12Archimedes, vol. iii, p. 232. 1S-17.Tannery, M&noires scientijiques, ii, 1912, pp. 447-54.

318 HERON OF ALEXANDRIAformed by the intersection of two cylinders with axes at rightangles inscribed in a cube, also measured in the Method (15),the five regular solids (16-19). Book III deals with the divisionof figures into parts having given ratios to one another,first plane figures (1-19), then solids, a pyramid, a cone and afrustum, a sphere (20-3).The Geometria or Geometrumena is a collection based uponHeron, but not his work in its present form. The addition ofa theorem due to Patrjcius * and a reference to him in theStereometric^ (1. 22) suggest that Patricias edited both works,but the date of Patricius is uncertain. Tannery identifieshim with a mathematical professor of the tenth century,Nicephorus Patricius if this is; correct, he would be contemporarywith the Byzantine writer (erroneously called Heron)who is known to have edited genuine works of Heron, andindeed Patricius and the anonymous Byzantine might be oneand the same person. The mensuration in the Geometry hasreference almost entirely to the same figures as thosemeasured in Book I of the Metrica, the difference being thatin the Geometry (1) the rules are not explained but merelyapplied to examples, (2) a large number of numerical illustrationsare given for each figure, (3) the Egyptian way ofwriting fractions as the sum of submultiples is followed,(4) lengths and areas are given in terms of particularmeasures, and the calculations are lengthened by a considerableamount of conversion from one measure into another.The first chapters (1-4) are of the nature of a general introduction,including certain definitions and ending with a tableof measures. Chaps. 6-99, Hultsch (= 5-20, 14, Heib.), thoughfor the most part corresponding in content to Metrica I,seem to have been based on a different collection, becausechaps. 100-3 and 105 (= 21, 1-25, 22, 3-24, Heib.) are clearlymodelled on the Metrica^ arid 101 is headed 'A definition(really measurement ' ')of a circle in another book of Heron '.Heiberg transfers to the Geometrica a considerable amount ofthe content of the so-called Liber Geeponicus, a badly orderedcollection consisting to a large extent of extracts from theother works. Thus it begins with 41 definitions identicalwith the same number of the Definitiovies. Some sections1Geometrica, 21 26 (vol. iv, p. 386. 23).

MENSURATION 319Heiberg puts side by side with corresponding sections of theGeometrica in parallel columns ;others he inserts in suitableplaces ; sections 78. 79 contain two important problems inindeterminate analysis (= Geom. 24, 1-2, Heib.). Heibergadds, from the Constantinople manuscript containing theMetrica, eleven more sections (chap. 24, 3-13) containingindeterminate problems, and other sections (chap. 24, 14-30 and37-51 ) giving the mensuration, mainly, of figures inscribed in orcircumscribed to others, e.g. squares or circles in triangles,circles in squares, circles about triangles, and lastly of circlesand segments of circles.The titereometrica I has at the beginning the title EicraycoyatrS>v orepo//rpoi;/*i>a>i> "Hpcoj/o? but, like the Geometrica,seems to have been edited by Patricius. Chaps. 1-40 give themensuration of the geometrical solid figures, the sphere, thecone, the frustum of a cone, the obelisk with circular base,the cylinder, the 'pillar', the cube, the crfyrjvicrKo? (also calledowg), the fjittovpov 7rpoeavcapt0i7io/oi>, pyramids, and frusta.Some portions of this section of the book go back to Heron ;thus in the measurement of the sphere chap.1 = MetricaII. 11, and both here and elsewhere the ordinary form offractions appears. Chaps. 41-54 measure the contents of certainbuildings or other constructions, e.g. a theatre, an amphitheatre,a swimming-bath, a well, a ship, a wine-butt, andthe like.The second collection, titereometrica II, appears to be ofByzantine origin and contains similar matter to Stereometrica I,parts of which are here repeated. Chap. 31 (27, Heib.) givesthe problem of Thales. to find the height of a pillar or a treeby the measurement of shadows the last sections; measurevarious pyramids, a prism, a fi&pfcKos (little altar).The Geodaesia is nob an independent work, but only containsextracts from the Geometry ;thus chaps. 116 = Geom.5-31, Hultsch (= 5, 2-12, 32, Heib.); chaps. 17-19 give themethods of finding, in any scalene triangle the sides of whichare given, the segments of the base made by the perpendicularfrom the vertex, and of finding the area direct by the wellknown'Metrica I. 5-8.formula of Heron ' ;i.e. we have here the equivalent ofLastly, the /zerpqcmy, or Mensurae, was attributed to Heron

320 HERON OF ALEXANDRIAin an Archimedes manuscript of tho ninth century, but cannotin its present form be due to Heron, although portions ofit have points of contact with the genuine works. Sects. 2-27measure all sorts of objects, e.g. stones of different shapes,a pillar,a tower, a theatre, a ship, a vault, a hippodrome but;sects. 28-35 measure geometrical figures, a circle and segmentsof a circle (cf.Metrica I), and sects. 36-48 on spheres, segmentsof spheres, pyramids, cones and frusta are closely connectedwith Stereom. I and Meti4ca II ;sects. 49-59, giving the mensurationof receptacles and plane figures of various shapes,seem to have a different origin. We can now take up theContents of the Metrics.Book I. Measurement of Areas.The preface records the tradition that the firstgeometryarose out of the practical necessity of measuring and distributingland '(whence the name geometry after which'),extension to three dimensions became necessary in order tomeasure solid bodies. Heron then mentions P]udoxus andArchimedes as pioneers in the discovery of difficult measurements,Eudoxus having been the first to prove that a cylinderis three times the cone on the same base and of equal height,and that circles are to one another as the squares on theirdiameters, while Archimedes first proved that the surface ofa sphere is equal to four times the area of a great circle in it,and the volume two-thirds of the cylinder circumscribingit.(a) Area of scale tie trianyle.After the easy cases of the rectangle, the right-angledtriangle and the isosceles triangle, Heron gives two methodsof finding the area of a scalene triangle (acute-angled orobtuse-angled) when the lengths of the three sides are given.The first method is based on Eucl. II. 12 and 13. If ct, b, cbe the sides of the triangle opposite to the angles A, Jl, Crespectively, Heron observes (chap. 4) that any angle, e.g. C, isacute, right or obtuse according as c2 < = or > a2 + // 2 ,and thisis< the criterion determining which of the two propositions isapplicable. The method is directed to determining,first thesegments into which any side is divided by the perpendicular

AREA OF SCALENE TRIANGLE 321from the opposite vertex, and thence the length of the perpendicularWe itself. have, in the cases of the triangle acuteangledat C and the triangle obtuse-angled at C respectively,or 67) =whence AD* (= b* CD*) is found, so that we know the areaIn the cases given in Melrica I. 5, the sides are (14, 15, 13)and (11, 13, 20) respectively, and AD is found to be rational(=12). But of course both CD (or HD) and AD may be surds,in which case Heron gives approximate values. Cf. Geom.53, 54, Hultsch (15, 1-4, Heib.) 3where we have a trianglein which a = 8, b = 4, c = 6, so that a 2 + '> 8 fl 2 = 44 andC7) = 44/10 = 2J. Thus 4/) 2 = 16-(2i)a = 16-7^= 8J I r^, and .47)= \/(8 f T^) = 2 approximately, whencethe area = 4x2|J=llj|.Heron then observes that we geta nearer result still if we multiply AD 2 by (|a) 2 beforeextracting the square root, for the area is then \/(16 x 8| ^)or \/(135), which is very nearly 11| -^ ^T or ll^f-So in Metrlca I. 9, where the triangle is 10, 8, 12 (10 beingthe base), Heron finds the perpendicular to be \/63, but heobtains the area as .\/(\AD" #(*). or \/(1575), while observingthat we can, of course, take the approximation to \/63, or1^ as7 I iV anc' multiply it by half 10. obtaining 39| | ythe area.Proof of the formula A = \/{ t s(s a) (s b) (s c)}.The second method is that known as the * formula ofHeron ', namely, in our notation, A = \/{s (s a) (s b) (s c) }.The proof of the formula is given in Metrlca I. 8 and also in1B23.2 Y

322 HERON OF ALEXANDRIAchap. 30 of the Dioptra ; but it is now known (from Arabiansources) that the propositionis due to Archimedes.Let the sides of the triangle ABC be given in length.Inscribe the circle DEF, and letJoin AO, KG, CO, DO, EG, FO.Then BG . 01) = 2 A 50(7,CA.GE= 2&COA,AB.GF=2AAOB;be the centre.whence, by addition,where p is the perimeter.Produce CB to H, so that BU = AF.Then, since AE = AF, BF = BD, and CE = CD, we haveThereforeCH.OD = A ABC.But CH.OD is the 'side' of the product C1I*.OD 2 ,i.e.so that. OD*),

PROOF OF THE'FORMULA OF HERON' 323Draw OL at right angles to 00 cutting BO in K9Join OL.and BL atright angles to BO meeting OL in L.Then, since each of the angles COL, OBL is right,COBL isa quadrilateral in a circle.Therefore L GOB + L CLB = 2 R.But /.COB + AOF = 27?, because AO, BO, 00 bisect theangles round 0, and the angles 00 B, AOF are together equalto the angles AOC, BOF, while the sum of all four anglesis equalto 4 R.ConsequentlyL AOF = Z OLB.Therefore the right-angled triangles AOF, OLB are similar ;therefore BO :and, alternately, CB :whence, componendo,:It follows thatOil* : OH.HB = BD . DOBL = AF: FO= Bll :07),Bit = BL : OD= BK : KDOH: HB = BD DK.:C7) . DK;= 7*7) . 7)

324 HERON OF ALEXANDRIASince', says Heron, 1 c 720 has not its side rational, we can'obtain its side within a very small difference as follows. Sincethe next succeeding square number is 729, which has 27 forits side, divide 720 by 27. This gives 26. Add 27 to this.The side of 720making 53f ,and take half of this or 26^.will therefore be very nearly 26^ ^. In fact, if we multiply26^^ by itself the product is 720^, so that the difference (in,the square)is -fe.1If we desire to make the difference still smaller than 3^, werather we should takeshall take 720^ instead of 729 [or26| instead of 27],and by proceeding in the same way weshall find that the resulting difference is much less than ^/In other words, if we have a non-square number A, and 2is the nearest square number to it,so that A = f/ 2 + />,have, as the first approximationto \/A.then wefor a second approximationwe take= a(i+-)'< 2 >and so on. 21Metrica, i. 8, pp. 18. 22-20. 5.2The method indicated by Heron was known to Barlaam and NicolasRhabdas in the fourteenth century. The equivalent of it was used byLuca Paciuolo (fifteenth-sixteenth century), and it was known to the otherItalian algebraists of the sixteenth century. Thus Luca Paciuolo gave2J, 2^ and 2ftffa as successive approximations to -/$ He obtained2 (2 1 ) 2 6the first as 2+ 5 o the second as 2* --^,It . Ct Z . Z.,and the third as**~ "' The above rule Kives -H2+ i)= 25.*(5+7> ! )= 2 5 )>The formula of Heron was again put forward, in modern times, byiBuzengeiger as a im ans of accounting for the Archimedean approximationto v3, apparently without knowing its previous history. 13ertrandalso stated it in a treatise on arithmetic (1853). The method, too, bywhich Oppermann and Alexeieff sought to account for Archimedea'sapproximations is in reality the same. The latter method depends onthe formulaAlexeieff separated A into two factors o ,7',and pointed out that if, say.>Z> .then, l(a.+ Jb)>v^> or

APPROXIMATIONS TO SURDS 325Substituting in (1) the value a?b for A, we obtainHeron does riot seem to have used this formula with a negativesign, unless in titereoru. I. 33 (34, Hultsch), where \/(63)and again, if J (\/3 > 1. We then haveand iiom this we derive successively1), or 2But, if we start trom $, obtained by the formula + v,

326 HERON OF ALEXANDRIAis given as approximately 8 TV In Metrica I, 9, as wehave seen, \/(63) is given as 7| T^, which was doubtlessobtained from the formula (1) asThe above seems to be the only classical rule which hasbeen handed down for finding second and further approximationsto the value of a surd. But, although Heron thusshows how to obtain a second approximation, namely byformula (2), he does not seem to make any direct use ofthis method himself, and consequently the question how theapproximations closer than the first which are to be found inhis works were obtained still remains an open one.(y)Quadrilaterals.It is unnecessary to give in detail the methods of measuringthe areas of quadrilaterals (chaps. 11-16). Heron deals withthe following kinds, the parallel-trapezium (isosceles or nonisosceles),the rhombus and rhomboid, and the quadrilateralwhich has one angle right and in which the four sides havegiven lengths. Heron points out that in the rhombus orrhomboid, and in the general case of the quadrilateral, it isnecessary to know a diagonal as well as the four sides. Themensuration in all the cases reduces to that of the rectangleand triangle.(8) The regular 'polygons with 3, 4, 5, 6', 7, #, ,9,10', 11,or 12 sides.Beginning with the equilateral triangle (chap. 17), Heronproves that, if a be the side and p the perpendicular froma vertex on the opposite side, a 2 :y a = 4 :3, whencea>:^2a 2 = 4:3 = 16: 12,so that a 4 :(A ABC)* =16:3,and (AABCJ* = T\a 4 . In the particular case taken a = 10and A 2 = 1875, whence A = 43^ nearly.Another method is to use an approximate value for

now J + ô= if = i"(fS)>THE REGULAR POLYGONS 327so ^h&i ê Approximation used byHeron for \/3 is here ff . For the side 10, the method givesthe same result as above, for %% . 100 = 43.The regular pentagonis next taken (chap. 18). Heronpremises the following lemma.Let ABC be a right-angled triangle, with the angle A equalfA to Produce AC to so that CO = ACIf now AO is divided in extreme andmean ratio, AB is equal to the greatersegment. (For produce AB to D so thatAD = 40, and join BO, DO.Then, sinceA DO is isosceles and the angle at A=.\R,ZA/)0 = AOJ) = $R t and, from theequality of the triangles4 #(7, OBG,LAOE = Z7J40 = f B. It follows thatthe triangle 4Z)0 is the isosceles triangle of Eucl. IV. 10, andAD is divided in extreme and mean ratio in B.) Therefore,says Heron, (BA + ACf = 5 AC*. [This is Eucl. XIII. 1.]Now, since Z BOG = f jK, if /JG f be produced to # so thatCE = -BO, 5A 7 subtends at an angle equal to \ R, and thereforeBE is the side of a regular pentagon inscribed in thecircle with as centre and OB as radius. (This circle alsopasses through 7), and BJ) is the side of a regular decagon inthe same circle.)If now BO = AB = r, OC = p, BE = a,we have from above, (r-hy;)2 = 5^, whence, since \/5 isapproximately f ,we obtain approximately r = f />,anda = />,so that />= |a. Hence ?pa = %a\of the pentagon = fa 2 .and the areaHeron adds that, if we take a closerapproximation to \/5 than |, we shall obtain the area stillmore exactly. In the Geometry l the formula is given as â 2 .The regular hejcayon (chap. 19)issimply 6 times theequilateral triangle with the same side. If A be the areaof the equilateral triangle with side a, Heron has provedthat A 2 = â 4 (Metrica I. 17), hence (hexagon) 2 = aâ4.If,= 67500, and (hexagon)= 259 nearly.e.g. a =210, (hexagon)In the Geometry 2 the formula is given as â-, while 'anotherbook' isquoted as giving 6 1 2(J + T Ty )c( it is added that the;latter formula, obtained from the area of the triangle, (^ + Yfr)tt2,represents the more accurate procedure, and is fully set out by1Geom. 102 (21, 14, Heib.). 2 Ib. 102 (21, 16, 17, Heib.).

&28 HERON OF ALEXANDRIAAs a matter of fact, however, 6 (^ + ^) = -^ exactly,Heron.and only the Metrioa gives the more accurate calculation.The regular heptagon.Heron assumes (chap. 20) that, if a be the side and r theradius of the circumscribing circle, a = |r, being approximatelyequal to the perpendicular from the centre of thecircle to the side of the regular hexagon inscribed in it (for -is the approximate value of ^ \/3).This theorem isquoted byJordanus Nemorarius (d. 1237) as an 'Indian rule'; he probablyobtained it from Abu'l Wafa (940-98). The Metricashows that it is of Greek origin, and, if Archimedes reallywrote a book on the heptagon in a circle, itmay be due tofrom the centre of thehim. If then p is the perpendicularcircle on the side (a) of the inscribed heptagon, r/(%a) = S/3%or 16/7, whence p 2 /(â}' 2 = - 2^V-, and p/î = (approximately)HJ/7 or 43/21. Consequentlyheptagon = 7 . %pa= 7 .f|a- = ff a'-.The regular octagon, decagon and dodecagon.the area of theIn these cases (chaps. 21, 23, 25) Heron findsp by drawingthe perpendicular 00 from 0, the centre of thecircumscribed circle, on a side AB> and then makingthe angle OAD equal to the angle AOD.For the octagon,A C B/.ADC = %R, and p = a(l+ v/2) = a(l + |)or | a . f approximately.For the decagon,LADG = f JJ, and -AD : DCAC =5:3, and p = %a (| +f =) fr*.hence AD :=5:4 nearly (see preceding page) ;For the dodecagon,LADG = ^Ji, and 7;= Ja(2-h V3)= Ja(2 + J)= V~ aAccordingly A% = - 2^-(t2the side in each ca&e.The regular enneagon and hendecagon.approximately.,^1 10 = \ 5 -a 2 ,J. 12= -\ 5 -^2 )where a isIn these cases (chaps. 22, 24) the Table of Chords (i e.

THE REGULAR POLYGONS 329presumably Hipparchus's Table) is appealed to.If AB be theside (a) of an enneagon or hendecagon inscribed in a circle, ACthe diameter through A9we are told that the Table of Chordsgives and -fa as the respective approximate values of theratio AB/AC. The angles subtended at the centre by theside AB are 10 and 32^8 T respectively,and Ptolemy's Table gives,as the chords subtended by angles of40 and 33 respectively, 41?' 2' 33"and 34^ 4' 55" (expressed in 120thparts of the diameter) ;Heron'sfigures correspond to40^ and 33^36' respectively. For the euueayoitAC* = SAB*, whence BC* = BAB*or approximately */-AB*, andBC = ~y a therefore; (area ofsa*. Forenueagon) = S &ABC=- .the hetulecayoib AC* = $/-AB* and BC* = - 5J/-AB~, so thatBC = tya, and urea of hendecagon = ^- AABC = -6â 2 . .An ancient formula for the ratio between the side of anyregular polygon and the diameter of the circumscribing circleispreserved in (jet^m. 147 sq. (= Pscudo-Dioph. 23-41),namely d n = /t- n . Nowothe ratio iut n /dntends to TT as thenumber ( H) of sides increases, and the formula indicates a timewhen TT was generally taken its = 3.(6)The Circle.Coming to the circle (Metrica 1. 26) Heron uses Archimedes'svalue for TT, namely - 2T^, making the circumference ofa4circle -\-r and the area2y-J(f where / is the radius and d,thediameter. It is here that he gives the more exact limitsfor TT which he says that Archimedes found in his work OnPiinthides and Cylinder &> but which are not convenient forcalculations. The limits, as we have seen, are given in thetext as ^^^\-

330 HERON OF ALEXANDRIA() Segment of a circle.According to Heron (Metrica I. 30) the ancients measuredthe area of a segment rather inaccurately, taking the areato be ^ (6 + h) h, where b is the base and h the height. Heconjectures that it arose from taking= TT 3, because,apply the formula to the semicircle, the area becomes ^. 3 r 2 ,where r is the radius. Those, he says (chap. 31), who haveif weinvestigated the area more accurately have added T 1^(^^) ito the above formula, making it 2%(b + h)h + -r*(i^) an^'^sseems to correspond to the value 3^ for TT, since, when appliedto the semicircle, the formula gives ^(3r2 + ^r2).He addsthat this formula should only be applied to segments ofa circle less than a semicircle, and not even to all of these, butonly in cases where b is not greater than 3 h. Suppose e.g.that b = 60, h = 1 ;in that case even T1^(|6)2 = ^.900 = 64f ,which is greater even than the parallelogram with 60, 1 assides, which again is greater than the segment. Where therefore6 > 3 h, he adopts another procedure.This is exactly modelled on Archimedes's quadrature ofa segment of a parabola. Heron proves (Metrica I. 27-29, 32)that, if ADB be a segment of a circle, and 1) the middle pointof the arc, and if the arcs AD, DB beAsimilarly bisected at E, F,A ADB < 4 (A AED + A DFB).Similarly, if the same construction bemade for the segments AED, BFD, eachof them is less than 4 times the sum of the two small trianglesin the segments left over.It follows that(area of segmt. ADB) > A ADB {\ *i + (i)2+ ...}'If therefore we measure thetriangle, and add one-third ofit, we shall obtain the area of the segment as nearly aspossible.' That is, for segments in which b > 3ft, Herontakes the area to be equal to that of the parabolic segmentwith the same base and height, or f bh.In addition to these three formulae for 8, the area ofa segment, there are yet others, namely8 = i (6 + h) h(l+ T ),Mensurae 29,l + , 31.

SEGMENT OF A CIRCLE 331The first of these formulae is applied to a segment greaterthan a semicircle, the second to a segmentless than a semicircle.In the Metrica the area of a segment greater than a semicircleis obtained by subtracting the area of the complementarysegment from the area of the circle.From the Geometrica lwe find that the circumference of thesegment less than a semicircle was taken to be V(h* + 4 h?) + $hor alternatively V(l?+W)+\v/(/; 2 -f 4/< 2 )-&}f-(77) Ellipse, parabolic wjiueut, surface ofcyliiuter, rightcone, sphere awl segment of sphere.After the area of an ellipse (Metrica I.34) and of a parabolicsegment (chap. 35), Heron gives the surface of a cylinder(chap. 36) and a right cone (chap. 37) ;in both cases he unrollsthe surface on a plane so that the surface becomes that of aparallelogram in the one case and a sector of a circle in theother. For the surface of a sphere (chap. 38) and a segment ofit (chap. 39) he simply uses Archimedes' s results.Book I ends with a hint how to measure irregular figures,plane or not. If the figure is plane and bounded by anirregular curve, neighbouring points are taken on the curvesuch that, if they are joined in order, the contour of thepolygonso formed is not much different from the curveitself, and the polygon is then measured byitdividing intotriangles.If the surface of an irregular solid figure is to befound, you wrap round it pieces of very thin paper or cloth,enough to cover and it, you then spread out the paper orcloth and measure that.Book II.Measurement of Volumes.The preface to Book II is interesting as showing howvague the traditions about Archimedes had already become.*After the measurement of surfaces, rectilinear or not, it isproper to proceed to the solid bodies, the surfaces of which wehave already measured in the preceding book, surfaces planeand spherical, conical and cylindrical, and irregular surfacesas well. The methods of dealing with these solids are, in1Cf. Geom., 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.).

332 HERON OF ALEXANDRIAview of their surprising character, referred to Archimedes bycertain writers who give the traditional account of theirorigin. But whether they belong to Archimedes or another,it is necessary to give a sketch of these methods as well/The Book begins with generalities about figures all thesections of which parallel to the base are equal to the baseand similarly situated, while the centres of the sections are ona straight linethrough the centre of the base, which may beeither obliquely inclined or perpendicular to the base ;whetherthe said straight line ('the axis ')is or is not. perpendicular tothe base, the volume is equal to the product of the area of thebase and the perpendicular height of the top of the figurefrom the base. The term ' height ' is thenceforward restrictedto the length of the perpendicular from the top of the figureon the base.(a) Cone, cylinder, parallelepiped (prism), pyramid, andfrustum.II. 1-7 deal with a cone, a cylinder, a ' parallelepiped ' (thenot restricted to the parallelogram but is inbase of which isthe illustration given a regular hexagon, so that the figure ismore properly a prism with polygonal bases), a triangularprism, a pyramid with base of any form, a frustum of atriangular pyramid the ; figures are in general oblique.(ft) Wedge-shaped solid (/Soo/LuV/coy orII. 8 is a case which is perhaps worth giving. It is that ofa rectilineal solid, the base of which is a rectangle ABCD andhas opposite to it another rectangle EFGH, the sides of whichare respectively parallel but not necessarily proportional tothose of ABCD. Take AK equal to EF, and BL equal to FO.Bisect BK, GL in V, W, and draw KRPU, VQOM parallel toAD, and LQRN, WOPT parallel to AB. Join FK, GR, LG,GU, HN.Then the solid is divided into (1) the parallelepiped withAR, EG as opposite faces, (2)the prism with KL as base andFG as the opposite edge, the (3) prism with NU as base andGH as opposite edge, and the (4) pyramid with RLCU as baseand G as vertex. Let h be the * height ' of the figure. Now

MEASUREMENT OF SOLIDS 333the parallelepiped (1) is on AR as base and has height h ;theprism (2) is equal to a parallelepiped on KQ as base and withheight h the ;prism (3) is equal to a parallelepiped with NPas base and height h; and finally the pyramid (4) is equal toa parallelepiped of height h and one-third of RC as base.Therefore the whole solid is equal to one parallelepipedwith height h and base equal to (AR + KQ + NP + RO + $RO)or AO + $RO.Now, if AB = a, #(7 = ft, EF = r, FG = d,AV = * ( 4- c),4T 7 = J(6 + rf), Q = i(c

334 HERON OF ALEXANDRIAThe method is the same mutatis mutandis as that used inII. 6 for the frustum of a pyramid with any triangle for base,and it is applied in II. 9 to the case of a frustum of a pyramidwith a square base, the formula for which iswhere a, a' are the sides of the larger and smaller basesrespectively, and h the height; the expressionis of courseeasily reduced to ^ h(a? + aa' + a' 2 ).(y) Frustum of cone, sphere, and segment thereof.A. frustum of a cone is next measured in two ways, (1) bycomparison with the corresponding frustum of the circumscribingpyramid with square base, (2) directly as the'difference between two cones (chaps. 9, 10). The volume ofthe frustum of the cone is to that of the frustum of thecircumscribing pyramid as the area of the base of the cone tothat of the base of the pyramidi.e. the volume of the frustumof the cone is TT, or ^J, times the above expression ; forthe frustum of the pyramid with a2 a' 2 as, bases, and itreduces to ^nh (d* + aa' + a' 2 where ),a, a' are the diametersof the two bases. For the sphere (chap, in Heron usesArchimedes's propositionthat the circumscribing cylinder is1^ times the sphere, whence the volume of the sphere= f .tZ.-H^2 or Ir^ 3 ;f r a segment of a sphere (chap. 12) helikewise uses Archimedes's result (On the Sphere and Cylinder,II. 4).(8) Anchor-ringor tore.The anchor-ring or tore is next measured (chap. 13) bymeans of a proposition which Heron quotes from Dionysodorus,and which is to the effect that, if a be the radius of eithercircular section of the tore through the axis of revolution, andc the distance of its centre from that axis,?ra 2 : ac = (volume of tore): ?rc 2 . 2 a[whence volume of tore = 2ir 2 ca 2 ].In the particular casetaken a = 6, c = 14, and Heron obtains, from the proportion113^:84 = F:7392, F= 9956f But he shows that he isaware that the volume is the product of the area of the

MEASUREMENT OF SOLIDS 335describing circle and the length of the pathof its centre*For, he says, since 1 4 is a radius (of the path of the centre),28 is its diameter and 88 its circumference. 'If then the torebe straightened out and made into a cylinder, it will have 88for its length, and the diameter of the base of the cylinder is12; so that the solid content of the cylinder is, as we haveseen, 9950$' (= 88 .-^(e) The tivo special solids of Archimedes 8 ' Method '.Chaps. 14,15 give the measurement of the two remarkablesolids of Archimedes's Method, following Archimedes's results.() The Jive regular solids.In chaps. 16-18 Heron measures the content of the fiveregular solids after the cube. He has of course in each caseto find the perpendicular from the centre of the circumscribingsphere on any face. Let p be this perpendicular, a theedge of the solid, r the radius of the circle circumscribing anyface. Then (1) for the tetrahedrona 2 = 3r 2 ,p* = a2 -a* =2.(2) In the case of the octahedron, which is the sum of twoequal pyramids on a square base, the content is one-thirdof that base multiplied by the diagonal of the figure,i.e. \ .a 2 . */2a or \/2.a 3 ;in the case taken a = 7, andHeron takes 10 as an approximation to \/(2. 7 2 or) \/98, theresult being |. 10.49 or 163J. (3) In the case of the icosahedronHeron merely says thatp:a = 93 : 127 (thereal value of the ratio is $7 7 + 3 s/5\^(4) In the case of the dodecahedron, Heron says thatp:a = 9 : 8 (the true value is ^ / > and, if \/5 isput equal to f ,Heron's ratio is readily obtained).Book II ends with an allusion to the method attributed toArchimedes for measuring the contents of irregular bodies byimmersing them in water and measuring the amount of fluiddisplaced.

336 HERON OF ALEXANDRIABook III. Divisions of figures.This book has much in common with Euclid's book On

DIVISIONS OF FIGURES 337should Apparently have 8j, since DO is immediately be 5^ (not 6).That is, in solving the equation= 0,stated towhich gives # = 7^(2^), Heron apparently substituted 2J or| for 2^, thereby obtaining 1^ as an approximationsurd.to the(The lemma assumed in this proposition is easily proved.Let m : be the ratio AF: FB = RD : DC = CE:EA.Then AF mc/(m + it),FB = nc/(m + n), CE = mb/(m + 71),KA = nb/(m + n), &c.Hence)is reduced (like III. 10) to the ; cutting-off* of an1Pappus, viii, pp. 1034-8.Of. pp. 430 2 post.

358 HERON OF ALEXANDRIAarea*. These propositions are ingenious and interesting.III. 11 shall be given as a specimen.Given any quadrilateral ABCD and a point E on the sideAD, to draw through E a straight line EF which shall cutthe quadrilateral into two parts inthe ratio of AE to ED. (We omitthe analysis.) Draw (76?. parallelto DA to meet AB produced in G.Join BE, and draw GH parallelto BE meeting BC in H.A i D Join CE, EH, EG.Then &GBE= AHBE and, adding &ABE to each, we haveAAGE= (quadrilateral ABHE).Therefore (quadr. ABIIE):&CED = A GAE: ACED= AE:ED.But (quadr. ABHE) and ACED are parts of the quadrilateral,and they leave over only the triangle EHC. We havetherefore only to divide &E1W in the same ratio AK-.ED bythe straight line EF. This is done by dividing HO at F inthe ratio AE: ED and joining EF.The next proposition (III. 1 2)is easilyreduced to this.If AE: ED is not equal to the given ratio, let F divide ADin the given ratio, and through Fdraw FG dividing the quadrilateralin the given ratio (III. 11).Join EG, and draw FH parallelto EG. Let FH meet BC in 7/,and join EH.A FED Then is EH the required straightline through E dividing the quadrilateralin the given ratio.For &FGE = AHGE. Add to each (quadr. GEDC).Therefore (quadr. CGFD) = (quadr. CHED).Therefore EH divides the quadrilateral in the given ratio,just as FG does.The case (III. 13) where E is hot on a side of the quadrilateral[(2) above] takes two different forms according as the

two opposite sidesDIVISIONS OF FIGURES 339which the required straight line cuts are(a) parallel or (6)not parallel.In the first case (a) theproblem reduces to drawing a straight line through E intersectingthe parallel sides in points F, G such that BF+AGis equal to a given length. In the second case (b) whereBC, AD are not parallel Heron supposes them to meet in //.The angle at // is then given, and the area ABH. It is thena question of cutting off from a triangle with vertex // atriangle IIF(f of given area by a straight line drawn from E,which is again a problem in Apollonius's Cutting-off of anHAGarea. The auxiliary problem in case (a) is easilyDsolved inIII. 16. Measure AH equal to the given length. Join BHand bisect it at M. Then EM meets BO, AD in points suchthat BF+ AG= the given length.BF = Gil.For. by congruent triangles,The same problems are solved for the case of any polygonin III. 14, 15. A sphereis then divided (III. 17) into segmentssuch that their surfaces are in a given ratio, by means ofArchimedes, On the Sphere and Cylinder, II. 3, just as, inIII. 23, Prop. 4 of the same Book is used to divide a sphereinto segments having their volumes in a given ratio.III. 18 is interesting because it recalls an ingenious propositionin Euclid's book On Divisions. Heron's problem is*To divide a given circle into three equal parts by two straightZ 2

340 HERON OF ALEXANDRIA'lines ',and he observes that, as the problem is clearly notrational, we shall, for practical convenience, make the division,as exactly as possible,in the followingway/ AE is the side of anequilateral triangleinscribed in thecircle. Let CD be the paralleldiameter, the centre of the circle,and join AO, BO, AD, DB. Thenshall the segment ABD be verynearly one-third of the circle. For,since AB is the side of an equilateraltriangle in the circle, thesector OAEB is one-third of thecircle. And the triangle AOB forming part of the sectoris equal to the triangle ADB\ therefore the segment AEBplus the triangle ABD is equal to one-third of the circle,the smalland the segment ABD only differs from this bysegment on BD as base, which may be neglected.Euclid'sproposition is to cut off one-third (or any fraction) of a circlebetween two parallel chords (see vol. i, pp. 429-30).III. 19 finds a point D within any triangle ABC such thatthe triangles DBC9 DCA, DAB are all equal and then Heron;passes to the division of solid figures.The solid figures divided in a given ratio (besides thesphere) are the pyramid with base of any form (III. 20),the cone (III. 21) and the frustum of a cone (III. 22), thecutting planes being parallel to the base in each case. Theseproblems involve the extraction of the cube root of a numberwhich is in general not an exact cube, and the point ofinterest is Heron's method of approximating to the cube rootin such a case. Take the case of the cone, and suppose thatthe portion to be cut off at the topis to the rest of the cone asm to n. We have to find the ratio in which the height or theedge is cut by the plane parallel to the base which cutsthe cone in the given ratio. The volume of a cone being^7rc 2 A, where c is the radius of the base and h the height,we have to find the height of the cone the volume of whichis -.virc * 2 h, and, as the height hf is to the radius cfhofits base as h is to c, we have simply to find // where

andDIVISIONS OF FIGURES 341= m/(m + 7&). Or, if we take the edges e, e' insteadof the heights, e' 3 /e= 3 m/(m + 7i).In the case taken byHeron m : u = 4 :1, and e = 5. Consequentlye' 3 = . 5 = 3 100.Therefore, says Heron, e = 4 4^ approximately, and in III. 20he shows how this is arrived at.'Approximation to the cube root of a iwu-cube number.Take the nearest cube numbers to 100 both above andbelow; these are 125 and 64.Then 125-100 = 25.and 100- 64 = 36.Multiply 5 into 36; this gives 180. Add 100, making 280.(Divide 180 by 280); this gives T\. Add this to the side ofthe smaller cube this :gives 4 X\. This is as nearly as possiblethe cube root ("cubic side") of 100 units.'We have to conjecture Heron's formula from this example.:iGenerally,if a:i < A < (a+ l),suppose that A a? = d l9and(a+l)* A = t/2 . The best suggestion that has been madeis Wertheim's, 1 namely that Heron's formula for the approximatecube root was a+ .*-", The 5 multiplied r(u + ljc^ + arfginto the 36 might indeed have been the square root of 25 orand the 100 added to the 180 in the denominator of thev/6/,,,fraction might have been the original number 100 (A) and not4 .25 or ad^ but Wertheim's conjecture is the more satisfactorybecause it can be evolved out of quite elementary considerations.This is shown by G. Enestrom as follows. 2 Using thesame notation, Enestrom further supposes that x is the exact1value of J/A, that U a)-= 8 V + =

342 HERON OF ALEXANDRIAand, solving for a; a,we obtainxa =or(aSince 8 V 8 2 are in any case the cubes of fractions, we mayneglect them for a first approximation, and we have(ac/i\Xl \III. 22, which shows how to cut a frustum of a cone in a givenratio by a section parallel to the bases, shall end our accountof the Metrica. I shall give the general formulae on the leftand Heron's case on the right. Let ABED be the frustum,let the diameters of the bases be a, a', and the height h.Complete the cone, and let the height of ODE be x.Suppose that the frustum has to be cut by a plane FG insuch a way that(frustum FB) = m : u.(frustum DG):In the case taken by Herona = 28, a' = 21, h = 12, m = 4, n = 1.Draw DH perpendicular to AB.

Since (DG) : (FB) = m :DIVISIONS OF FIGURES 343n,(DG):( ra) = 4:l,JNow(DB) = Y$7rh(Q? + (ji(Jif'and (DG) =*(DB) = 5698,(DG) = 4558|.Let ?/be the height (CM) of thecone CFG.Then DH:AH = CK:KA,or h:%(a = ')whence ic is known.i14.12x + h= -d 51- = 48,and x = 48 -12 = 36.(cone /).#) = 4158,(cone CFG) = 4158 + 4558=8716|,cone CAS= (CM) + (/)).(cone CAB) = 4158 + 5698 = 9856.Now, says Heron,''87169856 + 4158[He might have said simply= 87 16|.-^,Y4 4= 97805,whence y= 46 approximately.This gives y or CM,Therefore XJ/ = y-x = 10.whence LM is known.Now4^ = (3^ + 1 2-= 156},so that AD is known.and AD =12$.Thereforeknown.DFA.sTherefore

344 HERON OF ALEXANDRIAQuadratic eq f uuttions solved in, Heron.We have already met with one such equation (inMetricaIII. 4), namely a?2 14# + 46 = 0, the result only (x = 8|)being given. There are others in the Geometrica where theprocess of solution is shown.(1) Geometrica 24, 3 (Heib.). 'Given a square such that thesum of its area and perimeteris 896 feet: to separate the area'from the perimeter: i.e. x 2 4- 4 a; = 896. Heron takes half of4 and adds its square, completing the square on the left side.(2) Geometrica 21, 9 and 24, 46 (Heib.) give one and the sameequation, Geom. 24, 47 another like it. 'Given the sum ofthe diameter, perimeter and area of a circle, to find eachof them/The two equations areandOur usual method is to begin by dividing by ^ throughout,so as to leave d 2 as the first term. Heron's is to Multiply bysuch a number as will leave a square as the first term. In thiscase he multiplies by 154, giving H 2 d 2 + 58 . lid = 212 . 154or 67^,154 as the case may be. Completing the square,he obtains (lld-f 29) 2 = 32648 + 811 or 10395 + 841. Thuslld + 29 = V(33480) or ^(11236), that is, 183 or 106.Thus lid = 154 or 77, and d = 14 or 7, as the case may be.Indeterminate problems in the Geometrica.Some very interesting indeterminate problems are nowincluded by Heiberg in the Geometrical Two of them (chap.24, 1-2) were included in the Geeponicus in Hultsch's edition(sections 78, 79;; the. rest are new, having been found in theConstantinople manuscript from which Schone edited theMetrica. As, however, these problems, to whatever periodthey belong, are more akin to algebra than to mensuration,they will be more properly described in a later chapter onAlgebra.1Heronin Alexandrini opera, vol. iv, p. 414. 28 sq.

THE DIOPTRA 345The Dioptra (rreplSiOTTTpas).This treatise begins with a careful description of thedioptra, an instrument which served with the ancients forthe same purpose as a theodolite with us (chaps. 1-5). Theproblems with which the treatise goes on to deal are(a) problems of ' heights and distances ', (&) engineering problems,(c) problems of mensuration, to which is added(chap. 34) a description of a ' hodometer', or taxameter, consistingof an arrangement of toothed wheels and endlessscrews on the same axes working on the teeth of the nextwheels respectively. The book ends with the problem(chap. 37), 'With a given force to move a given weight bymeans of interacting toothed wheels', which really belongsto mechanics, and was apparently added, like some otherproblems (e.g. 31, 'to measure the outflow of, i.e. the volumeof water issuing from, a spring '),in order to make the bookmore comprehensive. The essential problems dealt with aresuch as the following. To determine the difference of levelbetween two given points (6), to draw a straight line connectingtwo points the one of which is not visible from the other(7), to measure the least breadth of a river (9),the distance oftwo inaccessible points (10), the height of an inaccessible point(12), to determine the difference between the heights of twoinaccessible points and the position of the straight line joiningthem (13), the depth of a ditch (14) to bore a tunnel ;througha mountain going straight from one mouth to the other (15), tosink a shaft through a mountain perpendicularly to a canalflowing underneath (1(5) given a subterranean canal of any;form, to find on the ground above a point from which avertical shaft must be sunk in order to reach a given pointon the canal (for the purpose e.g. of removing an obstruction)(20) to construct a harbour on;the model of a given segmentof a the ends circle,^giveii(17), to construct a vault so that itmay have a spherical surface modelled on a given segment(18). The mensuration problems include the following: tomeasure an irregular area, which is done by inscribing arectilineal figure and then drawing perpendiculars to thesides at intervals to meet the contour (23), or by drawing onestraight line across the area and erecting perpendiculars from

346 HERON OF ALEXANDRIAthat to meet the contour on both sides (24) ; giventhat allthe boundary stones of a certain area have disappeared excepttwo or three, but that the plan of the* area is forthcoming,to determine the position of the lost boundary stones (25).Chaps. 26-8 remind us of the Metrica: to divide a givenarea into given parts by straight lines drawn from one point(26) ; to measure a given area without entering it, whetherbecause it is thickly covered with trees, obstructed by houses,or entryis forbidden! (27) ; chaps. 28-30 = Metrica III. 7,III. 1, and I. 7, the last of these three propositions being theproof of the ' formula of Heron ' for the area of a triangle interms of the sides.Chap. 35 shows how to find the distancebetween Rome and Alexandria along a great circle of theearth by means of the observation of the same eclipse atthe two places, the analemma for Rome, and a concave hemisphereconstructed for Alexandria to show the position of thesun at the time of the said eclipse.It is here mentioned thatthe estimate by Eratosthenes of the earth's circumference inhis book On the Measurement of the Earth was the mostaccurate that had been made up to date. 1 Some hold thatthe chapter, like some others which have no particular connexionwith the real subject of the Dioptra (e.g. chaps. 31, 34,37-8) were probably inserted by a later editor, ' in order tomake the treatise as complete as possible V2The Mechanics.It is evident that the Mechanics, as preserved in the Arabic,is far from having keptits original form, especially inBook I. It begins with an account of the arrangement oftoothed wheels designed to solve the problem of moving agiven weight by a given force this account is the same as;that given at the end of the Greek text of the Diojrtra, and itis clearly the same description as that which Pappus 3 found intho work of Heron entitled Bapov\KO$ ('weight-lifter') andhimself reproduced with a ratio of force to weight alteredfrom 5 : 1000 to 4 : 160 and with a ratio of 2 : 1 substituted for5 : 1 in the diameters of successive wheels. It would appearthat the chapter from the Bapov\Ko$ was inserted in place of1Heron, vol. iii, p. 302. 18-17.3Pappus, viii, p. 1060 sq.2Ib , p.302. I).

THE MECHANICS 347the first chapter or chapters of the real Mechanics which hadbeen lost. The treatise would doiibtless begin with generalitiesintroductory to mechanics such as we find in the (muchinterpolated) beginning of Pappus, Book VIII. It must thenapparently have dealt with the properties of circles, cylinders,and spheres with reference to their importance in mechanics ;for in Book II. 21 Heron says that the circle is of all figuresthe most movable and most easily moved, the same thingapplying also to the cylinder and sphere, and he adds insupport of this a reference to a proof 'in the preceding Book '.This reference may be to I. 21, but at the end of that chapterhe says that * cylinders, even when heavy, if placed on theground so that they touch it in one line only, are easilymoved, and the same is true of spheres also, a matter whichive have already discussed'; the discussion may have comeearlier in the Book, in a chapter now lost.The treatise, beginning with chap. 2 after the passageinterpolated from the Bapov\Ko$, is curiously disconnected.Chaps. 2-7 discuss the motion of circles or wheels, equal orunequal, moving on different axes (e.g. interacting toothedwheels), or fixed on the same axis, much after the fashion ofthe Aristotelian Median teal problems.Aristotle'sWheel.In particular (chap. 7) Heron attempts to explain the puzzleof the ' Wheel of Aristotle ', which remained a puzzle up to quitemodern times, and gaverise to the proverb,crotam Aristotelisl 'magis torquere, quo magis torqueretur The '. question is ', saysthe Aristotelian problem 24, 6 why does the greater circle roll anequal distance with the lesser circle when they are placed aboutthe same centre, whereas, when they roll separately, as thesize of one is to the size of the other, so are the straight linestraversed by them to one another ' ? 2Let AC, BJ) be quadrantsof circles with centre bounded by the same radii, and drawtangents AE, BF at A and B. In the first case suppose thecircle BD to roll along BF till D takes the position //; thenthe radius ODC will be at right angles to AE, and C will beat (?, a point such that AG is equal12to BH. In the secondSee Van Capelle, Aristotelis quaestiones mechanicae, 1812, p. 263 sq.Avist. Mechanica, 855 a 28.

348 HERON OF ALEXANDRIAcase suppose the circle AC to roll along AE till ODG takesthe position O'FE; then D will be at F where AE = BF.And similarly if a whole revolution isperformed and OBA isagain perpendicular to AE. Contrary, therefore, to the principlethat the greater circle moves quicker than the smaller onthe same axis, it would appear that the movement of thesmaller in this case is as quick as that of the greater, sinceBH = AG, and BF'= AE. Heron's explanationis that, e.g.in the case where the larger circle rolls on AE, the lessercircle maintains the same speed as the greater because it hastivo motions; for if we regard the smaller circle as merelyfastened to the larger, and not rolling at all, its centre willmove to 0' traversing a distance 00' equal to AE and BF\-hence the greater circle will take the lesser with it over anequal distance, the rolling of the lesser circle havingupon this.The parallelogram of velocities.no effectHeron next proves the parallelogram of velocities (chap. 8) ;he takes the case of a rectangle, but the proof is applicablegenerally.The way it is putis this. Apoint moves with uniform velocityalong a straight line AB, from Ato JB, while at the same time ABmoves with uniform velocity alwaysparallel to itself with its extremityA describing the straight line AG.Suppose that, when the point arrives at B, thestraight line

LetTHE PARALLELOGRAM OF VELOCITIES 349reaches the position CD..EF be any intermediateposition of AB, and G the position at the same instantof the moving point on it. Thenclearly AE:AC=EG:EF;therefore AE:EG = AC:EF= AC:CD, and it follows that6r lies on the diagonal AD, which istherefore the actual pathof the moving point.Chaps. 9-19 contain a digression on the construction ofplane and solid figures similar to given figures but greater orless in a given ratio. Heron observes that the case of planefigures involves the finding of a mean proportional betweentwo straight lines, and 'the case of solid figures the finding oftwo mean proportionals in; chap.1 1 he gives his solution ofthe latter problem, which is preserved in Pappus and Eutociusas well, and has already been given above (vol. i, pp. 262-3).The end of chap. 19 contains, quite inconsequently, the constructionof a toothed wheel to move on an endless screw,after which chap. 20 makes a fresh start with some observationson weights in equilibrium on a horizontal plane buttending to fall when the plane isinclined, and on the readymobility of objects of cylindrical form which touch the planein one line onlyṀotion on an inclined plane.When a weightishanging freely by a rope over a pulley,no force applied to the other end of the rope less than theweight itself will keep it up, but, if the weight is placed on aninclined plane, and both the plane and the portion of theweight in contact with it are smooth, the case is different.Suppose, e.g., that a weight in the form of a cylinder is placedon an inclined plane so that the line in which they touch ishorizontal; then the force required to be applied to a ropeparallel to the line of greatest slope in the plane in order tokeep the weight in equilibrium is less than the weight. Forthe vertical plane passing through the line of contact betweenthe cylinder and the plane divides the cylinder into twounequal parts, that on the downward side of the plane beingthe greater, so that the cylinder will tend to roll down ;butthe force required to support the 'cylinder is the equivalent ',not of the weight of the whole cylinder, but of the difference

350 HERON OF ALEXANDRIAbetween the two portions into which the vertical planecuts it(chap. 23).On thecentre of gravity.This brings Heron to the centre of gravity (chap. 24). Herea definition by Posidonius, a Stoic, of the c centre of gravity 'or ' centre of inclination ' is given, namely ' a point such that,if the body is hung up at it, the bodyis divided into twoequal parts ' (he should obviously have said ' divided by anyvertical plane through the point of suspension into two equalBut, Heron says, Archimedes distinguished betweenparts ').the c centre of gravity ' and the ' point of suspension ', definingthe latter as a point on the body such that, if the body ishung up at it, all the parts of the body remain in equilibriumand do not oscillate or incline in any direction. ' "Bodies", saidArchimedes, " may rest (without inclining one way or another)with either a line, or only one point, in the body fixed "/ Thecentre of inclination ', says Heron, c is one single p*)int in any*particular body to which all the vertical lines through thepoints of suspension converge/ Comparing Simplicius's quotationof a definition by Archimedes in his Kei/rpojSapi/ca, tothe effect that the centre of gravity is a certain point in thebody such that, if the body is hung up by a string attached tothat point, it will remain in its position without inclining inany direction, 1 we see that Heron directly used a certaintreatise of Archimedes. So evidently did Pappus, who hasa similar definition. Pappus also speaks of a body supportedat a point by a vertical stick :if, he says, the bodyis inequilibrium, the line of the stick produced upwards must passthrough the centre of gravity. 2 Similarly Heron says thatthe same principles apply when the body is supported as whenit is suspended. Taking up next (chaps. 25-31) the questionof ' supports he considers cases of a heavy beam or a wall',supported on a number of pillars, equidistant or not, evenor not even in number, and projecting or not projectingbeyond one or both of the extreme pillars,and finds howmuch of the weight is supported on each pillar.He saysthat Archimedes laid down the principles in his Book ' on1Simplicius on De caelo, p. 543. 31-4, Heib.2Pappus, viii, p. 1032. 5-24.

ON THE CENTRE OF GRAVITY 351Supports '. As, however, the principles are the same whetherthe body is supported or hungit up, does not follow thatthis was a different work from that known as Trepl (vy&v.Chaps. 32-3, which are on the principles of the lever or ofweighing, end with an explanation amounting to the factthat ' greater circles overpower smaller when their movementis about the same centre ',a proposition which Pappus saysthat Archimedes proved in his work ncpl {vyS*?.1In chap. 32,too, Heron gives as his authority a proof given by Archimedesin the same work. With I. 33 may be compared II. 7,where Heron returns to the same subject of the greater andabout the same centre and states thelesser circles movingfact that weights reciprocally proportional to their radii arein equilibrium when suspended from opposite ends of thehorizontal diameters, observing that Archimedes proved theproposition in his work On the equalization(presumably I

352 HERON OF ALEXANDRIAdo great weightsfall to the ground in a shorter time thanlighter ones?', 'Why does a stick break sooner when oneputs one's knee against i^ in the middle ? '', Why do peopleuse pincers father than the hand to draw a tooth?', 'Whyis iteasy to move weights which are suspended?', and'Whyis it the more difficult to move such weights the fartherthe hand isaway from them, right up to the point of suspensionor a point near it ? V Why are great ships turned by a rudderalthough it is so small ? *', Why do arrows penetrate armouror metal plates but fail tp penetrate cloth spread out '?Problems on the centre of gravity, &c.II. 35, 36, 37 show how to find the centre of gravity ofa triangle, a quadrilateral and a pentagon respectively. Then,assuming that a triangle of uniform thickness is supported bya prop at each angle, Heron finds what weight is supportedby each prop, when (a) the props support the triangle only,(6) when they support the triangle plus a given weight placedat any point on it (chaps. 38, 39). Lastly, if known weightsare put on the triangle at each angle, he finds the centre ofgravity of the system (chap. 40) the ; problemis then extendedto the case of any polygon (chap. 41).Book III deals with the practical construction of enginesfor all sorts of purposes, machines employing pulleys withone, two, or more supports for lifting weights, oil-presses, &c.The Catoptrica.This work need not detain us long.Several of the theoreticalpropositions which it contains are the same as propositionsin the so-called Catoptrica of Euclid, which, as we haveseen, was in all probability the work of Theon of Alexandriaand therefore much later in date. In addition to theoreticalpropositions, it contains problems the purpose of which is toconstruct mirrors or combinations of mirrors of such shapeto make theas will reflect objects in a particular way, e.g.right side appear as the right in the picture (instead of thereverse), to enable a person to see his back or to appear inthe mirror head downwards, with face distorted, with threeeyes or two noses, and so forth. Concave and convex

THE CATOPTRTCA 353cylindrical mirrors play a part in these arrangements. Thewhole theory of course ultimately depends on the main propositions4 and 5 that the angles of incidence and reflectionare equal whether the mirror is plane or circular.Heron's proof of equality of angles of Incidence and reflection.Let AB be a plane mirror, C the eye, D the object seen.The argument rests on the fact that nature ' does nothing invain '. Thus light travels in a straight line, that is, by thequickest road. Therefore, evenwhen the ray is a line brokenat a point by reflection, it mustmark the shortest broken lineof the kind connecting the eyoand the object. Now, saysHeron, I maintain that theshortest of the broken lines(broken at the mirror) whichconnect C and D is the line, asCAD, the parts of which make equal angles with the mirror.Join T)A and produceit to meet in F the perpendicular fromC to AB. Let li be any point on the mirror other than A,and join FB, BD.NowLEAF = LBAD= L OAK, by hypothesis.Therefore the triangles AEF, AEG, having two angles equaland AK common, are equal in all respects.Therefore CA = AF, and OA + AD = DF.Since FE = EG, and BE is perpendicular to FC, BF = BC.Therefore CB + BD = FB + BDi.e.> FD,> CA+AD.The proposition was of course known to Archimedes.gather from a scholium to the Pseudo- Euclidean Catoptricathat he proved it in a different way, namely by reductio adWeabsurdum, thus: Denote the angles GAEy DAB by a, /9 respectively.Then, a is > = or < ft. Suppose ot > ft. Then,A a

354 HERON OF ALEXANDRIAreversing the ray so that the eye is at D instead of (7, and theobject at C instead of Z), we must have /8 > a. But /8 wasless than a, which is impossible. (Similarly it can be provedthat a is not less than jS.) Therefore oc = /8.In the Pseudo-Euclidean Catopt^ca the proposition ispractically assumed; for the third assumption or postulateat the beginning states in effect that, in the above figure,if Abe the point of incidence, CE:EA = J)H:HA (where DH isperpendicular to AB). It follows instantaneously (Prop. 1)thatLGAE^LDAH.If the mirror isthe convex side of a circle, the same resultfollows a fortiori. Let CA, AD meetthe arc at equal angles, and CB, BD atunequal angles. Let AE be the tangentat A, and complete the figure.Then, says Heron, (the angles GAG,BAD being by hypothesis equal), if wesubtract the equal angles GAE, BAFfrom the equal angles BAD OAC9 (bothpairs of angles being ' mixed ',be itobserved), we have LEAG = L FAD. Therefore CA+AD< CF+FD and a fortiori < CR + BD.The problems solved (though the text is so corrupt in placesthat little can be made of it)were such as the following:11, To construct a right-handed mirror (i.e.a mirror whichmakes the right side right and the left side left instead ofthe opposite); 12, to construct the mirror called polytheoron('with many images'); 16, to construct a mirror inside thewindow of a house, so that you can see in it (while insidethe room) everything that passes in the street; 18, to arrangemirrors in a given place so that a person who approachescannot actually see either himself or any one else but can seeany image desired (a 'ghost-seer ').

XIXPAPPUS OF ALEXANDRIAWE have seen that the Golden Age of Greek geometryended with the time of Apollonius of Perga. But the influenceof Euclid, Archimedes and Apollonius continued, and for sometime there was a succession of quite competent mathematicianswho, although not originating anything of capital importance,kept up the tradition. Besides those who were known forparticular investigations, e.g. of new curves or surfaces, therewere such men as Geminus who, it cannot be doubted, werethoroughly familiar with the great classics. Geminus, as wehave seen, wrote a comprehensive work of almost encyclopaediccharacter on the classification and content of mathematics,including the history of the development of each subject.But the beginning of the Christian era sees quite a differentstate of things. Except in sphaeric and astronomy (Menelausand Ptolemy), production was limited to elementary textbooksof decidedly feeble quality. In the meantime it wouldseem that the study of higher geometry languished or wascompletely in abeyance, until Pappus arose to revive interestin the subject. From the way in which he thinks it necessaryto describe the contents of the classical works belonging tothe Treasury of Analysis, for example, one would supposethat by his time many of them were, if not lost, completelyforgotten, and that the great task which he set himself wasthe re-establishment of geometry on its former high plane ofachievement. Presumably such interest as he was able toarouse soon flickered out, but for us his work has an inestimablevalue as constituting, after the works of the greatmathematicians which have actually survived, the. most importantof all our sources. A a 2

356 PAPPUS OF ALEXANDRIADate of Pappus.Pappus lived at the end of the third century A.D. Theauthority for this date is a marginal note in a Leyden manuscriptof chronological tables by Theon of Alexandria, where,opposite to the name of 'Diocletian, a scholium says, In histime Pappus wrote'. Diocletian reigned from 284 to 305,and this must therefore be the period of Pappus's literaryactivity. It is true that Suidas makes him a contemporaryof Theon of Alexandria, adding that they both lived underTheodosius I (379-395). But Suidas was evidently not wellacquainted with the works of Pappus; though he mentionsa description of the earth by him and a commentary on fourBooks of Ptolemy's Sytitaxis, he has no word about his greatestwork, the Synagoge. As Theon also wrote a commentary onPtolemy and incorporated a great deal of the commentary ofPappus, it is probable that Suidas had Theon's commentarybefore him and from the association of the two names wronglyinferred that they were contemporaries.Works (commentaries) other than the Collection.Besides the Si/nagoge, which is the main subject of thisa commentarychapter, Pappus wrote several commentaries, now lost except forfragments which have survived in Greek or Arabic. One wason the Elements of Euclid. This must presumablyhave been pretty complete, for, while Proclus (on Eucl. I)quotes certain things from Pappus which may be assumed tohave come in the notes on Book I, fragments of his commentaryon Book X actually survive in the Arabic (see above,vol. i, pp. 154-5, 209), and again Eutocius in his note on Archimedes,On the Sphere and Cylinder,I. 13, says that Pappusexplained in his commentary on the Elements how to inscribein a circle a polygon similar to a polygon inscribed in anothercircle, which problem would no doubt be solved by Pappus, asin a note on XII. 1. Some of the referencesit isby a scholiast,by Proclus deserve passing mention. (1) Pappussaid thatthe converse of Post. 4 (equality of all right angles) is nottrue, i.e. it is not true that all angles equal to a right angle arethemselves right, since the ' angle ' between the conterminousarcs of two semicircles which are equal and have their

WORKS OTHER THAN THE COLLECTION 357diameters at right angles and terminating at one pointisequal to, but is 1not, a right angle. (2) Pappus said that,in addition to the genuine axioms of Euclid, there were others ,on record about unequals added toequals and equals added to unequals.Others given by Pappus are (saysProclus) involved by the definitions,e.g. that 'all parts of the plane and ofthe straight line coincide with oneanother ',that ' a point divides a line,a line a surface, and a surface a solid ',and that ' the infiniteis (obtained) in magnitudes both by addition and diminution 1 2.(3) Pappus gave a pretty proof of Eucl. I. 5, which moderneditors have spoiled when introducingit into text-books. IfAB, AC are the equal sides in an isosceles triangle, Pappuscompares the triangles ABC and ACB (i.e.as if he were comparingthe triangle ABC seen from the front with the sametriangle seen from the back), and shows that they satisfy theconditions of I. 4, so that they are equal in all respects, whencethe result follows. 3Marinus at the end of his commentary on Euclid's Datarefers to a commentary by Pappus on that book.Pappus's commentary on Ptolemy's tfyibtaxis has alreadybeen mentioned (p. 274)it seems to have extended to six;Books, if not to the whole of Ptolemy's work. The Fihristsays that he also wrote a commentary on Ptolemy's PlanisjtJJiaeri'U/ni,which was translated into Arabic by Thabit b.Qurra. Pappus himself alludes to his own commentary onthe Anal&m/Hui of Diodorus, in the course of which he used theconchoid of Nicomedes for the purpose of trisecting an angle.We come now to Pappus's great work.The Synayoge orCollection,(a) Character of the work; luide raiuje.Obviously written with the object of reviving the classicalGreek geometry,it covers practically the whole field. It is,12Proclus on Eucl. I, pp. 189-90./&., pp. 197. 6-198. 15.3Ib., pp. 249. 20-250. 12.

358 PAPPUS OF ALEXANDRIAhowever, a handbook or guide to Greek geometry rather thanan encyclopaediait was; intended, that is, to be read with theoriginal works (where still extant) rather than to enable themto be dispensed with. Thus in the case of the treatisesincluded in the Treasury of Analysis there is a general introduction,followed by a general account of the contents, withlemmas, &c., designed to facilitate the reading of the treatisesthemselves.On the other hand, where the history of a subjectis given, e.g. that of the problem of the duplicationof thecube or the finding of the two mean proportionals, the varioussolutions themselves are reproduced, presumably because theywere not easily accessible, but had to bo collected from varioussources. Even when it is some accessible classic which isbeing described, the opportunityistaken to give alternativemethods, or to make improvements in proofs, extensions, andso on. Without pretending to great originality, the wholework shows, on the part of the author, a thorough grasp ofall the subjects treated, independence of judgement, masteryof technique; the style is terse and clear; in short, Pappusstands out as an accomplished and versatile mathematician,a worthy representative of the classical Greek geometry.(]8) List ofauthors mentioned.The immense range of the Collection can be gathered froma mere enumeration of the names of the various mathematiciansquoted or referred to in the course of it. The greatest ofthem, Euclid, Archimedes and Apollonius, are of course continuallycited, others are mentioned for some particularachievement, and in a few cases the mention of a name byPappus is the whole of the information we possess about theperson mentioned. In giving the list of the names occurringin the book, it will, I think, be convenient and may economizefuture references if I note in brackets the particular occasionof the reference to the writers who are mentioned for oneachievement or as the authors of a particular book or investigation.The list in alphabetical order is :Apollonius of Perga,Archimedes, Aristaeus the elder (author of a treatise in fiveBooks on the Elements of Conies or of ' five Books on SolidLoci connected with the conies'), Aristarchus of Samos (On the

THE COLLECTION 359sizes and distances of the sun and moo?*,), Autolycus (0/6 themoving sphere), Carpus of Antioch (who is quoted as havingsaid that Archimedes wrote only one mechanical book, thaton sphere-making, since he held the mechanical applianceswhich made him famous to be nevertheless unworthy ofwritten description:Carpus himself, who was known asmechanicus, applied geometryto other arts of this practicalkind), Charmandrus (who added three simple arid obvious locito those which formed the beginning of the Plane Loci ofApollonius), Conon of Samos, the friend of Archimedes (citedas the propounder of a theorem about the spiral in a planewhich Archimedes proved: this would, however, seem to bea mistake, as Archimedes says at the beginning of his treatisethat he sent certain theorems, without proofs, to Conon, whowould certainly have proved them had he lived), Demetrius ofAlexandria (mentioned as the author of a work called 'Linearconsiderations', ypa/jLfjLiKal CTrio-rao-eiy,i.e. considerations oncurves, as to which nothing more is known), Diriostratus,the brother of Menaechmus (cited,with Nicornedes, as havingused the curve of Hippias, to which they gave the name ofquadratriM, TeTpaya>viovora, for the squaring of the circle),Diodorus (mentioned as the author of an A nalemma), Eratosthenes(whose mean-finder, an appliance for finding two orany number of geometric means, is described, and who isfurther mentioned as the author of two Books On means ' 'and of a work entitled 'Loci witli reference to means'),Erycinus (from whose Pamdoxa are quoted various problemsseeming at first sight to be inconsistent with Eucl. I. 21, itbeing shown that straight lines can be drawn from two pointson the base of a triangle to a point within the triangle whichare together greater than the other two sides, provided that thepoints in the base may be points other than the extremities),Euclid, Geminus the mathematician (from whom is cited aremark 011 Archimedes contained in his book 'On the classificationof the mathematical sciences ',see above, p. 223), Heraclitus(from whom Pappus quotes an elegant solution of a vv

360 PAPPUS OF ALEXANDRIAwho is mentioned as having asked Pappus's opinion on theattempted solution by * plane 'methods of the problem of the twomeans, which actually gives a method of approximating toa solution 1 ), Hipparchus (quoted as practically adopting threeof the hypotheses of Aristarchus of Samos), Megethion (towhom Pappus dedicated Book V of his Collection), Menelausof Alexandria (quoted as the author of Sphaerica and as havingapplied the name irapd8oo$ to a certain curve), Nicomachus(on three means additional to the first three), Nicomedes,Pandrosion (to whom Book III of the Collection is dedicated),Pericles (editor of Euclid's Data), Philon of Byzantium (mentionedalong with Heron), Philon of Tyana (mentioned as thediscoverer of certain complicated curves derived from the interweavingof plectoid and other surfaces), Plato (with referenceto the live regular solids), Ptolemy, Theodosius (author of theSphaerica and On Days and Nights).The first(y)4Translations and editions.published edition of the Collection was the Latintranslation by Commandimis (Venice 1589, but dated at theend 'Pisauri apud Hicronymum Concordiam 1588'; reissuedwith only the title-page changed 'Pisauri... 1602'). Up to1876 portions only of the Greek text had appeared, namelyBooks VII, VIII in Greek and German, by C. J. Gerharclt, 1871chaps. 33-105 of Book V, by Eisenmann, Paris 1824, chaps45-52 of Book IV in losephi Torelli Veronensis Geometrica1769, the remains of Book II, by John Wallis (in Openinathematica, III, Oxford 1699); in addition, the restorersof works of Euclid and Apollonius from the indication!furnished by Pappus give extracts from the Greek texrelating to the particular works, Breton leChamp on Euclid'iPorisms, Halley in his edition of the Conies of Apolloniu(1710) and in his translation from the Arabic and restoratioirespectively of the De sectione rationis and De sections spatiof Apollonius (1706), Carnerer on Apolloiiius's Tactiones (1795Simson and Horsley in their restorations of Apollonius's PlanLoci and Inclinations published in the years 1749 and 177respectively. In the years 1876-8 appeared the only coin1See vol. i, pp. 268-70.

THE COLLECTION. BOOKS I, II, III 361plete Greek text, with apparatus, Latin translation, commentary,appendices and indices, by Friedrich Hultsch this;great edition is one of the first monuments of the revivedstudy of the history of Greek mathematics in the last halfof the nineteenth century, and has properly formed the modelfor other definitive editions of the Greek text of the otherclassical Greek mathematicians, e.g. the editions of Euclid,Archimedes, Apollonius, &c., by Heiberg and others. TheGreek index in this edition of Pappus deserves special mentionbecause itlargely serves as a dictionary of mathematicalterms Uvsecl not only in Pappus but by the Greek -mathematiciansgenerally.(8) tiunvinary of contents.At the beginning of the work, Book I and the first 13 propositions(out of 26) of Book II are missing. The first 13propositions of Book II evidently, like the rest of the Book,dealt with Apollonius's method of working with very largenumbers expressed in successive powers of the myriad, 10000.This system has already been described (vol. i,pp. 40, 54-7).The work of Apollouius seems to have contained 26 propositions(25 leading up to, and the 26th containing, the finalcontinued multiplication).Book III consists of four sections. Section (1) is a sort ofhistory of the problem of finding two mean proportionals, incontinued proportion, between two given straight lines.It begins with some general remarks about the distinctionbetween theorems and problems. Pappus observes that,whereas the ancients called them all alike 'by one name, someregarding them all as problems and others as theorems, a cleardistinction was drawn by those who favoured more exactterminology. According to the latter a problemis that inwhich it is proposed to do or construct something, a theoremthat in which, given certainhypotheses, we investigate thatwhich follows from and is necessarily implied by them.Therefore he who propounds a theorem, no matter how he hasbecome aware of the fact which is a necessary consequence ofthe premisses, must state, as the object of inquiry, the rightresult and no other. On the other hand, he who propounds

362 PAPPUS OF ALEXANDRIAa problem may bid us do something which is in fact impossible,and that without necessarily laying himself opento blame or criticism. For it is part of the solver's dutyto determine the conditions under which the problem ispossible or impossible, and, 'if possible, when, how, and inhow many ways it is possible '. When, however, a man professesto know mathematics and yet commits some elementaryblunder, he cannot escape censure. Pappus gives, as anexample, the case of an unnamed person who was 'thought tobe a great geometer' but who showed ignorance in that heclaimed to know how to solve the problem of the two meanproportionals by 'plane' methods (i.e. by using the straightline and circle only). He then reproduces the argument ofthe anonymous person, for the purpose of showing that itdoes not solve the problem as its author claims. We haveseen (vol. i, pp. 269-70) how the method, though not actuallysolving the problem, does furnish a series of successive approximationsto the real solution. Pappus adds a few simplelemmas assumed in the exposition.1Next comes the passage , already referred to, on the distinctiondrawn by the ancients between (1) plane pi'oblems orproblems which can be solved by means of the straight lineand circle, (2) solid problems, or those which require for theirsolution one or more conic sections, (3) linear problems, orthose which necessitate recourse to higher curves still, curveswith a more complicated and indeed a forced or unnaturalorigin (^^La(T^vr]v) such as spirals, quadratrices, cochloidsand cissoids, which have many surprising properties of theirown. The problem of the two mean proportionals, beinga solidfor its solution either conies or someproblem, requiredequivalent, and, as conies could not be constructed by purelygeometrical means, various mechanical devices were inventedsuch as that of Eratosthenes (the mean-finder), those describedin the Mechanics of Philon and Heron, and that of Nicomedes(who used the ' cochloidal ' curve). Pappus proceeds to give thesolutions of Eratosthenes, Nicomedes and Heron, and then addsa fourth which he claims as his own, but which is practicallythe same as that attributed by Eutocius to Sporus. All thesesolutions have been given above (vol.i, pp. 258-64, 266-8).1Pappus, iii, p. 54. 7-22.

THE COLLECTION. BOOK III 363Section (2). The theory of means.Next follows a section (pp. 69-105) on the theory of thedifferent kinds of means. The discussion takes its originfrom the statement of the ' second problem ', which was thatof 'exhibiting the three means' (i.e. the arithmetic, geometricand harmonic)'in. a semicircle'.Pappus first gives a constructionby which another geometer (aAXoy riy) claimed tohave solved this problem, but he does not seem to have understoodit, and returns to the same problem later (pp. 80-2).In the meantime he begins with the definitions of thethree means and then shows how, given any two of threeterms a, b, c in arithmetical, geometrical or harmonical progression,the third can be found. The definition of the mean(b) of three terms a, b, c in harmonic progression being that itsatisfies the relation a:c=a b:b c, Pappus gives alternativedefinitions for the arithmetic and geometric means in correspondingform ,namely for the arithmetic mean a a = a b : : b cand for the geometric a : b = a b : b c.The construction for the harmonic mean isperhaps worthgiving. Let AB, BG be two given straight lines. At A drawDAE perpendicular to AB, and make DA, AE equal. JoinDB, BE. From G draw GF&t rightangles to AB meeting DB in F. DJoin EF meeting AB in C. ThenBC is the required harmonic mean.For= EA:FG= AC:CG""= (AB-BC):(BC-BG).Similarly, by means of a like figure, we can find BG whenAB, BO are given, and AB when 5(7, BG are given (inthe latter case the perpendicular DE is drawn through Ginstead of A).Then follows a proposition that, if'CQthe three means and theseveral extremes are represented in one set of lines, there mustbe five of them at least, and, after a set of five such lines havebeen found in the smallest possible integers, Pappus passes toB

364 PAPPUS OF ALEXANDRIAthe problem of representing the three means with the respectiveextremes by six lines drawn in a semicircle.Given a semicircle on the diameter AC, and B any point onthe diameter, draw BD at right angles to A C.Let the tangentat D meet AC produced in G, and measure DH along thetangent equal to DG. Join HB meeting the radius OD in K.Let EF\}Q perpendicular to Of).Then, exactly as above, it is shown that OK is a harmonicmean between OF and OD. Also BD is the geometric meanbetween AB, BC, while OC (= OD)is the arithmetic meanbetween AB, BC.Therefore the six lines DO (= OC), OK, ^F, AB, BC, BDsupply the three means with the respective extremes.But Pappus seems to have failed to observe that the 'certainother geometer who has the same ',figure excluding the dottedlines, supplied the same in Jive lines. For he said that OFis 'a harmonic mean'. It is in fact the harmonic meanbetween AB, BC, as is easily seen thus.toSince ODB is a right-angled triangle, and BF perpendicularOD,DF:BD = BD:DO,or DF. DO = BD* = AB . BC.Buttherefore DF .Therefore AB .(DF-DO ^%(AB+BC\\(AB + BC) = 2AB. BC.BC) = BC.(AB- DF),that is, AB: BC = (AB - DF):(DF-BC),and DF is the harmonic mean between AB, BC.Consequently the Jive lines D0( = OC), DF, AB, BC, BDexhibit all the three means with the extremes.

THE COLLECTION. BOOK III 365Pappus does not seem to have seen this, for he observesthat the geometer in question, though saying that DF isa harmonic mean, does not say how it is a harmonic meanor between what straight lines.In the next chapters (pp. 84-104) Pappus, following Nicomachusand others, defines seven more means, three of whichwere ancient and the last four more modern, and shows howwe can form all ten means as linear functions of a, )8, y, wherea, /3, y are in geometrical progression. The exposition hasalready been described (vol. i, pp. 86-9).Section (3).The 'Paradoxes' of Erycinus.The third section of Book III (pp. 104-30) contains a seriesof propositions, all of the same sort, which are curious ratherthan geometrically important. They appear to have beentaken direct from a collection of Paradoxes by one Erycinus. 1The first set of these propositions (Props. 28-34) are connectedwith Eucl. I. 21, which says that, if from the extremitiesof the base of any triangle two strafght lines be drawn meetingat any point within the triangle, the straight lines are togetherless than the two sides of the triangle other than the base,but contain a greater angle. It is pointed out that, if thestraight lines are allowed to be drawn from points in the baseother than the extremities, their sum may be greater than theother two sides of the triangle.The firstcase taken is that of a right-angled triangle ABC7) on BC. Measureright-angled at B. Draw AD to any pointon it DE equal to A B, bisect A Ein F, and join F(J. Then shall ADF+FC be > BA + AC.For EF+FC=AF+FC>AC.Add DE and AB respectively,and we haveDF+FC> BA + AC.More elaborate propositions are next proved, such as thefollowing.1. In any triangle, except an equilateral triangle or an isosceles1Pappus, iii, p. 106. 5-9.

366 PAPPUS OF ALEXANDRIAtriangle with base less than one of the other sides, it is possibleto construct on the base and within the triangle two straightlines meeting at a point, the sum of which is equal to the sumof the other two sides of the triangle (Props. 29, 30).2. In any triangle in which it is possible to construct twostraight lines from the base to one internal point the sumof which is equal to the sum of the two sides of the triangle,it is also possible to construct two other such straight lines thesum of which is greater than that sum (Prop. 31).than either3. Under the same conditions, if the base is greaterof the other two sides, two straight lines can be so constructedfrom the base to an internal point which are respectivelygreater than the other two sides of the triangle ;and the linesmay be constructed so as to be respectively equal to the twosides, if one of those two sides is less than the other and eachof them is less than the base (Props. 32, 33),4. The linesmay be so constructed that their sum will bear tothe sum of the two sides of the triangle any ratio less than2 : 1 (Prop. 34).As examples of the proofs, we will take the case of thescalene triangle, and prove the first and Part 1of the third ofthe above propositions for such a triangle.In the triangle ABC with base BG let AB be greaterthan AC.Take D on BA such that BD = \ (BA + AC}.BHOn DA between D and A take any point E, and draw EFto BO. Let G be any point on EF\ draw GH parallelparallelto AB and join GO.

NowTHE COLLECTION. BOOK III 367JSA + AC > EF+ FCProduce> EG + GC and > GO, a fortiori.GO to K so that GK = EA + AC, and with G ascentre and GK as radius describe a circle. This circle willmeet HC and HG, because GH = EB > BD or DA + AC and> GK, a fortiori.Then HG + GL = BE+EA+AC = BA + AC.To obtain two straight lines #G',

PAPPUS368.OF ALEXANDRIA,the other, i.e. it is shown that broken lines, consisting ofseveral straight lines, can be drawn with two points on thebase of a triangle or parallelogram as extremities, and ofgreater total length than the remaining two sides of thetriangle or three sides of the parallelogram.Props. 40-2 show that triangles or parallelograms can beconstructed with sides respectively greater than those of a giventriangle or parallelogram but having a less area.Section (4).The inscribing of tie five regular solidsin a sphere.The fourth section of Book Til (pp. 132-G2) solves theproblems of iiibcribing each of the five regularsolids in agiven sphere. After some preliminary lemmas (Props. 43-53),Pappus attacks the substantive problems (Props. 54-8), usingthe method of analysis followed by synthesis in the case ofeach solid.(a) In order to inscribe a regular pyramidor tetrahedron inthe sphere, he finds two circular sections equal and parallelto one another, each of which contains^ one of two oppositeedges as its diameter. If d be the diameter of the sphere, theparallel circular sections have d / as diameter, where d 2 = fcZ' 2 .(b) In the case of the cube Pappus again finds two parallelcircular sections with diameter d' such that d 2 =%d' 2 ; a squareinscribed in one of these circles is one face of the cube and"the square with sides parallel to those of the first squareinscribed in the second circle is the opposite face.In (r)the case of the octahedron the same two parallel circularsections with diameter d' such that d* = f d' 2 are used ;anequilateral triangle inscribed in one circle is one face, and theopposite face is an equilateral triangle inscribed in the othercircle but placed in exactly the opposite way.(d) In the case of the icosahedron Pappus finds four parallelcircular sections each passing through three of the vertices ofthe icosahedron ;two of these are small circles circumscribingtwo opposite triangular faces respectively, and the other twocircles are between these two circles, parallel to them, andequalto one another. Thepairs of circles are determined in

THE COLLECTION. BOOKS III, LV 369this way. If d be the diameter of the sphere, set out twostraight lines x, y such that d, a, y are in the ratio of the sidesof the regular pentagon,, hexagon and decagon respectivelydescribed in one and the same circle. The smaller pair ofcircles have r as radius where r = 2 2# and the, larger pairhave r' as radius where r' = 2 2%x.(e) In the case of the dodecahedron the saw e four parallelcircular sections are drawn as in the case of the icosahedron.Inscribed pentagons set the opposite way are inscribed in thetwo smaller circles these ;pentagons form opposite faces.Regular pentagons inscribed in the larger circles with verticesat the proper points (and again set the opposite way) determineten more vertices of the inscribed dodecahedron.'The constructions are quite different from those inEuclidXIII. 13, 15, 14, 16, 17 respectively, where the problemis firstto construct the particular regular solid and then to com-'prehend it in a sphere ', i. e. to determine the circumscribingsphere in each case. I have set out Pappus's propositions indetail elsewhere. 1Book IV.At the beginning of Book IV the title and preface aremissing, and the first section of the Book begins immediatelywith an enunciation. The first section (pp. 176-208) containsPropositions 1-12 which, with the exception of Props. 8-10,seem to be isolated propositions given for their own sakes andnot connected by any general plan.Section (1).Extension of the theorem of Pythagoras.The first proposition is of great interest, being the generalizationof Eucl. I. 47, as Pappus himself calls it, which isby thistime pretty widely known to mathematicians. The enunciationis as follows.'If ABC be a triangle and on AB, AC any parallelogramswhatever be described, as ABDE, ACFG, and if DE, FOproduced meet in H and HA be joined, then the parallelogramsABDE, ACFG are together equal to the parallelogram1Vide notes to Euclid's propositions in The Thirteen Books of Euclid'sElements, pp. 473, 480, 477, 489-91, 501-3.1028.2 H b

370 PAPPUS OF ALEXANDRIAcontained by BC, HA in an angle which is equal to the sum ofthe angles ABC, DHA?Produce HA to meet BG in K, draw EL, CM parallel to KHmeeting DE in L and FG in M, and join LNM.Then BLHA is a parallelogram, and HA is equal andparallelto EL.BSimilarly HA, CM are equal and parallel ;therefore EL, CMare equal and parallel.Therefore BLMC is a parallelogram ;and its angle LBK isequal to the sum of the angles ABC, DHA.Now D ABDE = D BLHAtSimilarlyin the same parallels,= a BLNK, for the same reason.a ACFG = a 4(W# = D NKCM.a ABDE+ a 4(7^(7 = a JBUfCLTherefore, by addition,It has been observed (by Professor Cook Wilson 1 that the)parallelograms on AB, AC need not necessarily be erectedoutwards from AB, AC. If one of them, e.g. that on AC, bedrawn inwards, as in the second figure above, and Pappus'sconstruction be made, we have a similar result with a negativesign, namely,= ABDE-O ACFG.Again, if both ABDE and ACFG were -drawn inwards, theirsum would be equal to BLMC drawn outwards. Generally, ifthe areas of the parallelograms described outwards are regardedas of opposite sign to those of parallelograms drawn inwards,1Mathematical Gazette, vii, p. 107 (May 1913).

THE COLLECTION. BOOK IV 371we may say that the algebraic sum of the three parallelogramsis equal to zero.Though Pappus only takes one case, as was the Greek habit,I see no reason to doubt that he was aware of the resultsin the other possible cases.Props. 2, 3 are noteworthy in that they use the method andphraseology of Eucl. X, proving that a certain line in onefigure is the irrational called minor (see Eucl. X. 76), anda certain line in another figure is * the excess by which thebinomial exceeds the straight line which produces with arational area a medial whole ' (Eucl. X. 77). The propositions4-7 and 11-12 are quite interesting as geometrical exercises,but their isbearing not obvious :Props. 4 and 1 2 are remarkablein that they are cases of analysis followed by synthesisapplied to the proof of theorems. Props. 8-10 belong to thesubject of tangencies, being the sort of propositions that wouldcome as particular cases in a book such as that of ApolloniusOn Contacts', Prop. 8 shows that, if there are two equalcircles and a given point outside both, the diameter of thecircle passing through the point and touching both circlesis ' given ' the; proofis in many places obscure and assumeslemmas of the same kind as those given later a propos ofApollonius's treatise; Prop. 10 purports to show how, giventhree unequal circles touching one another two and two, tofind the diameter of the circle including them and touchingall three.Section (2).On circles inscribed in the('shoemaker's knife 9 ).The next section (pp. 208-32), directed towards the demonstrationof a theorem about the relative sizes of successivecircles inscribed in the dpftrjXos (shoemaker's knife), is extremelyinteresting and clever, and I wish that I had spaceto reproduce it completely. The dpfirjXos, which we havealready met with in Archimedes's 'Book of Lemmas', isformed thus. BO is the diameter of a semicircle BOO andBO is divided into two parts (in general unequal) at D;semicircles are described on BD, DO as diameters on the sameside of BO as BOO is ;semicircles is the &pf$ri\os.the figure included between the threeBb2

372 PAPPUS OF ALEXANDRIAThere is, says Pappus, on record an ancient proposition tothe follo^ng effect. Let successive circles be inscribed in thethe semicircles and one another as shown/3i7\oy touchingin the figure on p. 376, their centres being A, P, ... .Then, ifPi^ Pv Pz be the perpendiculars from the centres A, P, ...on BG and cZ a ,c 2,c? 3... the diameters of the correspondingcircles,pl = d l,p2 = 2(1 2 , ^3 = 3^3....He begins by some lemmas, the course of which I shallreproduce as shortly as I can.I. If (Fig. 1)two circles with centres A, C of which theformer is the greater touch externally at J5, and another circlewith centre G touches the two circles at K, L respectively,then KL produced cuts the circle BL again in D and meetsAC produced in a point E such that AB:BCAE:EC.This is easily proved, because the circular segments DL, LKare similar, and CD is parallel to AG. ThereforeAB:BC = AK:CD = AE: EC.AlsoKE.EL = E&.For AE:EC = AB : BC = AB: CF = (AE-AB):= BE:EF.(EC-CF)But AE :EC = KE : ED;FIG 1.therefore KE :Therefore KE .EL: EL. ED = BE 2 : BEED = BE :.EF.And EL .ED = BE. EF', therefore KE . EL = EB 2 .EF.

THE COLLECTION. BOOK IV 378II. Let (Fig. 2) BC, BD, being in one straight line, be thediameters of two semicircles BGC, BED, and let any circle asFGH touch both semicircles, A being the centre of the circle.Let M be the foot of the perpendicular from A on EC, r theradius of the circle FGH. There are two cases accordingas BD lies along BC or B lies between D and C\ i.e. in thefirst case the two semicircles are the outer and one of the innersemicircles of the ap/fyAoy, while in. the second case they arethe two inner semicircles; in the latter case the circle FGHmay either include the two semicircles or be entirely externalto them. Now, says Pappus,it is to be proved thatin case (1) BM:r = (BG+BD):(BC-BD),and in case (2) BM: r = (BC-BD):(BG+BD).We will confine ourselves to the first case, represented inthe figure (Fig. 2).Draw through A the diameter HF parallel to BC. Then,since the circles BOG, HGF touch at G, and BC, HF areparallel diameters, GHB, GFC are both straight lines.Let E be the point of contact of the circles FGH and BED]then, similarly, BEF, HED are straight lines.Let HK, FL be drawn perpendicular to BC.By the similar triangles BGC, BKH we haveBC:BG = BH: BK, or CB . BK =tGB.BH\and by the similar triangles ELF, BED= BD:BE, or DB.BL = FB.BE.

374 PAPPUS OF ALEXANDRIAButthereforeorGB.BH=FB.BE\CB.BK=DB. BL,BC:BD = BL:BK.Therefore (BG + BD):(BG- BD) = (BL + BK) (BL - BK):= 2BM:KLAnd KL = HF=2r;therefore BM:r = (BC+ BD):(BG-It is next proved that BK .LG = AM 2 .For, by similar triangles BKH, FLC,BK : KH=FL: LG, or BK . LCLastly, since BG :AlsoBD = BL :BD). (a)= KH.FL= AM 2 .(6)BK, from above,BC:CD = BL: KL, or BL.CD = BC.KL= BC.2r.(c)BD-.GD = BK:KL, or BK.GD =BD.KL= BD.2r. (d)III. We now (Fig. 3) take any two circles touching thesemicircles BGG, BED and one another.Let their centres beA and P, H their point of contact, d, d' their diameters respectively.Then, if AM, PN are drawn perpendicular to BG,Pappus proves that (AM+d):d = PN:d f .Draw BF perpendicular to BG and therefore touching thesemicircles BGG, BED at B. Join AP, and produceit tomeet BF in F.Now, by II. (a) above,(BG+ BD) (BC- BD) = BM-.AH,:and for the same reason = BN: PH;it follows that AH:PH= BM : BN= FA:FP.

THE COLLECTION. BOOK IV 375Therefore (Lemma I),if the two circles touch the semicircleBED in jR, E respectively, FRE is a straight line andEF. FR = FH\But EF .FR = FB* ;therefore FH = FB.and MA produced in 8, we have,If now BH meets PN inby similar triangles, FH:FB = PHiPO = A& :AS, whencePH = PO and &4 = AH, so that 0, S are the intersectionsof PN, AM with the respective circles.NFIG. 3.DMJoin BP, and produceitNowo meet MA in AT.BM:BN=FA:FP= AH: PH, from above,AndBMiBN=BK:BPTherefore Kti =circle EUG.,and KA =

376 PAPPUS OF ALEXANDRIAIV. We now come to the substantive theorem.Let FGH be the circle touching all three semicircles (Fig. 4).We have then, as in Lemma II,nsi T*ir z>r* -njJjLf . )/i = JjJJ JjJj,and for the same reason (regarding FGH as touching thesemicircles BGC, DUG)BC.CL'=GD.CK.From the first relation we haveBC:BD = BL:BK,S N K D M L CFIG. 4.whence DC:BD = KL :BK, and inversely BD : DC= BK :while, from the second relation, EG : CD = CK :CL,whence BD: DC = KL : CL.KL = KL :Consequently UK :or BK .LC = KL*.CL,KL,But we saw in Lemma II (6)that BK .LC = AM 2 .Therefore KL = AM, orpl = d rFor the second circle Lemma III gives uswhence, sincepld l ,pzFor the third circlewhencepa = 3d 3.And so on ad infinitum.

THE COLLECTION. BOOK IV 377The same proposition holds when the successive circles,instead of being placed between the large and one of the smallsemicircles, come down between the two small semicircles.where the twoPappus next deals with special cases (1)smaller semicircles become straight lines perpendicular to thediameter of the other semicircle at its extremities, (2)wherewe replace one of the smaller semicircles by a straight linethrough D at right angles to BC, and lastly (3)where insteadof the semicircle DUO we simply have the straightlineand make the first circle touch it and the two other semicircles.Pappus's propositions of course include as particular casesthe partial propositions of the same kind included in the Book''of Lemmas attributed to Archimedes (Props. 5, 6) cf.; p. 102.DCSections (3) and (4).Methods of squaring the circle, and oftrisecting (or dividing in any ratio) any given angle.The last sections of Book IV (pp. 234-302) are mainlydevoted to the solutions of the problems (1) of squaring orrectifying the circle and (2) of trisecting any given angleor dividingit into two parts in any ratio. To this end Pappusgives a short account of certain curves which were used forthe purpose.(a) The Archimedean spiral.He begins with the spiral of Archimedes, proving someof the fundamental properties. His method of finding thearea included (1)between the first turn and the initial line,(2) between any radius vector on the first turn and the curve,is worth giving because it differs from the method of Archimedes.It is the area of the whole first turn which Pappusworks out in detail. We will take the area up to the radiusvector OJB, say.With centreand radius OB draw the circle A'BCD.Let BC be a certain fraction, say 1 / nth, of the arc BCD A',and CD the same fraction, OC, OD meeting the spiral in F, Erespectively. Let A r $, 8V be the same fraction of a straightline KR, the side of a square KNLR.Draw ST, VW parallelto KN meeting the diagonal KL of the square in U, Q respectively,and draw MU> PQ parallel to KR.

378 PAPPUS OF ALEXANDRIAWith as centre and OE, OF as radii dr&w arcs of circlesmeeting OF, OB in H, G respectively.For brevity we will now denote a cylinder in which r is theradius of the base and h the height by (cyl. r, h)and the conewith the same base and height by (cone r, h).N T wBy the property of the spiral,whenceNowOB:BG = (arc A'DOB) := RK : KS= NK :KM,OB:OG = NK: NM.(arc CB)2 = JVTf 2 :(sector OBG) : (sector OGF) = 2: OG= (cyl.:KN, NT) (cyl. MN, NT).Similarly(sector 000) : (sector OEH) = (cyl.:ST, TW) (cyl. PT. TW),and so on.The sectors OBC, OCD... form the sector OA'DB, and thesectors OFG, OEH ... form a figure inscribed to the spiral.In like manner the cylinders (KN, TN), (XT, TW) ... form thecylinder (KN, NL), while the cylinders (MN, NT), (PTy TW) ...form a figure inscribed to the cone (KN> NL).Consequently(sector OA'DB) :(fig. inscr. in spiral)= (cyl. KN, NL) : (fig. inscr. in cone KN, NL).

THE COLLECTION. BOOK IV 379We have a similar proportion connecting a figurecircumscribedto the spiral and a figure circumscribed to the cone.By increasing n the inscribed and circumscribed figures canbe compressed together, and by the usual method of exhaustionwe have ultimately(sector OA'DB) : (area of = :spiral) (cyL KN, NL) (cone KN, NL)= 3:1,or (area of spiral cut off'by OB) = (sector OA'DB).The ratio of the sector OA'DB to the complete circle is thatof the angle which the radius vector describes in passing fromthe position OA to the position OB to four right angles, thatis, by the property of the spiral, r : a, where r = OB, a = OA.Therefore (area of spiral cut off by OB) = r irr*.aSimilarly the area of tfye/vector r' = 4anr'*.spiral cut off by any other radiusTherefore (as Pappus proves in his next proposition) thefirst area is to the second as r 3 to r' 3 .Considering the areas cut off by the radii vectores at thepoints where the revolving line has passed through anglesof TT, TT, ITT and %TT respectively, we see that the areas are inthe ratio of () 3 , () 3 3, (|)1 or, 1, 8, 27, 64, so that the areas ofthe spiral included in the four quadrantsof 1, 7, 19, 37 (Prop. 22).are in the ratio()3)The conchoid of N'icomedes.The conchoid of Nicomedes is next described (chaps. 26-7),and it is shown (chaps. 28, 29) how it can be used to find twogeometric means between two straight lines, and consequentlyto find a cube having a given ratio to a given cube (see vol. i,pp. 260-2 and pp. 238-40, where I have also mentionedPappus's remark that the conchoid which he describes is thefirst conchoid, while there also exist a second, a third and afourth which are of use for other theorems).(y)The quadratrix.The quadratrixis taken next (chaps. 30-2), with Sporus'scriticism questioning the construction as involving a petitio

380 PAPPUS OF ALEXANDRIAprinoipii. Its use for squaring the circle is attributed toDinostratus and Nicomedes. The whole substance of thissubsection is given above (vol. i, pp. 226-30).Tivo constructions for the quadratrix by means of1surface-loci '.In the next chapters (chaps. 33, 34, Props. 28, 29) Pappusgives two alternative ways of producing the quadratrix 'bymeans of surface-loci', for which he claims the merit thatthey are geometrical rather than c too mechanical' as thetraditional method (of Hippias) was.(1) The first method uses a cylindrical helix thus.Let ABC be a quadrant of a circle with centre B, andlet BD be any radius.Supposethat EF, drawn from a point Eon the radius BD perpendicularto BC, is (for all such radii) ina given ratio to the arc DC.'I say ', says Pappus, c that thelocus of E is a certain curve.'Suppose a right cylindererected from the quadrant anda cylindricalits surface. Let DH beuponthe generator of this cylinder through J) 9helix CGH drawnmeetingthe helixin H. Draw BL, El at right angles to the plane of thequadrant, and draw HIL parallel to BD.Now, by the property of the helix, EI( = DH) is to thearc CD in a given ratio. Also EF :(arc CD) = a given ratio.El is given. .And since EF, El areTherefore the ratio EF :given in position, Fl is given in position. But FI is perpendicularto BG. Therefore FI is in a plane given in position,and so therefore is 7.But I is also on a certain surface described by the line LH 9which moves always parallel to the plane ABC, with oneextremity L on BL and the other extremity// on the helix.Therefore / lies on the intersection of this surface with theplane through FI.

THE COLLECTION. BOOK IV 381Hence / lies on a certain curve. Therefore E, its projectionon the plane ABC, also lies on a curve.In the particular case where the given ratio of EF to thearc CD is equal to the ratio of BA to the arc CA, the locus ofE is a quadratrix.[The surface described by the straight line LH is a plectold.The shape of it is perhaps best realized as a continuous spiralstaircase, i.e. a spiral staircase with infinitely small steps.The quadratrixis thus produced as the orthogonal projectionof the curve in which the plectoid is intersected by a planethrough BC inclined at a given angle to the plane ABC.not difficult to verify the result analytically.]It is(2) The second method uses a right cylinder the base

382 PAPPUS OF ALEXANDRIAcurve, describes a certain plectoid, which therefore contains thepoint J.Also IE = EF, IF is perpendicular to BC, and hence IF, andtherefore /, lies on a fixed plane through BC inclined to ABOat an angle of ^TT.Therefore I, lying on the intersection of the plectoid and thesaid plane, lies on a certain curve. So therefore does theprojection of / on ABC, i.e. the point E.The locus of E is clearly the quadratrix.[This result can also be verified analytically.](S) Digression: a spiral on a sphere.Prop. 30 (chap. 35) is a digression on the subject of a certainspiral described on a sphere, suggested by the discussion ofa spiral in a plane.Take a hemisphere bounded by the great circle KLM,with H as pole. Suppose that the quadrant of a great circleabout the radius HO so that KHNK revolves uniformlydescribes the circle KLM and returns to its original positionat K, and suppose that a point moves uniformly at the sameNtime from H to K at such speed that the point arrives at Kat the same time that HK resumes its original position. Thepoint will thus describe a spiral on the surface of the spherebetween the points H and K as shown in the figure.Pappus then sets himself to prove that the portion of thesurface of the sphere cut off towards the pole between thespiral and the arc HNK is to the surface of the hemisphere in

THE COLLECTION. BOOK IV 383a certain ratio hown in the second figure where ABC i$a quadrant of a circle equal to a great circlein the sphere,namely the ratio of the segment ABC to the sector DABC.Draw the tangent CF to the quadrantat C. With C ascentre and radius CA draw the circle AEF meeting CF in F.Then the sect&r CAF is equal to the sector ADC (sinceCA* = 2AD\ while Z ACF = Z 4(7).It isrequired, therefore, to prove that, if S be the area cutoff by the spiral as above described,8 :(surface of hemisphere) = (segmt. A BC):(sector CAF).Let KL be a (small) fraction, say l/?ith, of the circum-of theference of the circle KLM, and let HPL be the quadrantgreat circle through H, L meeting the spiral in P. Then, bythe property of the spiral,(arc HP) : (arc HL) = :(arc KL) (circumf. of KLM)= 1 : 71.Let the small circleabout the pole H.NPQ passing through P be describedNext let FE be the same fraction, I/nth, of the arc FAand join ECthat KL is of the circumference of the circle KLMythe arc ABC in B. With C as centre and CB asmeetingradius describe the arc BG meeting GF in G.Then the arc CB is the same fraction, I/ nth, of the arcCBA that the arc FE is of FA (for it is easily seen thatLFGE = \LBDC, while LFCA = %CDA). Therefore, since(arc CBA) = (arc HPL), (arc CB) = (arc HP), and chord CB= chord HP.

384 PAPPUS OF ALEXANDRIANow (sector HPN on sphere):(sector HKL on sphere)=2(chord //P): (chord 7//,) 2(a consequence of Archimedes, On Sphere and Cylinder,I. 42).And HP 2 : HL = 2 OB 2 : GA2Therefore(sector HPN):(sector HKL) = (sector f77?(?):(sectorSimilarly, if the arc LI/ be taken equal to the arc KL andthe great circle through H, L / cuts the spiral in P', and a smallcirclein p ;described about H and through P' meets the arc HPLand if likewise the arc BB' is made equal to the arc Jt( \and CB' is produced to meet AF in E', while again a circulararc with G as centre and CB' as radius meets CE in 6,(sector /fP'?; on sphere) :And so on.(sector HLL' on sphere)= (sector CB'b):(sector CE'K).Ultimately then we shall get a figure consistingof sectorson the sphere circumscribed about the area H of the spiral anda figure consisting of sectors of circles circumscribed about thesegment GBA and in like manner we shall have inscribed;figures in each case similarly made up.The method of exhaustion will then give$: (surface of hemisphere) = (segint. ABC) : (sector CAP)= (segmt.:ABC) (sector DAG).[We may, as an illustration, give the analytical equivalentof this proposition.If p,o> be the spherical coordinates of Pwith reference to H as pole and the arc HNK as polar axis,the equation of Pappus's curve is obviously o> = 4 p.If now the radius of the sphereis taken as unity, we have asthe element of areadA = dco(l cosp)= 4rfp(l cosp).pi*Therefore A = 4dp (1 cosp) = 27r 4.Jo

THE COLLECTION. BOOK IV 385ThereforeA 27r-4 -4--^(surface of hemisphere)"~~2?r__ ~ (segment ABC) ^"(sector DABC) "JThe second part of the last section of Book IV (chaps. 36-41,of tri-pp. 270-302) is mainly concerned with the problemsecting any given angle or dividing it into parts in any givenratio. Pappus begins with another account of the distinctionbetween plane, solid and linear problems (cf. Book III, chaps.20-2) according as they require for their solution (1) the'straight line and circle only, (2) conies or their equivalent,(3) higher curves still, which have a more 'complicated andforced (or unnatural) origin, being produced from moreirregular surfaces and involved motions. Such are the curveswhich, are discovered in the so-called loci on surfaces, asdiscovered bymeans of the inter-well as others more complicatedstill and many in numberDemetrius of Alexandria in his Linear considerationsand by Philoii of Tyana bylacing of plectoids and other surfaces of all sorts, all of whichcurves -possess many remarkable properties peculiar to them.Some of these curves have been thought by the more recentone of them iswriters to be worthy of considerable discussion ;that which also received from Menelaus the name of thejxirado.rical curve. Others of the Fame class are flpirals,cjuadratrices, cochloids and cissoids.' He adds the often-quotedreflection -on the error committed by geometers w.hen theysolve a -problem by means of an 'inappropriate class* (ofthe use incurve or its equivalent), illustrating this byApollonius, Book V, of a rectangular hyperbola for finding thefeet of normals to a parabola passing through one point(where a circle would serve the purpose), and by the assumptionby Archimedes of a tolid i/ei/(7*9 in his book On Spirals(see above, pp. 65-8).Trisection (or division in any ratio] of any angle.The method of trisecting any angle based on a certain i/cOcnyis next described, with the solution of the vevvis itselfby1MS ?C C

386 PAPPUS OF ALEXANDRIAmeans of a hyperbola which has to be constructed from certaindata, namely the asymptotes and a certain point throughwhich the curve must pass (this easy construction is given inProp. 33, chap. 41-2). Then the problem is directly solved(chaps. 43, 44) by means of a hyperbola in two ways practicallyequivalent, the hyperbola being determined in the onecase by the ordinary Apollonian property, but in the other bymeans of the focus-directrix property. Solutions follow ofthe problem of dividing any angle in a given ra^tio by means(1) of the quadratrix, (2) of the spiral of Archimedes (chaps.45, 46). All these solutions have been sufficiently describedabove (vol. i, pp. 235-7, 241-3, 225-7).Some problems follow (chaps. 47-51) depending on thesenamely those of constructing an isosceles triangle inresults,which either of the base angles has a given ratio to the verticalangle (Prop. 37), inscribing in a circle a regular polygon ofany number of sides (Prop. 38), drawing a circle the circumferenceof which shall be equal to a given straight line (Prop.39), constructing on a given straight line AB a segment ofa circle such that the arc of the segment inay have a givenratio to the base (Prop. 40),and constructing an angle incommensurablewith a given angle (Prop. 41).Section (5).Solution of the j/cCo-iy of Archimedes, c On Spirals ',Prop. 8, by means of conies.Book IV concludes with the solution of the vevtns which,assumed inaccording ,to Pappus, Archimedes unnecessarilyOn Spirals, Prop.8. Archimedes's assumptionis this. Givena circle, a chord (BC) in it less than the diameter, and a pointA on the circle the perpendicular from which to BC cuts BCin a point D such that BD > DC and meets the circle againin E, it is possible to draw through A a straight line ARPcutting BC in R and the circle in P in such a way that RPshall be equal to DE (or, in the phraseology of vcvo-ei?, toplace between the straight line BC and the circumferenceof the circle a straight line equal to DE and vergingtowards A).Pappus makes the problem rather more general by notrequiring PR to be equal to DE, but making it of any given

THE COLLECTION. BOOK IV 387length (consistent with a real solution).The problemis bestexhibited by means of analytical geometry.If BD = a, DC = 6, AD = c (so that DE = ab/c), we haveto find the point R on BC such that AR produced solves theproblem by making PR equal to k, say.Let DR = x. Then, since BR . RC = PR.RA,we have(a-x)(b + x)= AAn obvious expedientis to 2put T/for \/('we have(a-x) (b + x)= ky,and = ifc.'2+x 2 .when(I)(2)These equations represent a parabola and a hyperbolarespectively, and Pappus does in fact solve the problem bymeans of the intersection of a parabola and a hyperbola ;oneof his preliminary lemmas is, however, again a little moregeneral. In the above figure y is represented by RQ.The first lemma of Pappus (Prop. 42, p. 298) states that, iffrom a given point A any straight line be drawn meetinga straight line BC given in position in R, and if RQ be drawnat right angles to'JSC and of length bearing a given ratioto AR, the locus of Q is a hyperbola.For draw AD perpendicular to BC and produceit to A 'so thatQR:RA = A'D:DA = the given ratio.cc 2

388 PAPPUS OF ALEXANDRIAMeasure DA" along DA equal to DA'.Then, if QN be perpendicular to AD,(AR*-AD*):(QR*-A'D*) = (const.),that is, QN 2 : A'N .A"N = (const.),and the locus of Q is a hyperbola.The equation of the hyperbola is clearlywhere p,is a constant. In the particular case taken byArchimedes QR = RA or9 p,= 1, and the hyperbola becomesthe rectangular hyperbola (2) above.The second lemma (Prop. 43, p. 300) proves that, if BC isgiven in length, and Q is such a point that, when QR is drawnperpendicular to BC, BR . RC = k .QR, where k is a givenlength, the locus of Q is a 'parabola.Let be the middle point of BC, and let OK be drawn atright angles to BC and of lengthsuch thatLet QN' be drawn perpendicular to OK.Then QN'* = OR 2= k .(KO - QJJ), by hypothesis,= k . KN'.Therefore the locus of Q is a parabola.The equation of the parabola referred to DB, DE as axes ofx and y is obviouslywhich easily reduces to(a x) (& + #)= ky, as above (1).In Archimedes's particular case k = ab/c.To solve the problem then we have only to draw the parabolaand hyperbola in question, and their intersection thengives Q, whence R, and therefore ARP, is determined.

THE COLLECTION. BOOKS IV, V 389Book V. Preface on the Sagacity of Bees.It is characteristic of the great Greek mathematicians that,whenever they were free from the restraint of the technicallanguage of mathematics, as when for instance they had occasionto write a preface, they were able to write in language ofthe highest literary quality, comparable with that of thephilosophers, historians, and poets. We have only to recallthe introductions to Archimedes' s treatises and the prefacesto the different Books of Apollonius's Conies. Heron, thoughseverely practical, is no exception when he has any generalexplanation, historical or other, to give. We have now tonote a like case in Pappus, namely the preface to Book V ofthe Collection. The editor, Hultsch, draws attention to theelegance and purity of the language and the careful writing ;the latter is illustrated 1by the studied avoidance of hiatus.The subject is one which a writer of taste arid imaginationwould naturally find attractive, namely the practical intelligenceshown by bees inselecting the hexagonal form for thecells in the honeycomb. Pappus does not disappoint us the;passage is as attractive as the subject,and deserves to bereproduced.4It is of course to men that God has given the best andmost perfect notion of wisdom in general and of mathematicalscience in particular, but a partial share in these things heallotted to some of the unreasoning animals as well. To men,as being endowed with reason, he vouchsafed that they shoulddo everything in the light of reason and demonstration, but tothe other animals, while denying them reason, he grantedthat each of them should, by virtue of a certain naturalinstinct, obtain just so much as is needful to support life.This instinct may be observed to exist in very many otherspecies of living creatures, but most of all in bees. In the firstplace their orderliness and their submission to the queens whorule in their state are truly admirable, but much more admirablestill is their emulation, the cleanliness they observe in thegathering of honey, and the forethought and housewifely carethey devote to its custody. Presumably because they knowthemselves to be entrusted with the task of bringing fromthe gods to the accomplished portion of mankind a share of1Pappus, vol. iii, p. 1233.

390 PAPPUS OF ALEXANDRIAambrosia in this form, they do not think itproper to pour itcarelessly on ground or wood or any other ugly and irregularmaterial ; but, first collecting the sweets of the moat beautifulflowers which grow on the earth, they make from them, forthe reception of the honey, the vessels which we call honeycombs,(with cells) all equal, similar and contiguous to oneanother, and hexagonal in form. And that they have contrivedthis by virtue of a certain geometrical forethought wemay infer in this way. They would necessarily think thatthe figures must be such as to be contiguous to one another,that is to say, to have their sides common, in order that noforeign matter could enter the interstices between them andso defile the purity of their produce. Now only three rectilinealfigures would satisfy the condition, I mean regularfigures which are equilateral and equiangular; for the 'beeswould have none of the figures which are not uniform. . . .There being then three figures capable by themselves ofexactly filling up the space about the same point, the bees byreason of their instinctive wisdom chose for the constructionof the honeycomb the figure which has the most angles,because they conceived that it would contain more honey thaneither of the two others.'Bees, then, know just this fact which is of service to themselves,that the hexagon is greater than the square and thetriangle and will hold more honey for the same expenditure ofmaterial used in constructing the different figures. We, however,claiming as we do a greater share in wisdom than bees,will investigate a problem of still wider extent, namely that,of all equilateral and equiangular plane figures having anequal perimeter, that which has the greater number of anglesisalways greater, and the greatest plane figure of all thosewhich have a perimeter equal to that of the polygonsis thecircle/Book V then is devoted to what we may callisoperivietry,including in the term not only the comparison of the areas ofdifferent plane figures with the same perimeter, but that of thecontents of different solid figures with equal surfaces.Section (1). Isoperimetry after Zetiodorus.The first section of the Book relating to plane figures(chaps. 1-10, pp. 308-34) evidently followed very closelythe exposition of Zenodorus Trepi /j/ cryrffidrîf (seepp. 207-13, above) but before;passing to solid figures Pappusinserts the proposition that of all circular segments having

THE COLLECTION. BOOK V 391the sa'fne circumference the semicircle is the greatest,with somepreliminary lemmas which deserve notice (chaps. 15, 16).(1)ABC is a triangle right-angled, at B. With C as centreand radius CA describe the arcAD cutting CB produced in D.To prove that (R denoting a rightangle)(sector CA D):(area ABD)>R:/.BCA.Draw AF at right angles to CA meeting CD produced in F9and dfaw BH perpendicular to AF.AB as radius describe the arc GBE.Now (area:EBF) (area EBH) > (area:EBF)With A as centre andand, component, &FBH:(EBH) > &ABF: (ABE).But (by an easy lemma which has just preceded)whenceandAFBU:(E/UI) = &ABF:(ABD),&ABF: (ABD) > &ABF: (ABE),(ABE) > (ABD).(.sector ABE),Therefore (ABE) (ABG) > (ABD) : :(ABG)> (ABD) :^ ABC, a fortiori.Therefore L BAF: L BAC > (ABD):A ABC: (ABD) > L BAC: L BAF.(A BD) > R :whence, inversely,and, comiwneiuto, (sector ACD):&ABC,L BCA.[If a be the circular measure of LEG A, this gives (if AC=b)iaft 2 :(a& 2 ---| sin a cosa.fr2 )> TT :a,or 2a:(2a sin2a) > 7r:2a;that is, 0/(0-sin0) > ir/0,where < < TT.](2) ABC is again a triangle right-angled at B. With C ascentre and CA as radius draw a circle AD meeting BC producedin D. To prove that(sector CAD):(area ABD) > R : LAGD.,

392 PAPPUS OF ALEXANDRIADraw AE at right angles to AC.' With A as centre andAC as radius describe the circle FOE meeting AB producedin F&ud AEmE.Then, since LACD > LVAE, (sector ACD) > (sector ACE).Therefore (AGO) : &ABC> (ACE): &ABG> (ACE):(ACF), a fortiori,> LEAC-.LGAB.Inversely, (ACD):(ABD)Inversely,A ABC: (ACD) < LGABÊAC,and, componendoy:(ABD) (ACD)< L EAB :> Z EAC : L EABL EAC.We come now to the application of these lemmas to theproposition comparing the area of a semicircle with that ofother segments of equal circumference (chaps. 17, 18).A semicircle is the greatest of all segments of circles whichhave the same circumference.Let ABC be a semicircle with centre 0, and DEF anothersegment of a circle such that the circumference DEF is equalto the circumference ABC. I saygreater than the area of DEF.Let H be the centre of the circle DEF.that the area of ABC isDraw EUK, BG atright angles to DF, AC respectively. Join D1I, and drawLHM parallelto DF.

THE COLLECTION. BOOK V 393Then LH:AG = (arc LE) : (arc AB)Also Lll* : AG*= (sector LHE):= (sector LHE):(sector(sectorTherefore the sector LHE is to the sector AGB in theratio duplicate of that which the sector LHE has to thesectorDUE.Therefore(sector LHE):(sector DUE) = (sector DHE):(sector EDH):(sector AGB).Now (1) in the case of the segment less than a semicircleand (2) in the case of the segment greater than a semicircle(EDK) > R : Z DHE,by the lemmas (1) and (2) respectively.That is,(sector EDH ):(EDK) > Z LHE: Z DHE> (sector LHE):> (sector EDH):(sector DHE)(sector AGB),from above.Therefore the half segment EDK is less than the halfsemicircle A(1B, whence the semicircle ABC is greater thanthe segment DEh\We have already described the content of Zenodorus'streatise (pp. 207-13, above) to which, so far as plane figuresare concerned, Pappus added nothing except the above propositionrelating to segmentsof circles.JSection (2). Comparison of volumes of solids liaviny theirsurfaces equal. Case of the spfiere.The portion of Book V dealing with solid figures begins(p. 350. 20) with the statement that the philosophers whoconsidered that the creator gave the universe the form of asphere because that was the most beautiful of all shapes alsoasserted that the sphereis the greatest of all solid figures

394 PAPPUS OF ALEXANDRIAwhich have their surfaces equal ; this, however, they had notproved, nor could it be proved without a long investigation.Pappus himself does not attempt to prove that the sphere isgreater than all solids with the same surface, but only thatthe sphere is greater than any of the five regular solids havingthe same surface (chap. 19) and also greater than either a coneor a cylinder of equal surface (chap. 20).Section (3). Digression on the semi-regular solidsof Archimedes.He begins (chap. 19) with an account of the thirteen tseniiregularsolids discovered by Archimedes, which are containedby polygons all equilateral and all equiangular but not allsimilar (see pp. 98-101, above), and he shows how to determinethe number of solid angles and the number of edges whichthey have respectively he then gives them the go-by for his;present purpose because they are not completely regularstill;less does he compare the sphere with any irregular solidhaving an equal surface.The sphere is greater than any of the regularsolids whichhas its surface equal to that of the sphere.The proof that the sphere is greater than any of the regularsolids with surface equal to that of the sphereis the same asthat given by Zenodorus. Let P be any one of the regular solids,S the sphere with surface equal to that of P. To prove thatS>P.Inscribe in the solid a sphere s, and suppose that r is itsradius. Then the surface of P is greater than the surface of s,and accordingly, if It is the radius of /S,R > r. But thevolume of 8 is equal to the cone with base equal to the surfaceof S yand therefore of P, and height equal to R and the volume;of P is equal to the cone with base equal to the surface of Pand height equal to r. Therefore, since R >?*, volume of tf >volume of P.Section (4).'Propositions on the lilies of Archimedes,On the Sphere and Cylinder '.For the fact that the volume of a sphere is equal to the conewith base equal to the surface, arid height equal to the radius,

ofTHE COLLECTION. BOOK V 395the sphere, Pappus quotes Archimedes, On the Sphere andCylinder, but thinks proper to add a series of propositions(chaps, 20-43, pp. 362-410) on much the same lines as those ofArchimedes and leading to the same results as Archimedesobtains for the surface of a segment of a sphere and of the wholesphere (Prop. 28), and for the volume of a sphere (Prop. 35).Prop. 36 (chap. 42) shows how to divide a sphere into twosegments such that their surfaces are in a given ratio andProp. 37 (chap. "43) proves that the volume as well as thesurface of the cylinder circumscribing a sphere is 1^ timesthat of the sphereitself.Among the lemmatic propositions in this section of theBook Props. 21, 22 may be mentioned. Prop. 21 proves that,if C, E be two points on the tangent at H to a semicircle suchthat CH = HE, and if CD, EF be drawn perpendicular to thethen (CD + EF) CE-AB. DF\ Prop. 22 provesa like result where C, E are points on the semicircle, CD, EFdiameter AByare as before perpendicular to AB, and EH is the chord ofthe circle subtending the arc which with CE makes up a semicircle;in this case (CD + EF) CE = EH .DF.Both resultsare easily seen to be the equivalent of the trigonometricalformulasin (x + y) + sin (xy) = 2 sin x cos yyor, if certain different angles be taken as x, y,sin x 4- sin y , , , v= cot 4(8 y).cos //cos xSection (5).Of regular solufs with surfaces equal, that isgreater which has more faces.Returning to the main problem of the Book, Pappus showsthat, of the five regular solid figures assumed to have theirsurfaces equal, that is greater which has the more faces, sothat the pyramid, the cube, the octahedron, the dodecahedronand the icosahedron of equal surface are, as regards solidcontent, in ascending order of magnitude (Props. 3-56).Pappus indicates (p. 410. 27) that 'some of the ancients' hadworked out the proofs of these propositions by the analyticalmethod; for himself, he will give a method of his own by

396 PAPPUS OF ALEXANDRIAsynthetical deduction, for which he claims that it is clearerand shorter. We have first propositions (with auxiliarylemmas) about the perpendiculars from the centre of thecircumscribing sphere to a face of (a) the octahedron, (6) theicosahedron (Props. 39, 43), then the proposition that, if adodecahedron and an icosahedron be inscribed in the samesphere, the same small circle in the sphere circumscribes boththe pentagon of the dodecahedron and the triangle of theicosahedron (Prop. 48) ;this last is the proposition proved byHypsicles in the so-called Book XIV ' of Euclid ',Prop. 2, andPappus gives two methods of proof, the second of which (chap.56) corresponds to that of Hypsicles. Prop. 49 proves thattwelve of the regular pentagons inscribed in a circle are togethergreater than twenty of the equilateral triangles inscribed inthe same circle. The final propositions proving that the cubeis greater than the pyramid with the same surface, the octahedrongreater than the cube, and so on, are Props. 52-6(chaps. 60-4). Of Pappus's auxiliary propositions, Prop. 41is practically contained in Hypsicles's Prop. 1, and Prop. 44in Hypsicles's last lemma; but otherwise the expositionisdifferent.Book VI.On the contents of Book VI we can be brief.It ismainlyastronomical, dealing with the treatises included in the socalledLittle Astronomy, that is, the smaller astronomicaltreatises which were studied as an introduction to the greattiyntaxis of Ptolemy. The preface says that many of thosewho taught the Treasury of Astronomy through a careless>understanding of the propositions, added some things as beingnecessary and omitted others as unnecessary. Pappus mentionsat this point an incorrect addition to Theodosius, Sphaerica,III. 6, an omission from Euclid's Phaenomena, Prop. 2, aninaccurate representation of Theodosius, On Days and Nights,Prop. 4, and the omission later of certain other things asbeinunnecessary. His object is to put these mistakesright. Allusions are also found in the Book to Menelaus'sSpftaertca, e.g. the statement (p. 476. 16) that Menelaus inhis Sphaerica called a spherical triangle TpiirXtvpov, three-side.

THE COLLECTION. BOOKS V, VI 397The Sphaerica of Theodosius is dealt with at some length(chaps. 1-26, Props. 1-27), and so are the theorems ofAutolycus On the moving Sphere (chaps. 27-9), TheodosiusOn Days and Nights (chaps. 30-6, Props. 29-38), Aristarchus0??, the sizes and distances of the Sun and Moon (chaps. 37-40,including a proposition, Prop. 39 with two lemmas, which iscorrupt at the end and is not really proved), Euclid's Optics(chaps. 41-52, Props. 42-54), and Euclid's Phaenomena (chaps.53-60, Props. 55-61).Problem arising out of Euclid's 'Optws*.There is little in the Book of general mathematical interestexcept the following propositions which occur in the section onEuclid's Optics.Two propositions are fundamental in solid geometry,namely :(a) If from a point A above a plane AB be drawn perpendicularto the plane, and if from B a straight line BD bedrawn perpendicular to any straight line EF in the plane,then will AD also be perpendicular to EF (Prop. 43).(/>)If from a point A above a plane A B be drawn to the planebut not at right angles to it, and AM be drawn perpendicularto the plane (i.e.the plane), the angle ABM is the least of all the angles whichAB makes with any straight lines through B, as BP, in theplane the angle A BP increases as BP moves away from BM;on either side* ; and, given any straight line BP makinga certain angle with BA> only one other straight line in theif BM be the orthogonal projection of BA onplane will make the same angle with BA, namely a straightline Rl v on the other side of BM making the same angle withit that BP does (Prop. 44).These are the first of a series of lemmas leading up to themain problem, the investigation of the apparent form ofa circle as seen from a point outside its plane. In Prop. 50(= Euclid, Optics, 34) Pappus proves the fact that all thediameters of the circle will appear equal if the straight linedrawn from the point representing the eye to the centre ofthe circle is either (a) at right angles to the plane of the circleor (6),if not at right angles to the plane of the circle, is equal

398 PAPPUS OF ALEXANDRIAin length to the radius of the circle. In all other cases(Prop. 51 = Eucl. Optics, 35) the diameters will appear unequal.Pappus's other propositions carry farther Euclid's remarkthat the circle seen under these conditions will appeardeformed or distorted (TrapcerTraoy-eli'os), proving (Prop. 53,pp. 588-92) that the apparent form will be an ellipse with itscentre not, 'as some think', at the centre of the circle butat another point in it, determined in this way. Given a circleABDE with centre 0, let the eye be at a point F above theplane of the circle such that FO is neither perpendicularto that plane nor equal to the radius of the circle. Draw FGperpendicular to the plane of the circle and let ADG be thediameter through G. Join AF, DF, and bisect the angle AFDby the straight line FC meeting AD in C. Through C drawBE perpendicular to AD, and let the tangents at B, E meetAG produced in K. Then Pappus proves that C (not 0)is thecentre of the apparent ellipse, that AD, BE are its major andminor axes respectively, that the ordinates to AD are parallelto BE both really and apparently, and that the ordinates toBE will pass through K but will appear to be parallel to AD.Thus in the figure,G being the centre of the apparent ellipse,it is proved that, if LCM is any straight line through (7, LC isapparently equal to CM (it is practically assumed a propositionproved later in Book VII, Prop. 156 that, if LK meetthe circle again in P, and if PM be drawn perpendicular toAD to meet the circle again in M, LM passes through 0).

THE COLLECTION. BOOKS VI, VII 399The test of apparent equality is of course that the two straightlines should subtend equal angles at F.The main points in the proof are these. The plane throughOF, CK ishence OF isperpendicular to the planes BFJB, PFM and LFR;perpendicular to BE, QF to PM and HF to LR,whence BG and CE subtend equal angles at F: so do LH, HR,and PQ, QM.Since FG bisects the angle AFD, and AC:GD = AK :KD(by the polar property), Z.CFK is a right angle.And GF isthe intersection of two planes at right angles, namely AFKand BFJS, in the former of which FK lies; therefore KF isperpendicular to the plane BFE, and therefore to FN. Sincetherefore (by the polar property) LN : NP = LK :KP, itfollows that the angle LFP is bisected by FN] hence LN, NPare apparently equal.Again LC : CM = LN:NP = LF: FP = i^:mTherefore the angles LFC, GFM are equal,and LC, CMareapparently equal.LastlyLR:PM=LK:KP=LN:NP=LFiFP\ thereforethe isosceles triangles FLR, FPM are equiangular; thereforethe angles PFM, LFR, and consequently PFQ, LFH, areequal. Hence LP, RM will appear to be parallel to AD.We have, based on this proposition, an easy method ofsolving Pappus's final problem (Prop, 54).*Given a circleABDE and any point within it, to find outside the plane ofthe circle a point from which the circle will have the appearanceof an ellipse with centre C!We have only to produce the diameter AD through C to thepole K of the chord BE perpendicular toAD and then, inthe plane through AK perpendicular to the plane of the circle,to describe a semicircle on CK as diameter.this semicircle satisfies the condition.Any point F onBook VII. On the ' Treasury of A nalysis'.Book VII is of much greater importance, since it gives anaccount of the books forming what was called the Treasury ofAnalysis (avaXvontvos r6iro$) and, as regards those of the bookswhich are now lost, Pappus's account, with the hints derivablefrom the large collection of lemmas supplied by him to each

400 PAPPUS OF ALEXANDRIAbook, practically constitutes our only source of information.The Book begins (p. 634) with a definition of analysis andsynthesis which, as being the most elaborate Greek utteranceon the subject, deserves to be quoted in full.cThe so-called Hi/aAv6/zi/o9 is, to put it shortly, a specialbody of doctrine provided for the use of those who, afterfinishing the ordinary Elements, are desirous of acquiring thepower of solving problems whicli may be set them involving(the construction of) lines, and it is useful for this alone.It isthe work of three men, Euclid the author of the Elements,Apollonius of Perga and Aristaeus the elder, and proceeds byway of analysis and synthesis.'Definition of Analysis and Synthesis.cAnalysis, then, takes that which is sought as if it wereadmitted and passes from itthroughits successive consequencesto something which is admitted as the result ofsynthesis: for in analysis we assume that which is soughtas if it were already done (yeyoi/os), and we inquire what it isfrom which this results, and again what is the antecedentcause of the latter, and so on, until by so retracing our stepswe come upon something already known or belonging to theclass of first principles, and such a method we call analysisas being solution backwards (avaira\iv Xvo-iv).1But in synthesis, reversing the process, we take as alreadydone that which was last arrived at in the analysis and, byarranging in their natural order as consequences what beforewere antecedents, and successively connecting them one withanother, we arrive finally at the construction of what wassought and this we call synthesis.;'Now analysis is of two kinds, the one directed to searchingfor the truth and called theoretical, the other directed tofinding what we are told to find and called problematical.(1) In the theoretical kind we assume what is sought as ifit were existent and true, after which we pass throughitssuccessive consequences, as if they too were true and establishedby virtue of our hypothesis, to something admitted: then(a), if that something admitted is true, that which is soughtwill also be true and the proof will correspond in the reverseorder to the analysis, but (b),if we come upon somethingadmittedly false, that which is sought will also be false.(2) In the jyroUematical kind we assume that which is propoundedas if it were known, after which we pass throughits

THE COLLECTION. BOOK VII 401vsuccessive consequences, taking them as true, up to somethingadmitted : if then (a) what is admitted is possible and obtainable,that is, what mathematicians call given, what wasoriginally proposed will also be possible, and the proof willagain correspond in the reverse order to the analysis, but if (b)we come upon something admittedly impossible, the problemwill also be impossible/This statement could hardly be improved upon except thatitought to be added that each step in the chain of inferencein the analysis must be unconditionally convertible] that is,when in the analysis we say that, if A is true, B is true,we must be sure that each statement is a necessary consequenceof the other, so that the truth of A equally followsfrom the truth of B. This, however, is almost implied byPappus when he says that we inquire, not what it is (namelyB) which follows from A, but what it is (B) from which Afollows, and so on.List of works in the ' Treasury of Analysis \Pappus adds a list, in order, of the books forming theSy namely:*Euclid's Data, one Book, Apollonius's Cutting-off of a ratio,two Books, Cutting-off of an area, two Books, DeterminateSection, two Books, Contacts, two Books, Euclid's Por/sms,three Books, Apollonius's Inclinations or Vergings (vvat$),two Books, the same author's Plane Loci, two Books, andConies, eight Books, Aristaeus's Solid Loci, five Books, Euclid'sSurface-Loci, two Books, Eratosthenes's On means, two Books.There are in all thirty-three Books, the contents of which upto the Conies of Apolloniua I have set out for your consideration,including not only the number of the propositions, thediorismi and the cases dealt with in each Book, but also thelemmaa which are required; indeed I have not, to the bestof my belief, omitted any question arising in the study of theBooks in question.'Description of thetreatises.Then follows the short description of the contents of thevarious Books down to Apollonius's Conies; no account isgiven of Aristaeus'& Solid Loci, Euclid's Surface-Loci and1531.2 D d

402 PAPPUS OF ALEXANDRIAEratosthenes's On rrtâns, nor are there any lemmas to theseworks except two on the Surface-Loci at the end of the Book.The contents of the various works, including those of thelost treatises so far as they can be gathered from Pappus,have been described in the chapters devoted to their authors,and need not be further referred to here, except for anaddendum to the account of Apollonius's Conies which isremarkable. Pappus has been speaking of the ' locus withrespect to three or four lines ' (which is a conic), and proceedsto say (p. 678. 26) that we may in like manner have loci withreference to five or six or even more lines ;these had not upto his time become generally known, though the synthesisof one of them, not by any means the most obvious, had beenworked out and its utility shown. Suppose that there arefive or six lines, and that p l , p2 #, p^ or , , , p6 p^ t p29 p.^p4t pr, , pQare the lengths of straight lines drawn from a point to meetthe five or six at given angles, then,if in the first casePiP^'Ps == ^PiP& a (where X is a constant ratio and a a givenlength), and in the second case Pip^p^ = Aj04 7>6the locus^6 ,of the pointis in each case a certain curve given in position.The relation could not be expressed in the same form ifthere were more lines than six, because there are only threedimensions in geometry, although certain recent writers hadallowed themselves to speak of a rectangle multiplied bya square or a rectangle without giving any intelligible idea ofwhat they meant by such a thing (is Pappus here alluding toHeron's proof of the formula for* the area of a triangle interms of its sides given on pp. 322-3, above ?).But the systemof compounded, ratios enables it to be expressed for anynumber of lines thus, 2-1 .*-? ^ for? r^\ = A. Pappus/>2 PI a \ pn /proceeds in language not very clear (p. 680. 30) but the;gistseems to be that the investigation of these curves had notattracted men of light and leading, as, for instance, the oldgeometers and the best writers. Yet there were other importantdiscoveries stillremaining to be made. For himself, henoticed that every one in his day was occupied with the elements,the first principles and the natural origin of the subjectmatterof investigation ;ashamed to pursue such topics, he hadhimself proved propositions of much more importance and

THE COLLECTION. BOOK VII 403utility.In justification of this statement and ' in order thathe may not appear empty-handed when leaving the subject ',he wijl present his readers with the following.(Anticipation of Guldins Theorem.)The enunciations are not very clearly worded, but thereis no doubt as to the sense.cFigures generated by a complete revolution of a plane figureabout an axis are in a ratio compounded (1) of the ratioof the areas of the figures, and (2) of the ratio of the straightlines similarly drawn to (i.e. drawn to meet at the same angles)the axes of rotation from the respective centres of gravity.Figures generated by incomplete revolutions are in the ratiocompounded (1) of the ratio of the areas of the figures and(2) of the ratio of the arcs described by the centres of gravityof the respective figure*, the latter ratio being itself compounded(a) of the ratio of the straight lines similarly drawn (fromthe respective centres of gravity to the axes of rotation) and(b) of the ratio of the angles contained (i.e. described) aboutthe axes of revolution by the extremities of the said straightlines (i.e. the centres of (jravity).9Here, obviously, we have the essence of the celebratedtheorem commonly attributed to P. Guldin (1577-1643),'quantitas rotunda in viam rotationis ducta producit PotestatemRotundam uno grado altiorem Potestate sive QuantitateRotata'. 1Pappus adds that'these propositions, which are practically one, include anynumber of theorems of all sorts about curves, surfaces, andsolids, all of which are proved at once by one demonstration,and include propositions both old and new, and in particularthose proved in the twelfth Book of these Elements.'Hultsch attributes the whole passage (pp. 680. 30-682. 20)to an interpolator,I do not know for what reason; but itseems to me that the propositions are quite beyond whatcould be expected from an interpolator, indeed I know ofno Greek mathematician from Pappus'sday onward exceptPappus himself who was capable of discovering such a proposition.1Centrobaryca, Lib. ii, chap, viii, Prop. 3. Viennae 1641.Dd2

404 PAPPUS OF ALEXANDRIAIf the passage is genuine, it seems to indicate, what is notelsewhere confirmed, that the Collection originally contained,or was intended to contain, twelve Books.Lemmas tothe different treatises.and full command of all theAfter the description of the treatises forming the Treasuryof Analysis come the collections of lemmas given by Pappusto .assist the student of each of the books (except Euclid'sData) down to Apollonius's Conies, with two isolated lemmasto the Surface-Loci of Euclid. It is difficult to give anysummary or any general idea of these lemmas, because theyare very numerous, extremely various, and often quite difficult,requiring first-rate abilityresources of pure geometry. Their number is also greatlyincreased by the addition of alternative proofs, often requiringlemmas of their own, and by the separate formulation ofparticular cases where by the use of algebra and conventionsthewith regard to sign we can make one proposition cover allcases. The style isadmirably terse, often so condensed as tomake the argumentdifficult to follow without some littlefilling-out; the hand is that of a master throughout. Theonly misfortune is that, the books elucidated being lost (exceptthe Conies and the Cutting-off of a ratio of Apollonius),it isdifficult, often impossible, to see the connexion of the lemmaswith one another and the problems of the book to which theyrelate. In the circumstances, all that I can hope to do is toindicate the types of propositions included in the lemmas and,by way of illustration, now and then to give a proof where itout of the common.is sufficientlywith Lemmas to the tiectio rationiv and(a) Pappus begins Sectio spatii of Apollonius (Props. 1-21, pp. 684-704). Thefirst two show how to divide a straight line in a given ratio,and how, given the first, second and fourth terms of a proportionbetween straight lines, to find the third term. Thenext section (Props. 3-12 and 16) shows how to manipulaterelations between greater and less ratios by transformingthem, e.g. componendo, convertendo, &c., in the same wayas Euclid transforms equal ratios in Book V; Prop. 1 6 provesthat, according as a : b > or < c : d, ad > or < be. Props.

THE COLLECTION. BOOK VII 40517-20 deal with three straight lines a, 6, c in geometricalprogression, showing how to mark on a straight line containinga, 6, c as segments (including the whole among 'segments'),the lengths are of course equallengths equal to a + c + 2 v/(ac) ;to a + c2b respectively. These lemmas are preliminary tothe problem (Prop. 21), Given two straightlines AB, BC(C lying between A and B) to find a pointD on BA producedysuch that BD:DA = CD :(AB + %C-2 SAB7BCJ.This is,of course, equivalent to the quadratic equation (a + x):x= (a c + x):(a + c 2 \/ac), and, after marking AE off alongAD equal to the fourth term of this proportion, Pappus solvesthe equation in the usual way by applicationof areas.(ft) Lemmas to the 'Determinate Section 9 of Apollonius.The next set of Lemmas (Props. 22-64, pp. 704-70) belongsto the Determinate Section of Apollonius.As we have seen(pp. 180-1, above), this work seems to have amounted toa Theory of Involution. Whether the applicationof certainof Pappus's lemmas corresponded to the conjectureof Zeuthenor not, we have at all events in this set of lemmas someremarkable applications of * geometrical algebra '. They maybe divided into groups as follows1. Props. 22, 25, 29If in the figure AD. DC = BD .DE, then= AB.BC:AE. EC.The proofs by proportionsare not difficult. Prop. 29 is analternative proof by means of Prop. 26 (see below). Thealgebraic equivalent may be expressed thus if ax = : 6y, then6II. Props. 30, 32, 34.If in the same figure AD . DE = BD .DC, then= AB.Bti:EC.CA.

406 PAPPUS OF ALEXANDRIAProps. 32, 34 ar alternative proofs based on other lemmasThe algebraic equivalent may be(Props. 31, 33 respectively).i. * j *u -e L *ubstated thus : if ax = by, " then - =yIII. Props. 35, 36.If AB.BE=CB. BD, then AB : BE = DA . ACand GB:BD = AC. CE: AD .following : if ax = by, then:.CE ED,DE, results equivalentto thea _ ~ (a y) (ab) , b_ (ab) (bx)x (b^x^T-x) anIV. Props. 23, 24, 31, 57, 58.~y (a-y)(y-x)'IfAB = CD, and E isany point in GD,AC.CD = AE . ED + BE .EC,and similar formulae hold for other positionsof E. If E isbetween B and C, AG . GD = AE.ED-BE. EC; and if Eis on AD produced,BE. EC = AE. ED + BD . DC.V. A small group of propositions relate to a triangle ABCwith two straight lines AD, AE drawn from the vertex A topoints on the base BC in accordance with one or other of theconditions (a) that the angles BAG, DAE are supplementary,(b)that the angles BAE, DAG are both right angles or, as wemay add from Book VI, Prop. 12, (c)that the angles BAD,EAG are qual. The theorems are :BG.CD: BE ED = CA* : AE' 2 ,In case (a).(b)BG.CE-.BD. DE =CA Z :A D'\CE EB BD = AC 3 : A&.(c)DC .:.

THE COLLECTION. BOOK vii 407Two proofs are given of the first theorem. Wewill give thefirst (Prop. 26) because it is a case of theoretical analysisfollowed by synthesis. Describe a circle about ABD :produceEA, CA to meet the circle again in F, G, and join BF, FG.Substituting GC . CA for BC . CD and FE . EA for BE. ED,we have to inquire whether GC . CA : CA = 2 FE . EA :AE*,i.e.whether GC:CA = FE-.EA,AC = FA :i.e. whether GA :i.e.AE,whether the triangles GAP, CAE are similar or, in otherwords, whether GF is parallelto BC.But GF is parallel to BC, because, the angles BAC, DAEbeing supplementary, Z DAE = Z GAB = Z GFB, while at thesame time Z DAE = suppt.

408 PAPPUS OF, ALEXANDRIAVI. Props. 37, 38.If AB:BC = AD Z :DC*, whether AB be greater or lessthan AD, then[E in the 'figure is a pointsuch that ED =A (E)096(E)09c B P (E)The algebraical equivalent is: If - = ,~c (o T c)2 , then ac=6 2 .These lemmas are subsidiary to the next (Props. 39, 40),being used in the first proofs of them.Props. 39, 40 prove the following:If AGDEB be a straight line, and ifBA . AE: BD.DE = AC* :CD*,then AB.BD:AE.ED = BC* : CE* ;if, again, AC. CB : AE. EB = CD 2 :DE*,then EA .AC:CB.BE = AD* : DB\If AB = a, BC = I),BD c, BE = d, the algebraic equivalentsare the following.a-d)_(a-Z ~~ (b-cf (a-d)(c-d)~ (b-dc(c-

THE COLLECTION. BOOK VII 409The algebraical equivalents for Figures (1) and (2) respectivelymay be written (if a = AD, b = DC, c = BD,d =-DE):If ab = cd, then (a + d + c + b)a = (a + c) (a d),(ad + cb) b = (cb)(b + d),(ad + cb) c = (c +Figure (3) gives other varieties of sign. Troubles aboutsign can be avoided by measuring all lengths in one directionfrom an origin outside the line. Thus, if A = a, OB = 6,&c., the proposition may be as follows :If (d a) (d c)= (6 d) (e d) and /c = e a + 6c,then k(d a) = (6 a)(e a), k(d c) = (6 c)(e c),d) = (& )(& f) and &(e d) = (e a)(0 c).VIII. Props. 45-56.More generally, if AD . DC = BD.DE and /k = AEBC,then, if J^Tbe any point on the line, we have, according to theposition of F in relation to A, B, (7, /), J,V + EF. FB = k. DF.Algebraically, if OA = a, OjS = 6 ... 0^T = a;,: If (d-tt)(ti-c) = (6-d)(6~d!), and & = (e- a)then (x a)(xthe equivalentBy making .f = a, 6, c, e successively in this equation, weobtain the results of Props. 41-3 above.IX. Props. 59-64.In this group Props. 59, 60, 63 are lemmas required for theremarkable propositions (61, 62, 64) in which Pappus investigates' singular and minimum ' values of the ratioAP.PJ):BP.PC,where (A, D), (B tC) are point-pairs on a straight line and Pis another point on the straight line. He finds, not only whenthe ratio has the 'singular and minimum (or maximum)'value,

410 PAPPUS OF ALEXANDRIA'but also what the value is, for three different positions of P inrelation to the four given points.I will give, as an illustration, the first case, on account of itselegance. It depends on the following Lemma. AEB beinga semicircle on AB as diameter, (7, D any two points on AB,and CE, DF being perpendicular to AB, let EF be joined andX C D Bproduced, and let BG be drawn perpendicular to EG. Toprove that CB.BD = BG Z , (I)(2)Join GC, GD, FB, EB, AF.0)(1) Since the angles at G, D are right, F, G, B, D are concyclic.Similarly E, G, B, C are concyclic.Therefore= Z FAB= Z FEB, in the same segment of the semicircle,= Z GOB, in the same segment of the circle EGBC.And the triangles GOB, DGB also have the angle, CHGcommon ;therefore they are similar, and CJi : B(J = BG :BD,or(2) We have AB . BDCR.U1) = Mi*.= BF* ;therefore, by subtraction, AC. DB = BF*-BG* = FG*.(3) Similarly AB .BG= /JJ5?2therefore, by subtraction, from the same result (1),AD.BC= BE' - 2 BG* = W.;Thus the lemma gives an extremely elegant construction forsquares equal to each of the three rectangles.

PC-THE COLLECTION. BOOK VII 411Now suppose (A, D), (B, C) to be two point-pairs on astraight line, and let P, another point on it, be determined bythe relationthen, says Pappus, the ratio AP .a minimum, and is equal toPD:BP.PCis singular andAD* : V'ACTBD- VAB .( CD)\On AD as diameter draw a circle, and draw BF, CG perpendicularto AD on oppositesides.Then, by hypothesis, A Ii . BD:AC.therefore HF* : CG* = BP- :orBF:CG = BP:CP,CD - BP- :CP*.CP-;whence the triangles FBP, GCP arc similar and thereforeequiangular. .o that FPG is a straight line.Produce G( 'to meet the circle in 11, join Fll, and draw DKperpendicular to Fll produced. Draw the diameter FL andjoin Lll.Now, by the lemma, VK* = AC .therefore FH = FK - UK = V(AC.rightBD,and HK*= ABt. CDBD) - .S(AB;CD).C arethe triangles are similar, andSince, in the triangles FHL, PCG, the angles at //,and LFLH LPGC>GP :PC = FL:F1I = AD: FH= AD:{ V(AC.ButGP:PC=FP:PBtherefore(U* .':= FP . PG BP PC= AP.P1):BP.PC.BD) - V(AB . CD)}..

412 PAPPUS OF ALEXANDRIAThereforeAP. PD : BP.PC= AD 2 :.{ V(AC BD)- V(AB .CD)}*.The proofs of Props. 62 and 64 are different, the formerbeing long and involved. The results are :Prop. 62.then the ratio AP .If P is between C and J9, andAD.DB:AC.CB = DP 2 : PCPB:GP.PD2 ,is singularand is equal to .{ J(AG BD) + .V(ADProp. 64.If P is on AD produced, andAB . BD AC CD::..BC) }* : DCand a minimum2.= BP* :CP\then the ratio AP . PD BP PC is singular and a maximum,and is equal to AD2 2:.{ V(AC BD) + . .V(AB CD) }(y) Lemmas on the Neva-eis of Apollonius.After a few easy propositions (e.g. the equivalent of theproposition that, if ax = -f #2 2by + y , then, according as a >or < b, a + x > or < b + y), Pappus gives (Prop. 70) thelemma leading to the solution of the j/eSo-iy with regard tothe rhombus (see pp. 190-2, above),and afterthat the solutionby one Heraclitus of the same problem with respect toa square (Props. 71, 72, pp. 780-4). The problem is, Given asquare ABCD, to draw through B a straight line, meeting CDin H and AD produced in E, such that HE isajual to a givenlength.The solution depends on a lemma to the effect that, ifanystraight line BHE through B meets CD in // and AD pro-Hduced in E, and if EF be drawp perpendicular to BE meetingBC produced in F9then

THE COLLECTION. BOOK VII 413Draw EG perpendicular to BF.Then the triangles BCH, EOF are similar and (sinceBO = EG) equal in all respects : therefore EF = BH.Nowor BO . BF+ BF . FC = BH .BE+ BE.EH+ EF\But, the angles HCF, HEF being right, H, C,F9Eand BC.BF= BH BE.concyclic,.areTherefore, by subtraction,= FB.BC+EH*.Taking away the common part, BC .CF, we haveNow suppose that we have to draw BHE through B insuch a way that HE = k. Since BC, EH are both given, wehave only to determine a length x such that x 2 = BC* + k 2 ,produce BC to F so that OF = x, draw a semicircle on BF asdiameter, produce AD to meet the semicircle in E, and joinBE. BE is thus the straight line required.Prop. 73 (pp. 784-6) proves that, if 7) be the middle pointof BC, the base of an isosceles triangle ABC, then BC is theshortest of all the straight lines through D terminated bythe straight lines AB, AC, and the nearer to .66' is shorter thanthe more remote.There follows a considerable collection of lemmas mostlyshowing the equality of certain intercepts made on straightlines through one extremity of the diameter of one of twosemicircles having their diameters in a straight line, eitherone including or partly including the other, or wholly externalto one another, on the same or opposite sides of thediameter.

414 PAPPUS OF ALEXANDRIAI need only draw two figures by way of illustration.In the first figure (Prop. 83), ABC, DEF being the semicircles,BEKC isany straight line through C cutting both ;FG is made equal to AD AB is ; joined GU is drawn per-;pendicular to BK produced. It isrequired to prove thatLPIG. 1.BE = KH. (This is obvious when from L, the centre of thesemicircle DEF, LM is drawn perpendicular to BK.) If EyKcoincide in the point M' of the semicircle so that B'Gll' isa tangent, then B'M' = M'H' (Props. 83, 84).In the second figure (Prop. 91) D is the centre of thesemicircle ABC and is also the extremity of the diameterof the semicircle DEF. If BEGF be any straight line throughA C PFIG. 2.F cutting both semicircles, BE = EG.is perpendicular to BG.This is clear, since DEThe only problem of any difficulty in this section is Prop.85 (p. 796). Given a semicircle ABC on the diameter ACand a point D on the diameter, to draw a semicircle passingthrough D and havingits diameter along DC such that, ifCEB be drawn touchingitABO in JS, BE shall be equal to AD.at E and meeting the semicircle

THE COLLECTION. BOOK VII 415The problemis reduced to a problem contained in Apollonius'sDeterminate Section thus.Suppose the problem solved by the semicircle DEF> BEto AD. Join E to the centre G of the semicirclebeing equalF CDEF. Produce DA to 77, making HA equal to AD. Let Kbe the middle point of DC.Since the triangles ABC, GEC are similar,= AD*:EG\ by hypothesis,7r'-- 7W* (since DG = (Therefore= //trf:= AD*:2DC.GK.Take a straight line Z such that 4J9- = L.2DC:= A : Âtherefore //(; : DGr ,or //tf . GA' = L . DG.Therefore, given the two straight lines //D, DK (or thethree points 77, D, K on a straight line), we have to finda point G between D and 7C such thatwhich is the second epitagma of the third Problem in theDeterminate Section of Apollonius, and therefore may betaken as solved. (The problem is the equivalent of the

416 PAPPUS OF ALEXANDRIAsolution of a certain quadratic equation.) Pappus observesthat the problem is always possible (requires no Siopurfios),and proves that it has only one solution,(8) Lemmas on the treatise On contacts ' by Apollonius.These lemmas are all pretty obvious except two, which areimportant, one belonging to Book I of the treatise, and the otherto Book II. The two lemmas in question have already been setout & propos of the treatise of Apollonius (see pp. 1 82-5, above).As, however, there are several cases of the first (Props. 105,107, 108, 109), one case (Prop. 108, pp. 836-8), different frotoithat before given, may be put down here: Given a circle andtwo points D, E within it, to draiv straight lines through 7), Eto a point A on the circumference in such a way that, if theymeet the circle again in B, (7, BG shall be parallel to DE.We proceed by analysis. Suppose the problem solved andDA,EA drawn ('inflected') to A in such a way that, if AD,AE meet the circle again in B, C,BC is parallel to DE.Draw the tangent at B meetingED P>dcedin F.Then Z FBD = L AGB = LAED;therefore A, E, B, F are concyclic,and consequentlyFD.DE=AD.DB.But the rectangle AD . DB is given, since it depends onlyon the position of D in relation to the circle, and the circletherefore B is given. Therefore EDA is givenis given.Therefore the rectangle FD . DE is given.And DE is given therefore FD is given, and therefore F.;If follows that the tangent FB is given in position, andand consequentlyAE also.To solve the problem, therefore, we merely take F on EDproduced such that FD . DE = the given rectangle made bythe segments of any chord through D, draw the tangent FB,join BD and produceit to A, and lastly draw AE through toC ;BC is then parallel to DE.

THE COLLECTION. BOOK VIT 417The other problem (Prop. 117, pp. 848-50) is, as we haveseen, equivalent to the following: Given a circle arid threepoints D, E, F in a straight line external to it, to inscribe inthe circle a triangle ABO such that its sides pass severallythrough the three points D, E, F. For the solution, seepp. 182-4, above.(e) The Lemmas to the Plane Loci of Apollonius (Props.119-26, pp. 852-64) are mostly propositions in geometricalalgebra worked out by the methods of Eucl., Books II and VI.We may mention the following:Prop. 122 is the well-known proposition that,middle point of the side BG in a triangle ABC,if D be theBA* + AC = 2 2 (AD* + DC*).PropiS. 123 and 124 are two cases of the same proposition,the enunciation being marked by an expression which is alsofound in Euclid's Data. Let AB : BC be a given ratio, andA P E c Blet the rectangle CA .AD\w given; then,if BE is a meanproportional between DB, BC, 'the square on AE isgreaterby the rectangle CA .square on EC \ by which is meant thatAD than in the ratio of AB to BO to theorThe algebraical equivalent may be expressed thus (ifAB= a,TC //-rr i.i (a + 2.r) -(a b)c nIf x = v/(a-c)6, then =Prop. 125 is remarkable If :C, D be two points on a straightline AB,AD* .+ 7)52 = AC*+ AC . GB+

418 PAPPUS OF ALEXANDRIAThis isequivalent to the general relation between fourpoints on a straight line discovered by Simson and thereforewrongly known as Stewart's theorem :AD* . BC+BD* CA + CD* . AB + BC.CA .AB = 0..(Simson discovered this theorem for the more general casewhere D is a point outside the line ABG.)An algebraical equivalentis the identity- - (d a) 2 (6 r) + (d- - 26) (r a) + (d-

. Thei.e. (ifDA .THE COLLECTION. BOOK VII 419AC be subtracted from each side)that AD . DC + FD . DBi.e. (ifAF . CD be subtracted from each side)thator= AC. DB + AF. CD,FD. DC+FD. DB = AC. DB,FD.CBÂC.DB:FD:DB- AC: CB.which is true, since, by (1) above,() Lemmas fo the ' Porisms' of Endid.38 Lemmas to the Por turns of Euclid form an importanthas been included in one form orcollection which, of course,other in the ' restorations ' of the original treatise. Chasles lin particular gives a classification of them, and we cannotdo better than use it in this place:'23 of the Lemmas relateto rectilineal figures,7 refer to the harmonic ratio of fourpoints, and 8 have reference to the circle.1Of the 23 relating to rectilineal figures,6 deal with thequadrilateral cut by a transversal; 6 with the equality ofthe anharmonic ratios of two systems of four points arisingfrom the intersections of four straight lines issuing fromone point with two other straight lines ;4 may be regarded asexpressing a property of the hexagon inscribed in two straightlines; 2 give the relation between the areas of two triangleswhich have two angles equal or supplementary 4 others refer;to certain systems of straight lines; and the last is a caseof the problem of the 9Cutting-off of an area.The lemmas relating to the quadrilateral and the transversalare 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133).Prop. 130 is a general proposition about any transversalwhatever, and is equivalent to one of the equations by whichwe express the involution of six points. If A, A' \ B, B'\C, C' be the points in which the transversal meets the pairs of1Chasles, Les trois ?iv>-es de Ponsmes d'Eitclide, Paris, 1860, pp. 74 sq.

420 PAPPUS OF ALEXANDRIAopposite sides and the two diagonals respectively, Pappus'sresult is equivalent to AB^B'C _ CAA r B r .BC'~ C'A''Props. 127, 128 are particular cases in which the transversalis parallel to a side; in Prop. 131 the transversal passesthrough the points of concourse of opposite sides, and theresult is equivalent to the fact that the two diagonals divideinto proportional parts the straight line joining the points ofconcourse of opposite sides; Prop. 132 is the particular caseof Prop. 131 in which the line joining the points of concourseof opposite sides is parallel to a diagonal; in Prop. 133 thetransversal passes through one only of the points of concourseof opposite sides and is parallel to a diagonal, the result beingCA* = CB .CB'.Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16,19) establish the equality of the anharmonic ratios whichfour straight lines issuing from a point determine on twotransversals ;but both transversals are supposed to be drawnLetfrom the same point on one of the four straight lines.AB, AC, AD be cut by transversals HBCD, HEFO. It isrequired to prove thatHE.FG _HB.CDHQ.EF~*HD.BCPappus gives (Prop. 129) two methods of proof which arepractically equivalent. The followingis'the proof by compoundratios '.Draw HK parallelto AF meeting DA and AE produced

THE COLLECTION. BOOK VII 421in K, L ;and draw LM parallel to AD meeting GH producedin M.nHE.FG _ HE FG_LU AF _ LH'HG . EF~ EF HG ~ 'AF//# ~~ UKIn exactly the same way,if JDY/ produced meets LAI in it/'we prove thatThereforeHB . CD LHUD.BC~ UKHE.FG UB.CDUG.EF~ HD.BC(The proposition is proved for HBCD and any other transversalnot passing through // by applying our propositiontwice, as usual.)Props. 136, 142 are the reciprocal; Prop. 137 is a particularcase in which 6ne of the transversals is parallelto one of thestraight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145another case of Prop. 129.The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) areequivalent to the property of the hexagon inscribed in twostraight lines, viz. that, if the vertices of a hexagon aresituate, three and three, on two straight lines, the points ofconcourse of opposite sides are in a straight line; in Props.138, 141 the straight lines are parallel, in Props. 139, 143 notparallel.Lemmas 20, 21 (Props. 146, 147) prove that, when one angleof one triangle is equal or supplementary to one angle ofanother triangle, the areas of the triangles are in the ratiosof the rectangles contained by the sides containing the equalor supplementary angles.The seven Lemmas 22, 23. 24, 25, 26, 27, 34 (Props. 148-53and 160) are propositions relating to the segments of a straightline on which two intermediate points arc marked. Thus :Props. 148, 150.If (7,D be two points on AB, then'(a) if 2AB . CD = CB* tAD*= AC 2 + DB*;A C D B(6) if 2AC.BD = CD\ AB* = A1P + U1P.

422 PAPPUS OF ALEXANDRIAProps. 149, 161.If AB .BC= BD*,then (AD DC) BD = AD .DC,(ADDC)BO= DC 2 ,Band (4 DC) BA = AD*.Props. 152, 153.If AB:BC=AD*:DC*, then A B . BC_5__B4-= BD*.Prop. 160.If AB : BC=AD:DC, then, if E be the middle point of AC,BE. ED = EC\BD.DE = AD. DC,E D CThe Lemmas about the circle include the harmonic proper-of the pole and polar, whether the pole is external to thetiescircle (Prop. 154) or internal (Prop. 161). Prop. 155 is aproblem, Given a segment of a circle on AB as base, to inflectstraight lines AC, BC to the segment in a given ratio to oneanother.Prop. 156 is one which Pappus has already used earlierin the Collection. It proves that the straight lines drawnof thefrom the extremities of a chord (DE) to any point (F)circumference divide harmonically the diameter (AB) perpendicularto the chord. Or, if ED, FK be parallel chords, andEF, DK meet in (?, and EK> DF in //, then

THE COLLECTION. BOOK VII 423Since AB bisects ])E perpendicularly, farcAE) = (arc4D)and LEFA = Z AFD, or AF bisects the angle EFD.Since the angle AFB is right, FB bisects Z.1IFG, the supplementof Z EFD.Therefore (Eucl. VI. 3)GB : BH = GF: FH =and, alternately and inversely, AH :IIB = AG :GB.Prop. 157 is remarkable in that (without any mention ofa conic) it is practically identical with Apollonius's ConiesIII. 15 about the foci of a central conic. Pappus's theoremis as follows. Let AB be the diameter of a semicircle, andfrom A yB let two straight lines AM, BI) be drawn at rightangles to AB. Let any straight line meet the two A)AT perpendicularsin D, E and the semicircle in F. Further, let FG bedrawn at right angles to DE, meeting AB produced in Or.It is to be proved thatAG.GB = AE.BDSince F, D, G> B are concyclic, Z BDG = Z BFG.

424 PAPPUS OF ALEXANDRIAAnd, since AFB, EFG are both right angles, LBFG^LAFE.But, since A, E, G, F are concyclic, LAFE = LAGE.ThereforeL BDG = L AGE]and the right-angled triangles DBG, GAE are similar.orThereforeAG:AE = BD: GB,AG.GB = AE.DB.In Apollonius G and the corresponding point G' onproduced which is obtained by drawing F'G' perpendicular toED (where DE meets the circle again in F')are the fociof a central conic (in this case a hyperbola), and DE isanytangent to the conic the; rectangle AE . BD is of course equalto the square on half the conjugate axis.The Lemmas to the Conies of (rj) Apollonius (pp. 918-1004)extended notice. There arc a large numberdo not call for anyof propositions in geometrical algebra of the usual kind,relating to the segments of a straight line marked by a numberof points on it ; propositions about lines divided into proportionalsegments and about similar figures; two propositionsrelating to the construction of a hyperbola (Props. 204, 205)and a proposition (208) proving that two hyperbolas with thesame asymptotes do not meet one another. There are alsotwo propositions (221, 222) equivalent to an obvious trigono-E'-'_^ * ~ i,0metrical formula. Let ABCD be a rectangle, and letanystraight line through A meet DC produced in E and BC(produced if necessary) in F.Then EA .AF = ED . DC + CB .BF.

THE COLLECTION. BOOK VII 425ForAlsoEA* + AF* = ED* + DA 2 + A B* + BF*= #D 2 4- 5C 2 +

426 PAPPUS OF ALEXANDRIAis fond of giving, to cover a class of propositions.The'enunciation may be translated as follows : If AB be a straightline, and CD a straight line parallel to a straight line given inposition, and if the ratio AD . DB : D(7 2 be given, the pointlies on a conic section. If now AB be no longer given inposition, and the points A, B are no longer given but lie(respectively) on straight lines AE, EB given in position, thelies onpoint raised above (the plane containing AE, EB)a surface given in position.And this was proved/ Tannerywas the first to explain this intelligibly ;and his interpretation only requires thevery slight change in the text of substitutingv0tai$ for evdeTa in the phraseytvrfrai 5e rrpoy 0trei tvOtia rats AJE, EB.It is not clear whether, when AB ceasesto be given in position, it is stillgivenin length. If it is given in length and A, B move on the linesAE, EB respectively, the surface which is the locus of C is/complicated one such as Euclid would hardly have beenin a position to investigate. But two possible cases areindicated which he may have discussed, (1) that in which ABmoves always parallel to itself and varies in length accordingly,(2) hat in, which the two lines on which A, B move areparallel instead of meeting at a point. The loci in these twocases would of course be a cone and a cylinder respectively.The second Lemma is still more important, since it is thefirst statement on record of the focus-directrix property ofthe three conic sections. The proof, after Pappus,set out above (pp. 119-21).has been() An uutdlocated Lem f ma.Book VII ends (pp. 1016-18) with a lemma which is notgiven under any particular treatise belonging to the Treasuryof Analysis, but is simply called 'Lemma to the 'AvaXvofiei'os'.If ABC be a triangle right-angled at B, and AB, BG bedivided at F, G so that AF : FB = BG : GO = AB :BC, andif AEG, GEF be joined and BE joined and produced to D,then shall BD be perpendicular to AC.The text is unsatisfactory, for there is a long interpolationcontaining an attempt at a proof by reductio ad absurdum ;

THE COLLECTION. BOOKS VII, VIII 427but the genuine proof is indicated, althoughbefore it is quite complete.it breaks offSinceAF: FB = BG:GC,orAB:BC = FB:GC.But, by hypothesis, AB:BC=BG: GC;thereforeBF = BG.From this point the proof apparently proceeded by analysis.1Suppose it done ' (yeyoi>eTa>), i.e. suppovse the proposition true,and BED perpendicular to AC.Then, by similarity of triangles,AD :DB = AB : BC;therefore AF: FB = AD: DB, and consequently the angleADB is bisected by DP.Similarly the angle BDC is bisected by DG.Therefore each of the angles BDF, BDG is half a rightangle, and consequently the angle FDG is a right angle.Therefore J5, (7, D, F are concyclic ; and, since the anglesFDB, BDG are equal,FB = BG.This is of course the result above proved.Evidently the interpolator tried to clinch the argument byproving that the angle BDA could not be anything but a rightangle.BookVIII.Book VIII of the Collection is mainly on mechanics, althoughit contains, in addition, some propositionsof purely geometricalinterest.

428 PAPPUS OF ALEXANDRIAHistorical preface.It begins with an interesting prefaceon the claim oftheoretical mechanics, as distinct from the merely practicalor industrial, to be regarded as a mathematical subject.Archimedes, Philon, Heron of Alexandria are referred to asthe principal exponents of the science, while Carpus of Antiochis also mentioned as having applied geometry to 'certain(practical) arts'.The date of Carpus is uncertain, though it is probable thathe caine after Geminus;the most likely date seems to be thefirst or second century A.D. Simplicius gives the authority oflamblichus for the statement that Carpus squared the circleby means of a certain curve, which he simply called a curve'generated by a double motion.1 Proclus calls him Carpus thewriter on mechanics (o fAriyaviKos) ',and quotes from a work ofhis on Astronomy some remarks about the relation betweenproblems and theorems and the 'priority in order' of theformer. 2 Proclus also mentions him as having held that anangle belongs to the category of quantity (TTOOW), since itrepresents a sort of ' distance ' between the two lines formingit, this distance being 'extended one way' (40* &6orei>y)though in a different sense from that in which a line representsextension one way, so that Carpus's view appeared to be/ thegreatest possible paradox ' ;5;Carpus seems in reality to havebeen anticipating the modern view of an angle as representingdivergence rather than distance, and to have meant by

THE COLLECTION. BOOK VIII 429mathematics and the practical work, he who has not the formermust perforce use the resources which practical experience inhis particular art or craft gives him. Other varieties ofmechanical work included by the ancients under the" generalterm mechanics were (1) the use of the mechanical powers,or devices for moving or lifting great weights by means ofa small force, (2) the construction of engines of war forthrowing projectiles a long distance, the (3) pumping of waterfrom great depths, (4) the devices of 'wonder-workers'(OavfjLao-iovpyot), some depending on pneumatics (like Heronin the Pneumatwa), some using strings, &c., to produce movementslike those of living things (like Heron in Automata and*Balancings'), some employing floating ladies (like Archimedesin ' Floating Bodies others'), using water to measure time''(like Heron in his Water-clocks'), and lastly sphere-making ',or the construction of mechanical imitations of the movementsof the heavenly bodies with the uniform circular motion ofwater as the motive power. Archimedes, says Pappus, washeld to be the one person who had understood the cause andthe reason of all these various devices, and had applied hisextraordinarily versatile genius and inventiveness to all thepurposes of daily life, and yet, although this brought himunexampled fame the world over, so that his name was onevery one's lips, he disdained (according to Carpus) to writeany mechanical work save a tract on sphere-making, butdiligently wrote all that he could in a small compass of themost advanced parts of geometry and of subjects connectedwith arithmetic. Carpus himself, says Pappus, as well asothers applied geometry to practical arts, and with reason:'for geometry is in no wise injured, nay it is by naturecapable of giving substance to many arts by being associatedwith them, and, so far from being injured, itmay be said,while itself advancing those arts, to be honoured and adornedby them in return.' The object of the Book.Pappus then describes the object of the Book, namelyto set out the propositions which the ancients established bygeometrical methods, besides certain useful theorems discoveredby himself, but in a shorter and clearer form and

430 PAPPUS OF ALEXANDRIAin better logical sequence than his predecessors had attained.The sort of questions to be dealt with are a (1) comparisonbetween the force required to move a given weight alonga horizontal plane and that required to move the same weightupwards on an inclined plane, (2) the finding of two meanproportionals between two unequal straight lines, (3) givena toothed wheel with a certain number of teeth, to find thediameter of, and to construct, another wheel with a given numberof teeth to work on the former. Each of these things, he says,will be clearly understood in its proper place if the principleson which the ' centrobaric doctrine ' is built up are first set out.

THE COLLECTION.BOOK VIII431Now, by hypothesis,whence CA :AE= BC :CD,and, if we halve the antecedents,AK:AE=HC:CD;therefore A K : EK = HC : HDor BH: HD,whence, cvmpoiMtido, CE: EK = BD :But AF:FB= BD : DC = (BD:DH)DH.(1):(DH DC).= (CE:EK).(DH:DC). (2)Now, ELD being a transversal cutting the sides of thetriangle KJIC, we haveHL : KL = (CE:EK). :(DH DC). (3)[This is 'Menelaus's theorem ' ; Pappus does not, however,quote it,but proves the relation ad hoc in an added lemma byThe.LK = (HL.LM) (LM LK)= (HD:DC).(CE:EK).]drawing CM parallel to DE to meet HK produced in M.proof is easy, for HL.It follows from (2)and (3) thatA r*. ETO TIT . T \7^LjT . r/> = fiJLj .Z/A,and, since AB is parallel to 7//i, and AH, BK are straightlines meeting in G, FGL is a straight line.[This is proved in another easy lemma by reductio adabsurdumJ]

432 PAPPUS OF ALEXANDRIAWe have next to prove thatEL = LD.Now [again by 'Menelaus's theorem', proved ad hoc bydrawing CN parallel to HK to meet ED produced inN]EL : LD = (EK: .KG) (CH:HD). (4)But, by (1) above, CE:EK = BD :DH\therefore OK : KE = BH : HD = CH :HD,so that (EK:KC).(CH:HD)=\, and therefore, from (4),EL = LD.It remains to prove that FG = 2 GL, which isparallels, since FG :GL = AG :Gil = 2 : 1.obvious byTwo more propositions follow with reference to the centreof gravity. The first is, Given a rectangle with AB, BC asadjacent sides, to draw from G a straight line meeting the sideopposite BC in a point D such that, if the trapezium ADOB ishung from the point D, it will rest with AD, BG horizontal.In other words, the centre of gravity must be in DL drawnperpendicular to BC. Pappus proves by analysis thatCL 2 = 3BL 2 so that the , problemis reduced to that ofdividing BC into parts BL, LC such that this relation holds.The latter problem is solved (Prop. 6) by taking a point,say X, in CB such that (JX = 3 XB, describing a semicircle onBC toBO as diameter and drawing XY at right angles tomeet the semicircle in F, so that XY* = T\ B

THE COLLECTION. BOOK VIII 483with its extremities on AC, AB and so that AC :ratio,BD is a giventhen the centre of gravity of the triangle ADC will lieon a straight line.Take Eythe middle point of AC, and F a point on DE suchthat DF = 2 FE. Also let // be a point on BA such thatBH =2 HA. Draw FG parallel to AC.Then 4(? = ^/_), and AH=%AB\' *therefore HG = $BD.Also FG = AE = 4(7. Therefore,since the ratio AC:BD is given, theratio 6r// : GF is given.And the angle FGH (A) is given ;therefore the triangle FGH is given inspecies, and consequently the angle GHFis given. And // is a given point.Therefore HF is a given straight line, and it contains thecentre of gravity of the triangle ADC.The inclined plane.Prop. 8 is on the construction of a plane at a given inclinationto another plane parallel to the horizon, and with thisPappus loaves theory and proceeds to the practical part.Prop. 9 (p. 1054. 4 sq.) investigates the problem 'Givena weight which can be drawn along a plane parallel to thehorizon by a given force, and a plane inclined to the horizonat a given angle, to find the force required to draw the weightupwards on the inclined plane'. This seems to be the firstor only attempt in ancient times to investigate motion onan inclined plane, and as such it is curious, though of novalue.Let A be the weight which can be moved by a force C alonga horizontal plane. Conceive a sphere with weight equal to Aplaced in contact at L with the given inclined plane the circle;OGL represents a section of the sphere by a vertical planepassing through E its centre and LK the line of greatest slopedrawn through the point L.Draw EGH horizontal and thereforeparallel to MN in the plane of section, and draw LFperpendicular to EH. Pappus seems to regard the planeas rough, since he proceeds to make a system in equilibrium

434 PAPPUS OF ALEXANDRIAabout FL as if L were the fulcrum of a lever. Now theweight A acts vertically downwards along a straight linethrough E. To balance it, Pappus supposes a weight Battached with its centre of gravity at G.Then A:B=GF:EF= (EL-EF):EF[= (1 -sin 6): sin 6,whereKMN = ff\\and, since /.KMNis given, the ratio EF: EL,and therefore the ratio (EL - EF):EF,MNisgiven thus B is found.;Now, says Pappus,if D is the force which will move Balong a horizontal plane, as C is the force which will moveA along a horizontal plane, the sum of C and D will be theforce ^required to move the sphere upwards on the inclinedplane. He takes the particular case where 6 = 60. Thensin0 is approximately f -J (he evidently uses -|.ff for -1^/3),and 4:5 = 16:104.Suppose, for example, that A = 200 talents; then B is 1300talents. Suppose further that C is 40 man-power; then, sinceC = B :A, D = 260 man-power and it will take D; + C, orD :300 man-power, to move the weight up the plane!Prop. 10 gives, from Heron's Barulcus, the machine consistingof a pulley, interacting toothed wheels, and a spiralscrew working on the last wheel and turned by a handle;Pappus merely alters the proportions of the weight to theforce, and of the diameter of the wheels. At the end ofthe chapter (pp. 1070-2) he repeats his construction for thefinding of two mean proportionals.Construction of a conicthrough jive paints.Chaps. 13-17 are more interesting, for they contain thesolution of the problem of constructing a conic through fivegiven points. The problem arises in this way. Suppose weare given a broken piece of the surface of a cylindrical columnsuch that no portion of the circumference of either of its base

THE COLLECTION. BOOK VIII 435is left intact, and let it be required to find the diameter ofa circular section of the cylinder. We take- any two points-A, B on the surface of the fragment and by means of these wefind five points on the surface all lying in one plane section,in general oblique. This is done by taking five different radiiand drawing pairs of circles with A, B as centres and witheach of the five radii successively. These pairs of circles withequal radii, intersecting at points on the surface, determinefive points on the plane bisecting AB at right angles. The fivepoints are then represented on any plane by triangulation.Suppose the points are A, B,C9 D, E and are such thatno two of the lines connecting the different pairs are parallel.This case can be reduced to the construction of a conic throughto AB.the five points A, B, D, E, F where EF is parallelThis is shown in a subsequent lemma (chap. 16).For, if EF be drawn through E parallel to AB and if CDymeet AB in and EF in 0', we have, by the well-knownproposition about intersecting chords,CO .OD:AO.OB = CO' . O'D : EO' . Q'F,whence O'F is known, and F is determined.We have then (Prop. 13) to construct a conic through A, B,D,Ey F, where EF is parallel to AB.Bisect AB, EF at V, W\ then VW produced both waysis a diameter. Draw DR, the chord through 7) parallel

436 PAPPUS OF ALEXANDRIAto this diameter. Then li is determined byrelationin this way.Join DBymeans of theRG.GD:BG.GA = RH.HD:FH.HE (1)RA, meeting EF in K, L respectively.Then, by similar triangles,RG . GD:BG.Therefore, by (1),FH .GA= (RH: 1IL).(DH:HK)= RH.HD\KH.HL.HE = KH.HL,whence HL is determined, and therefore L. The intersectionof AL, DH determines R.Next, in order to find the extremities P, P' of the diameterthrough V, W, we draw ED, RF meeting PP' in M, N respectively.Then, as before,FW. WE-.P'W. WP = FH. HE: RH. HD, by the ellipse,= FW.WE-.NW.WM, by similar triangles.Therefore P' W. WP = NW . WM;and similarly we can find the value of P'V'. VP.Now, says Pappus, since l^W.WP and P'KFP are givenareas and the points V, W are given, P, P 7 are given. Hisdetermination of P, P' amounts (Prop. 14 following) to anelimination of one of the points and the finding of the otherby means of an equation of the second degree.Take two points Q, Q' on the diameter such thatP'V.VP=WV.VQ,(a)Q, Q' are thus known, while P, P' remain to be found.By (a)P'V: VW= QV: VP,whence P'W:VW= PQ:PV.Therefore, by means of (j8),PQ:PV=Q'W:WP 9

THE COLLECTION. BOOK VIII 437so thatorPQ:(JV=Q'W:PQ',PQ.1>W=QV.VW.Thus P can be found, and similarly P'.The conjugate diameter is found by virtue of the relation(conjugate diam.)* : PP'* = p : PI*.where p is the latus rectum to PPf determined by the propertyof the curveV,PP>= AV*:PV.V1*.Problem, Given two conjugate diameters of an ellipse,to find theaxes.Lastly, Pappus shows (Prop. 14, chap. 17) how, when we aregiven two conjugate diameters, we can find the axes. Theconstruction is as follows. Let AB> CD be conjugate diameters(CD being the greater), E the centre.Produce EA to II so thatKA.AH = DE*.Through A draw FG parallel to CD. Bisect Ell in Kydraw KL at right angles to EH meeting FG in L.andWith L as centre, and LE as radius, describe a circle cuttingGFmG, F.Join EF, EG, and from A draw AM, AN parallel to EF, EGrespectively.

438 PAPPUS OF ALEXANDRIATake points P, R on EG, EF such thatEP* = GE. EM, and ER* = FE. EN.Then EP is half the major axis, and ER half the minor axis.Pappus omits the proof.Problem of seven hexagonsin a circle.Prop. 19 (chap. 23) is a curious problem.equal regular hexagons in a circle in such a wayTo inscribe seventhat oneis about the centre of the circle, while six others stand on itssides and have the opposite sides in each case placed as chordsin the circle.Suppose GUKLNM to be the hexagon so described on HK,a side of the inner hexagon OKL will then be a ;straight line.Produce OL to meet the circle in P.Then OK = KL = LN. Therefore, in the triangle OLN,OL = 2LN, while the included angle OLN (= 120)is alsogiven. Therefore the triangle is given in species; thereforethe ratio ON :the side NLNL is given, and, since ON is given,of each of the hexagons is given.Pappus gives the auxiliary construction thus.Let AF botaken equal to the radius OP. Let AC = AF, and on AC asbase describe a segment of a circle containing an angle of 60.Take GE equal to AC, and draw EB to touch the circle at B.

THE COLLECTION. BOOK VIII 439Then he proves that, if we join AB, AB is equal to the lengthof the side of the hexagon required.Produce BC to D so that BD = BA, and join DA. ABDis then equilateral.Since EB is a tangent to the segment, AE.EC = EB* orAE: EB = EB :EG, and the triangles EAB, EBC are similar.Therefore BA 2 : BC* = 4A* : JEB* = A E : EC = 9 : 4 ;and BC = |4 = #A so that JW; = 2C7ABut CF=2CA; therefore AC:CF= D(J:CB, andare parallel.Therefore BFi AD = BC:CD = 2:1, so thatAlso ^^J?(7 = ZJWXA = 60, so that LABF= 120, andthe triangle ABF is therefore equal and similar to the requiredtriangle NLO.Construction of toothed wheels and indented screw*.The rest of the Book is devoted to the construction (1) ofto those oftoothed wheels with a given number of teeth equala given wheel, (2) of a cylindrical helix, the cocM-ias, indentedso as to work 011 a toothed wheel. The text is evidentlydefective, and at the end an interpolator has inserted extractsabout the mechanical powers from Heron's Mechanics.

XXALGEBRA:DIOPHANTUS OF ALEXANDRIABeginnings learntfrom Egypt.IN algebra, as in geometry, the Greeks learnt the beginningsfrom the Egyptians. Familiarity on the part of the Greekswith Egyptian methods of calculation is well attested. (1)These methods are found in operation in the Heronian writingsand collections. (2) Psellus in the letter published by Tanneryin his edition of Diophantus speaks of *the method of arithmeticalcalculations used by the Egyptians, by which problemsin analysis are handled'; he adds details, doubtless takenfrom Anatolius, of the technical terms used for different kindsof numbers, including the powers of the unknown quantity.(3) The scholiast to Plato's Cliarmides 165 E says that 'partsof Xoyia-TiKrj, the science of calculation, are the so-called Greekand Egyptian methods in multiplications and divisions, andthe additions and subtractions of fractions '. Plato himselfin the Laws 819 A-c says that free-born boys should, as is thepractice in Egypt, learn, side by side with reading, simplemathematical calculations adapted to their age, which shouldbe put into a form such as to combine amusement withinstruction :problems about the distribution of, say, apples orgarlands, the calculation of mixtures, and other questionsarising in militaryor civil life.(4)'Hau '-calculations.The Egyptian calculations here in point (apart from theirmethod of writing and calculating in fractions, which, withthe exception of,were always decomposed and writtenas the sum of a diminishing series of aliquot parts or submultiples)are the /tau-calculations. Hau, meaning a heap, isthe term denoting the unknown quantity, and the calculations

HAU '-CALCULATIONS 441in terms of it are equivalent to the solutions of simple equationswith one unknown quantity. Examples from the PapyrusRhind correspond to the following equations:= 19,x = 33,= 10.The Egyptians anticipated, though only in an elementaryform, a favourite method of Diophantus, that of the 'falsesupposition' or 'regula falsi'. An arbitrary assumption ismade as to the value of the unknown, and the true valueis afterwards found by a comparison of the result of substitutingthe wrong \&alue in the original expression with theactual data. Two examples may be given. The first, fromthe Papyrus Rhind, is the problem of dividing 100 loavesthat the shares are inamong five persons in such a wayarithmetical progression, and one-seventh offirst three shares is equala + 4d, a+3d, a + 2t/, a + d, a be the shares, thenthe sum of theto the sum of the other two. Iford = 5ja.Alnnes says, without any explanation, 'make the difference,as it is, 5 1', and then, assuming a = 1, writes the series23, 17, 12, 6^, 1. The addition of these gives 60, and 100 isIf times 60. Ahmes says simply * multiply If times' and,thus gets the correct values 38, 29, 20, 10, If.The second example (taken from the Berlin Papyrus 6619)is the solution of the equationsx 2 + y2= 100,x:y= 1:, or y = #.x is first assumed to be 1, and #- + 7/2 is thus found to be f.In order to make 100, ff has to be multiplied by 64 or 8 2 .The true value of x is therefore 8 times 1, or 8.Arithmetical epigrams in the Greek Anthology.The simple equations solved in the Papyrus Rhind are justthe kind of equations of which we find many examples in the

442 ALGEBRA: DIOPHANTUS OF ALEXANDRIAarithmetical epigrams contained in the Greek Anthology. Mostof these appear under the name of Metrodorus, a grammarian,probably of the time of the Emperors Anastasius I (A.D. 491-518) and Justin I (A.D. 518-27). They were obviously onlycollected by Metrodorus, from ancient as well as more recentsources. Many of the epigrams (46 in number) lead to simpleequations, and several of them are problems of dividing a numberof apples or nuts among a certain number of persons, thatis to say, the very type of problem mentioned by Plato. Forexample, a number of apples has to be determined such that,if four persons out of six receive one-third, one-eighth, onefourthand one-fifth respectively of the whole number, whilethe fifth person receives 10 apples, there is one apple left overfor the sixth person,i.e.= x.Just as Plato alludes to bowls (id\at)of different metals,there are problems in which the weights of bowls have tobe found. We are thus enabled to understand the allusions ofProclus and the scholiast on Charmides 165 E to fjLrjXiTaiand îa\lraL dpidpoi,'numbers of apples or of bowls'.It is evident from Plato's allusions that the origin of suchsimple algebraical problems dates back, at least,to the fifthcentury B.C.The following is a classification of the problems in theAnthology. (1) Twenty-three are simple equations in oneunknown and of the type shown above; one of these is anepigram on the age of Diophantus and certain incidents ofhis life (xiv. 1 26). (2) Twelve are easy simultaneous equationswith two unknowns, like Dioph. I. 6 ; they can of course bereduced to a simple equation with one unknown by means ofan easy elimination. One other (xiv. 51) gives simultaneousequations in three unknownsand one (xiv. 49) gives four equations in four unknowns,40, # + 0=45, o; + u = 36, x + y + z + u = 60.With these may be compared Dioph. I. 16-21, as well as thegeneral solution of any number of simultaneous linear equa-

EPIGRAMS IN THE GREEK ANTHOLOGY 443tions of this type with the same number of unknown quantitieswhich was given by Thymaridas, an early Pythagorean, and'was called the ewdvOrjua, flower ' or ' bloom ' of Thymaridas(see vol. i, pp. 94-6). (3) Six more are problems of the usualtype about the filling and emptying of vessels by pipes ; e.g.(xiv. 130) one pipe fills the vessel in one day, a second in twoand a third in three ;how long will all three running togethertake to fill it? Another about brickmakers (xiv. 136) is ofthe same sort.Indeterminate equations of the firstdegree.The Anthology contains (4)two indeterminate equations ofthe first degree which can be solved in positive integers in aninfinite number of ways (xiv. 48, 144) ;the first is a distributionof apples, 3x in number, into parts satisfying the equationx 3y = T/,where yis not less than 2; the second leads tothree equations connecting four unknown quantities:x l=the general solution of which is x = 4&, y = k, x l= 3k = t2k. These yl very equations, which, however, are madedeterminate by assuming that x + y = x l+ yl= 100, are solvedin Dioph.I. 1 2.Enough has been said to show that Diophantus was notthe inventor of Algebra. Nor was he the first to solve indeterminateproblems of the second degree.Indeterminate equations of second degree beforeDiophantus.Take first the problem (Dioph. II. 8) of dividing a squarenumber into two squares, or of finding a right-angled trianglewith sides in rational numbers. We have already seen thatPythagoras is credited with the discovery of a general formulafor finding such triangles, namely,

441 ALGEBRA: DIOPHANTUS OF ALEXANDRIAwhere n isany odd number, and Plato with another formulaof the same sort, namely (2?&) 2 + (7&2I) 3 = (/& a +l) a . Euclid(Lemma following X. 28) finds the following more generalformulam 2 M 2 2 2;p g= { 4 (>np 2 2+ mnq*) }[ 4 (w ft^22.mitq*) }The Pythagoreans too, as we have seen (vol.i,pp. 91-3),solved another indeterminate problem, discovering, by meansof the series of ' side- ' and ' diameter-numbers ', any numberof successive integral solutions of the equations-7l a = +1.Diophantus does not particularly mention this equation,but from the Lemma to VI. 15 it is clear that he knew howto find any number of solutions when one is known. Thus,seeing that 2 a;2 1 = y2 is satisfied by x = 1, y = 1, he wouldput2(1 + a;)21 = a square1)-, say;whence x = (4 + 2p)/(jr 2).Take the value p = 2, and we have x == 4, and .+ 1 =. 5;in this case 2 . 5 2 1 = 49 = 7 2 .Putting jj + 5 in place of a;,we can find a still higher value, and so on.Indeterminate equations in the Heronian collections.Some further Greek examples of indeterminate analysis arenow available. They come from the Constantinople manuscript(probably of the twelfth century) from which Scheme editedthe Metrica of Heron ; they have been published and translatedby Heiberg, with comments by Zeuthen. 1Two of the problems(thirteen in number) had been published in a less completeform in Hultsch's Heron (Geeponicus, 78, 79); the othersare new.I. The first problem is to find two rectangles such that theperimeter of the second .is three times that of the first, andthe area of the first is three times that of the second. The1 Bibliotheca mathematics, viii 8 , 1907-8, pp. 118-34. See now Geom.24. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 411-26.

HERONIAN INDETERMINATE EQUATIONS 445number 3 is of course only an illustration, and the problem isequivalent to the solution of the equations(1)(2) xy = n.uvThe solution given in the text is equivalent toU = >fc(4>63 ~ 2),V = 71Zeuthen suggests that the solution may have been obtainedthus. As the problem is indeterminate, it would be naturalto start with some hypothesis, e.g. to put v = n. It wouldfollow from equation (1) that u is a multiple of n, say nz.We have thenwhile, by (2), ,/?/ = M 3 0,whence xy = it 3 (^ + #) >r,or (x-/t 3 ) (2;/t a ) = /t3 (u 3 1 ).An obvious solution isu 1 u :i =7t:>>1, y./t 3 = >/ :i ,which gives z = 2?i31 -f 2 /t :j 1 = 4 H, 3 2, so that-u = nz = n(4 u 3 2).II. The second is a similar problem about two rectangles,equivalent to the solution of the equations(1) x+ y = u+v(2) a:?/= u . wan

446 ALGEBRA: DIOPHANTUS OF ALEXANDRIAwhen equation (1)would give= 2(n l)x (7il)it,a solution of which is x = n 2 1, u = TI 1.III. The fifth problem is interesting in one respect.We areasked to find a right-angled triangle (in rational numbers)with area of 5 feet. We are told to multiply 5 by some36. This makes 180,square containing 6 as a factor, e.g.and this is the area of the triangle (9, 40, 41). Dividing eachside by we 6, have the triangle required. The author, then,is aware that the area of a right-angled triangle with sides inwhole numbers is divisible byformula for a right-angled triangle, making6. If we take the Euclideanthe sides a .win,a .^(m 2 ft 2 a ), ^(m 2, 4- ?6 2 where a ), is any number, and m, nare numbers which are both odd or both even, the area is^mn (m n) (m + n) a 2 ,and, as a matter of fact, the number mn(m n) (m + n)isdivisible by 24, as was proved later (for another purpose) byLeonardo of Pisa.IV. The last four problems (10 to 13) are of great interest.They are different particular cases of one problem, that offinding a rational right-angled triangle such that the numericalsum of its area and its perimeter is a given number. Theauthor's solution depends on the following formulae, wherea, b are the perpendiculars, and c the hypotenuse, of a rightangledtriangle, 8 its area, r the radius of the inscribed ciixsle,and s =S = rs = -|a&, r + 8 = a -f b, c = s r.(The proof of these formulae by means of the usual figure,namely that used by Heron to prove the formulais easy.)Solving the first two equations, in order to find a and ft,we havewhich formula is actually used by the author for finding a

HERONIAN INDETERMINATE EQUATIONS 447and b. The method employedis to take the sum of the areaand the perimeter +28, separated into its two obviousfactors s(r+2), to put s(r+2) = A (the given number), andthen to separate A into suitable factors to which s and r + 2may be equated. They must obviously be such that sr, thearea, is divisible by 6. To take the first example whereA 280 : the possible factors are 2 x 140, 4 x 70, 5 x 56, 7 x 40,8 x 35, 10 x 28, 14 x 20. The suitable factors in this case are?* + 2 = 8, s = 35, because r is then equal to 6, and rs isa multiple of 6.The author then says thatrt = 4[G + 35-

448 ALGEBRA: DIOPHANTUS OF ALEXANDRIANumerical solution of quadratic equations.The geometrical algebra of the Greeks has been in evidenceall through our history from the Pythagoreans downwards,and no more need be said of it here except that its arithmeticalapplication was no new thing in Diophantus. It is probable,for example, that the solution of the quadratic equation,discovered firstby geometry, was applied for the purpose offinding numerical values for the unknown as early as Euclid,if not earlier still. In Heron the numerical solution ofequations is well established, so that Diophantus was not thefirst to treat equations algebraically. What he did was totake a step forward towards an algebraic notation.The date of DIOPHANTUS can now be fixed with fair certainty.He was later than Hypsicles, from whom he quotes a definitionof a polygonal number, and earlier than Theon of Alexandria,who has a quotation from Diophantus's definitions. Thepossible limits of date are therefore, say, 150 B.C. to A.D. 360.But the letter of Psellus already mentioned says that Anatolius(Bishop of Laodicea about A.D. 280) dedicated to Diophantusa concise treatise on the Egyptian method of reckoning;hence Diophantus must have been a contemporary, so that lieflourished A.D. 250 or not much later.probablyAn epigram in the Anthology gives some personal particulars:his boyhood lasted th of his life his beard ; grew after T^thmore ;he married after ^th more, and his son was born 5 yearslater; the son lived to half his father's ago, and the fatherdied 4 years after his son. Thus, if x was his aye whenhe died,= X,which gives x = 84.Works of Diophantus.The works on which the fame of Diophantusrests are :(1) the Arithmetica (originally in thirteen Books),(2) a tract OH Polygonal Numbers.

WOKKS 449Six Books only of the former and a fragment of the lattersurvive.Allusions in the Arithmetica imply the existence of(3)A collection of propositions under the title of Porisms ;in three propositions (3, 5, 16) of Book V, Diophantus quotesas known certain propositions in the Theory of Numbers,*to the statement of them the words We have it inprefixingthe Porisms that . . /A scholium on a passage of lamblichus, where lamblichuscites a dictum of certain Pythagoreans about the unit beingthe dividing line (neQopiov) between number and aliquot parts,says 'thus Diopliantus in the Moriastica .... for he describesas "parts" the progression without limit in the direction ofless than the unit '. The Moriastica may be a separate workby Diophantus giving rules for reckoning with fractions but;I do not feel sure that the reference may not simply be to thedefinitions at the beginning of the Arithmetica.The Arithmetica.The seven lostHooks and their place.None of the manuscripts which we possess contain morethan six Books of the Arithmetica, the only variations beingthat some few divide the six Books into seven, while one ortwo give the fragment on Polygonal Numbers as VIII. Themissing Books were evidently lost at a very early date.Tannery suggests that Hypatia's commentary extended onlyto the first six Books, and that she left untouched the remainingseven, which, partly as a consequence, were first forgottenand then lost (cf.the case of Apollonius's Conic*, where theonly Books which have survived in Greek, I-IV, are thoseon which Eutocius commented). There is no sign that eventhe Arabians ever possessed the missing Books. The Fakhn,an algebraical treatise by Abu Bekr Muh. b. al-Hasan al-Karkhi (d. about 1029), contains a collection of problems indeterminate and indeterminate analysis which not only showthat their author had deeply studied Diopliantus but in manycases are taken direct from the Arithmetica, sometimes witha change in constants; in the fourth section of the work,1523.2 G g

450 DIOPHANTUS OF ALEXANDRIAbetween problems corresponding to problems in Dioph. IIand III, are 25 problems not found in Diophantus, butinternal evidence, and especially the admission of irrationalresults (which are always avoided by Diophantus), excludethe hypothesis that we have here one of the lost Books.Nor is there any sign that more of the work than we possesswas known to Abu'l Wafa al-Buzjam (A'.D. 940-98) who wrotea 'commentary on the algebra of Diophantus', as well asa Book 'of proofs of propositions used by Diophantus in hiswork'. These facts again pointto the conclusion that thelost B6oks wore lost before the tenth century.The old view of the place originally occupied by the lostseven Books is that of Nesselmann, who arguedit with greatability. 1 According to him (1) much less of Diophantus iswanting than would naturally be supposed on the basis ofthe numerical proportion of 7 lost to 6 extant Books, (2) themissing portion came, not at the end, but in the middle ofthe work, and indeed mostly between the first and secondBooks. Nesselmann's general argumentis that,if we carefullyread the last four Books, from the third to the sixth,we shall find that Diophantus moves in a rigidly defined andlimited circle of methods and artifices, and seems in fact to beat the end of his resources. As regards the possible contentsof the lost portion on this hypothesis, Nesselmann can onlypoint to (1) topics which we should expect to find treated,either because foreshadowed by the author himself or asnecessary for the elucidation or completion of the wholesubject, (2) the Porisms', under head (1) come, (a) determinateequations of the second degree, and (b) indeterminateequations of the first degree.Diophantus does indeed promiseto show how to solve the general quadratic ax~ + bx c = sofar as it has rational and positive solutions ;for this would have been between Books I and II.the suitable placeBut thereis nothing whatever to show that indeterminate equationsof the first degree formed part of the writer's plan. HenceNesselmann is far from accounting for the contents of sevenwhole Books; and he is forced to the conjecture that the sixBooks may originally have been divided into even more thanseven Books ;there is, however, no evidence to support this.1Nesaelmann, ,AI(jebra der GHechen, pp. 264-73.

RELATION OF WORKS 451Relation oftlte c Porisms ' to the Arithmetica.Did the Porisms form part of the Arithmetica in its originalform ? The phrase in which they are alluded to, and whichoccurs three times, 'We have it in the Porisms that . . .' suggeststhat they were a distinct collection of propositions concerningthe properties of certain numbers, their divisibility into acertain number of squares, and so on ;and it is possible thatit was from tho same collection that Diophantus took thenumerous other propositions which he assumes, explicitly orimplicitly. If the collection was part of the Arithmetica, itwould be tstrange to quote the propositions under a separatetitle 'The Porisms' when it would have been more naturalto refer to particular propositions of particular Books, andmore natural still to say TOVTO yap TrpoStSeiKTai, or some suchphrase, ' for this has been proved', without any reference tothe particular place where the proof occurred. The expression'We have it in the PorismH* (in the plural) would be stillmore inappropriateif the Porisms had been, as Tannerysupposed, not collected together us one or more Books of theArithmetica, but scattered about in the work as corollaries toparticular propositions. Hence I agree with the view ofHultscli that the Porisms were not included in the Arithmeticaat all, but formed a separate work.If this is right, we cannot any longer hold to the view ofNesselmann that the lost Books were in the middle and not atthe end of the treatise ; indeed Tannery produces strongarguments in favour of the contrary view, that it is the lastand most difficult Books which are lost. He replies first tothe assumption that Diophantus could not have proceededto problems more difficult than those of Book V. 'If thefifth or the sixth Book of the Arithmetica had been lost, who,pray, among us would have believed that such problems hadever been attempted by the Greeks ? It would be the greatesterror, in any case in which a thing cannot clearly be provedto have been unknown to all the ancients, to maintain thatitcould not have been known to some Greek mathematician.If we do not know to what lengths Archimedes brought thetheory of numbers (to say nothing of other things), let usadmit our ignorance. But, between the famous problem of theo ir

452 DIOPHANTUS OF ALEXANDRIAcattle and the most difficult of Diophantus's problems,is therenot a sufficient gap to require seven Books to fill it? And,without attributing to the ancients what modern mathematicianshave discovered, may not a number of the thingsattributed to the Indians and Arabs have been drawn fromGreek sources? May not the same be said of a problemsolved by Leonardo of Pisa, which isvery similar to those ofDiophantus but is not now to be found in the Arithmetica ?In fact, itmay fairly be said that, when Chasles made hisreasonably probable restitution of the Porisms of Euclid, he,notwithstanding that he had Pappus's lemmas to help him,undertook a more difficult task than he would have undertakenif he had attempted to fillup seven Diophantine Books withnumerical problems which the Greeks may reasonably belsupposed to have solved.'It is not so easy to agree with Tannery's view of the relationof the treatise On Polygonal Numbers to the Arithmetica.According to him, just as Serenus's treatise on the sectionsof cones and cylinders was added to the mutilated Conies ofApollonius consisting of four Books only, in order to make upa convenient volume, so the tract on Polygonal Numbers wasadded to the remains of the Arithmetica, though forming nopart of the larger work. 2 Thus Tannery would seem to denythe genuineness of the whole tract on Polygonal Numbers,though in his text he only signalizes the portion beginningwith the enunciation of the problem 'Given a number, to findin how many ways it can be a polygonal number ' as ' a vainattempt by a commentator ' to solve this problem. Hultsch,on theconclude that Dio-other hand, thinks that we mayphantus really solved the problem. The tract begins, likeBook I of the Arithmetica) with definitions and preliminarypropositions; then comes the difficult problem quoted, thediscussion of which breaks off in our text after a few pages,and to these it would be easy to tack on a great variety ofother problems.The name of Diophantus was used, aswere the names ofEuclid, Archimedes and Heron in their turn, for the purposeof palming oft* the compilations of much later authors.1Diophantus, ed. Tannery, vol. ii, p. xx./&., p. xviii.

RELATION OF WORKS 453Tannery includes in his edition three fragments under theheading ' Diophantus Pseudepigraphus The '. first, which isnot from ' the Arithmetic of Diophantus ' as itsheading states,is worth notice as containing some particulars of one of two*methods of finding the square root of any square number';we are told to begin by writing the number 'according tothe arrangement of the Indian method', i.e. in the Indiannumerical notation which reached us through the Arabs. Thesecond fragment is the work edited by C. Henryin 1879 asOjiusculum de multiplicatioiie et divisions sexagesimalibusDiophanto vel Pwppo attribuendum. The third, beginningwith AiofydvTov C7n7r5o/*erpi/ca is a Byzantine compilationfrom later reproductions of the yeo>/zT/)orf/zJ>a and o-HeTpovfAcva of Heron. Not one of the three fragments hasanything to do with Diophantus.Commentators from Hypatia dowmvards.The first commentator on Diophantus of whom we hearisHypatia, the daughter of Theon of Alexandria; shewas murdered by Christian fanatics in A.D. 415. I havealready mentioned the attractive hypothesis of Tannery thatHypatia's commentary extended only to our six Books, andthat this accounts for their survival when the rest were lostIt is possible that the remarks of Psellus (eleventh century) atthe l>eginning of his letter about Diophantus, Anatolius andthe Egyptian method of arithmetical reckoning were takenfrom Hypatia's commentary.Georgius Pachy meres (1240 to about 1310) wrote in Greeka paraphrase of at least a portion of Diophantus. Sections25-44 of this commentary relating to Book I, Dcf. 1 to Prop11, survive. Maximus Plamides (about 1260-1310) also wrotea systematic commentary on Books I, II. Arabian commentatorswere Abu'l Wafa al-Bfizjam (940-98), Qusta b. Liiqaal-Ba'labakki (d. about 912) and probably Ibn al-Huithaui(about 965-1039).Translations and editions.To Regiomontanus belongs the credit of being the first tccall attention to the work of Diophantus as being extant in

454 DIOPHANTUS OF ALEXANDRIAGreek. In an Oratio delivered at the end of 1463 as anintroduction to a course of lectures on astronomy which hegave at Padua in 1463-4 he observed: 'No one has yettranslated from the Greek into Latin the fine thirteen Booksof Diophantus, in which the very flower of the whole ofarithmetic lies hid, the ars rei et cemus which to-day theycallby the Arabic name of Algebra/ Again, in a letter datedFebruary 5, 1464, to Bianchini, he writes that he has found atVenice 'Diofantus, a Greek arithmetician who has not yetbeen translated into Latin '. Rafael Bombelli was the first tofind a manuscript in the Vatican and to conceive the idea ofpublishing the work; this was towards 1570, and, withAntonio Maria Pazzi, he translated five Books out of theseven into which the manuscript was divided. The translationwas not published, but Bombelli took all the problems of thefirst four Books and some of those of the fifth and embodiedthem in his Algebra (1572), interspersing them with his ownproblems.The next writer on Diophantus was Wilhelm Holzmann,who called himself Xylander, and who with extraordinaryindustry and care produced a very meritorious Latin translationwith commentary (1575). Xylander was an enthusiastfor Diophantus, and his preface and notes are often delightfulreading. Unfortunately the book is now very rare. Thestandard edition of Diophantus till recent years was that ofBachet, who in 1621 published for the first time the Greektext with Latin translation and notes. A second edition(1670) was carelessly printed and is untrustworthy as regardsthe text ;on the other hand it contained the epoch-makingnotes of Fermat; the editor was S. Format, his son. Thegreat blot on the work of Bachet is his attitude to Xylander,to whose translation he owed more than he was willing toavow. Unfortunately neither Bachet nor Xylander was ableto use the best manuscripts that used ;by Bachet was Parisinus2379 (of the middle of the sixteenth century), with the helpof a transcription of part of a Vatican MS. (Vat. gr. 304 ofthe sixteenth century), while Xylander's manuscript was theWolfenbiittel MS. Guelferbytanus Gudianus 1 (fifteenth century).The best and most ancient manuscriptis that ofMadrid (Matritensis 48 of the thirteenth century) which was

TRANSLATIONS AND, EDITIONS 455unfortunately spoiled by corrections made, especially in BooksI, II, from some manuscript of the Planudean ' ' class ;wherethis is the case recourse must be had to Vat. gr. 191 whichwas copied from it befqre it had suffered the general alterationreferred to : these are the first two of the manuscripts used byTannery in his definitive edition of the Greek text (Teubner,1893, 1895).Other editors can only be shortly enumerated. In 1585Simon Stevin published a French version of the first fourBooks, based on Xylander. Albert Girard added the fifth andsixth Books, the complete edition appearing in 1625. Germantranslations were brought out by Otto Schulz in 1822 and byG. Wertheim in 1890. Poselger translated the fragment onPolygonal Numbers in 1810. All these translations dependedon the text of Bachet.A reproduction of Diophaiitus in modern notation withintroduction and notes by the present writer (second edition1910) is based on the text of Tannery and may claim to be themost complete and up-to-date edition.My account of the Arithmetica of Diophantus will be mostconveniently arranged under three main headings (1) thenotation and definitions, (2) the principal methods employed,the nature of theso far as they can be generally stated, (3)contents, including the assumed Porisms, with indications ofthe devices by which the problems are solved.Notation and definitions.In his work Die Algebra der Griecheii Nesselmaim distinguishesthree stages in the evolution of algebra. (1) Thefirst stage he calls 'Rhetorical Algebra' or reckoning bymeans of complete words. The characteristic of this stageis the absolute want of all symbols, the whole of the calculationbeing carried on by means of complete words and formingin fact continuous prose. This first stage is represented bysuch writers as lamblichus, all Arabian and Persian algebraists,and the oldest Italian algebraists and their followers, includingRegiomontanus. (2) The second stage Nesselmann calls the4Syncopated Algebra', essentially like the first as regards

456 DIOPHANTUS OF ALEXANDRIAvliterary style, but marked by the use of certain abbreviationalsymbols for* constantly recurring quantities and operations.To this stage belong Diophantus and, after him, all the laterEuropeans until about the middle of the seventeenth century(with the exception of Vieta, who was the first to establish,under the name of Logistica speciosa, as distinct from Logist'icanumerosa, a regular system of reckoning with letters denotingmagnitudes as well as numbers). (3)To the third stageNesselmann gives the name of 'Symbolic Algebra', whichuses a complete system of notation by signs having no visibleconnexion with the words or things which they represent,a complete language of symbols, which entirely supplants the'rhetorical ' system, it being possible to work out a solutionwithout using a single word of ordinary language with theexception of a connecting word or two here and there used forclearness' sake.Sign for the unknown (= x), and its origin.Diophantus's system of notation then ismerely abbreviational.We will consider first the representation of theunknown quantity (our x). Diophantus defines the unknownquantity as ccontaining an indeterminate or undefined multitudeof units ' (7rA?J0oy novdSoov dopurrov), adding that it iscalled aped/to?, i.e. number simply, and is denoted by a certainsign. This sign is then used all through the book. In theearliest (the Madrid) MS. the sign takes the form ij,inMarcianus 308 it appears as S. In the printed editions ofDiophantus before Tannery's it was represented by the finalsigma with an accent, $\ which is sufficiently like the secondof the two forms. Where the symbol takes the place ofinflected forms aptd/iop, dpi6p,ovt&c., the termination was putabove and to the right of the sign like an exponent, e.g. $p fordpidjjLov as f* for rd*>, y o0 for dpiOpovl the symbol was, inaddition, doubled in the plural cases, thus ss

NOTATION AND DEFINITIONS 457that the sign was not duplicated for the plural, although suchduplication was the practice of the Byzantines. That thesign was merely an abbreviation for the word dpiOpos and noalgebraical symbol is shown by the fact that it occurs in themanuscripts for aptd/*6? in the ordinary sense as well as fordpiOfjLos in the technical sense of the unknown quantity. Noris it confined to Diophantus. It appears in more or lesssimilar forms in the manuscripts of other Greek mathematicians,e.g. in the Bodleian MS. of Euclid (D'Orville 301)of the ninth century (in the forms 9 99,or* as a curved linesimilar to the abbreviation for /ecu), in the manuscripts ofthe Sand-reckoner of Archimedes (in a form approximatingto y),where again there is confusion caused by thesimilarity of the signs for dpiO/jiSs and tat, in a manuscriptof the Geodaesia included in the Heronian collections editedby Hultsch (where it appears in various forms resemblingsometimes,sometimes p,sometimes o,and once,withcase-endings superposed) and in a manuscript of Theon ofSmyrna.What is the originof the sign? It is certainly not thefinal sigma, as is proved by several of the forms which ittakes. I found that in the Bodleian manuscript of Diophantusit is written in the form ?c , larger than and quite unlike thefinal sigma. This form, combined with the fact that in oneplace Xylander's manuscript read ap for the full word, suggestedto me that the sign might be a simple contraction of the firsttwo letters of dpiOpos. This seemed to be confirmed byGardthausen's mention of a contraction for ap, in the form vpoccurring in a papyrus of A.D. 154, since the transition to theform found in the manuscripts of Diophantus might easilyhave been made through an intermediate form ^>.The loss ofthe downward stroke, or of the loop, would give a closeapproximation to the forms which we know. This hypothesisas to the origin of the sign has not, so far as I know, beenimproved upon. It has the immense advantage that it makesthe sign for aptd/io? similar to the signs for the powers ofthe unknown, e.g.A Y for Swapis, K Y for /crfoy, and to theosign M for the unit, the sole difference being that the twoletters coalesce into one instead of being separate.

458 DIOPHANTUS 'OF ALEXANDRIAISignsfor the powers of the unknown and their reciprocals.The powers of the unknown, corresponding to our # 2 , x* . . . XQ ,are defined and denoted as follows :x 2 is 8vva/ju$ and is denoted by A Y ,# 3 Kvpos K Y ,X 4 AK ,of* /o/jSo/cf/Jos' K K.Beyond the sixth power Diophantus does not go. It shouldbe noted that, while the terms from KV&OS onwards may beused for the powers of any ordinary known number as well asfor the powers of the unknown, Swa/us is restricted to thesquare of the unknown wherever a;particular square numberisspoken of, the term isrer/oayooi/oy dpiO/ios.The termSwapoSwaiJus occurs once in another author, namely in theMetrica of Heron, 1 where it is used for the fourth power ofthe side of a triangle.Diophantus has also terms and signs for the reciprocals ofthe various powers of the unknown, i.e. for I/ x, l/x*....As an aliquot part was ordinarily denoted by the correspondingnumeral sign with an accent, e.g. y'= ^, *a'= -rr, Diophantushas a mark appended to the symbols for x, x* ... to denote thereciprocals; this, which is used for aliquot parts as well, isprinted by Tannery thus, *.With Diophantus then, denoted by s*, is equivalent to 1/ir,and so on. .The coefficient of the term in x, x l ... or I/ x, I/ x* ... isexpressed by the ordinary numeral immediately following,AK Ye.g. *r = 26x\ A YX or =2250/a.Diophantus does not need any signs for the operations ofmultiplication and division. Addition is indicated by merejuxtaposition ;thus K Y a A Y ty y e corresponds to XA + 13# 2 + 5x.1Heron, Metrica, p. 48. 11, 19, Schone.'

NOTATION AND DEFINITIONS 459When there are units in addition, the units are indicated byoothe abbreviation M thus K Y ;aA Y ty$M corresponds toThe sign (A) for minus and its meaning.For subtraction alone is a sign used. The full term forwanting is Aen/^9, as opposed to virapfcis, a fortlicominf/,which denotes a yxwitive term. The symbol used to indicatea ivanting, corresponding to our sign for minus, is A, whichis described in the text as a ^ ' turned downwards andtruncated ' (W cAAiTrey Araro) vtvov). The description is evidentlyinterpolated, and it is now certain that the sign has nothingto do with \lr.Nor is it confined to Diophantus, for it appearsin practically the same form in Heron's Metrical where in oneplace the reading of the manuscript is fjiovd8a>v oS T I'tf,74 T^. In the manuscripts of Diophantus, when the signis resolved by writing the full word instead of it, it isgenerally resolved into Xttyet, the dative of Aen/riy,but inother places the symbol is used instead of parts of the verb\tiirttv, namely AITTOJ/ or AeM/ray and once even \irr. On the whole,therefore, it isprobable that in Diophantus, and wherever elseit occurred, A is a compendium for the root of the verb Ae/Tra*',in fact a A with Iplaced in the middle (cf. A, an abbreviationfor rdXavTov). This is the hypothesis which I put forwardin 1885, and it seems to be confirmed by the fresh evidencenow available as shown above.1Heron, Metrica, p. 156. 8, 10.

460 DIOPHANTUS OF ALEXANDRIAAttached to the definition of minus is the statement that'a wanting (i.e,a minus) multiplied by a wanting makesa forthcoming (i.e.a plus) ;and a wanting (a minus) multipliedby a, forthcoming (a plus) makes a wanting (a minus) '.Since Diophantus uses no .sign for plus, he has to put allthe positive terms in an expression together and write allnegative terms together after the sign for minus', e.g. for# 3 5x 2 + 8x- 1 he necessarily writes K Y a s 17A A Y M a.The Diophantine notation for fractions as well as for largenumbers has been fully explained with many illustrationsin Chapter II above. It is only necessary to add here that,when the numerator and denominator consist of compositeexpressions in terms of the unknown and its powers, he putsthe numerator first followed by kv poptip or popiov and thedenominator.Thus A Y Mx kv pop!? A Y A a M ^ A A Ythe= (60a>2 + 2520)/(a4 + 900-60a 2 ), [VI. 12]and A Y ie A M X? h /topfpA Y A a M A

NOTATION AND DEFINITIONS 461in three terms is clearly assumed in several places of theArithmetica, but Diophantus never gives the necessary explanationof this case as promised in the preface.Before leaving the notation of Diophantus, we may observethat the form of itlimits him to the use of one unknown ata time. The disadvantageis obvious. For example, wherewe can begin with any number of unknown quantities andgradually eliminate all but one, Diophantus has practically toperform his eliminations beforehand so as to express everyquantity occurring in the problem in terms of only oneunknown. When he handles problems which are by natureindeterminate and would lead in our notation to an indeterminateequation containing two or three unknowns, he hasto assume for one or other of these some particular numberarbitrarily chosen, the effect being to make the problemdeterminate. However, in doing so, Diophantusis carefulto say that we may for such and such a quantity put anynumber whatever, say such and such a number; there istherefore (as a rule)no real loss of generality. The particulardevices by which he contrives to expressall his unknownsin terms of one unknown are extraordinarily various andclever. He can, of course, use the same variable y in thesafrneproblem with different significations successively, aswhen it is necessary in the course of the problem to solvea subsidiary problem in order to enable him to make thecoefficients of the different terms of expressions in x suchas will answer his purpose and enable the original problemto be solved. There are, however, two cases, II. 28, 29, wherefor the proper working-out of the problem two unknowns areimperatively necessary. We should of course use x and y\Diophantus calls the first y as usual; the second, for wantof a term, he agrees to call in the first instance 'one unit 9 ,i.e. 1. Then later, having completed the part of the solutionnecessary to find x, he substitutes its value and uses y overagain for what he had originally called 1. That is, he has toput his finger on the place to which the 1 has passed, so asto substitute y for it. This is a tour deforce in the particularcases, and would be difficult or impossible in more complicatedproblems,

462 DIOPHANTUS OF ALEXANDRIAThe methods of Diophantus.It should be premised that Diophantus will have in hissolutions no numbers whatever except 'rational' numbers;he admits fractional solutions as well as integral, but heexcludes not only surds and imaginary quantities but alsonegative quantities. Of a negative quantity per se, i.e. withoutsome greater positive quantity to subtract it from, hehad apparently no conception. Such equations then as leadto imaginary or negative roots he regards as useless for hispurpose; the solution is in these cases a&Ji/aroy, impossible.So we find him (V. 2) describing the equation 4 = 4*0 + 20 asLOTTOS, absurd, because it would give x = 4. He does, it istrue, make occasional use of a quadratic which would givea root which is positive but a surd, but only for the purposeof obtaining limits to the root which are integers or numericalfractions ;he never uses or tries to express the actual root ofsuch an equation. When therefore he arrives in the courseof solution at an equation which would give an 'irrational'result, he retraces his steps, finds out how his equation hasarisen, and how he may, by altering the previous work,substitute for it another which shall give a rational result.This gives rise in general to a subsidiary problem the solutionof which ensures a rational result for the problem itself.It is difficult to give a complete account of Diophantus'smethods without setting out the whole book, so great is thevariety of devices and artifices employed in the differentproblems. There are, however, a few general methods whichdo admit of differentiation and description, and these we proceedto set out under subjects.I.Diophantus's treatment of equations.(A)Determinate equations.Diophantus solved without difficulty determinate equationsof the first and second degrees;of a cubic we find only oneexample in the Arithmetica, and that is a very special case.(1) Pure determinate equations.Diophantus gives a general rule for this case without regardto degree. We have to take like from like on both sides of an

DETERMINATE EQUATIONS 463equation ahd neutralize negative terms by adding to bothsides, then take like from like again, until we have one termleft equal to one term. After these operations have beenperformed, the equation (after dividing out, if both sidescontain a power of x, by the lesser power) reduces to Axm = JB,and is considered solved. Diophantus regards this as givingone root only, excluding any negative value as 'impossible'.No equation of the kind is admitted which does not givea ' rational ' value, integral or fractional. The value x = isignored in the case where the degree of the equationis reducedby dividing out by any power of x.(2) Mixed quadratic equations.Diophantus never gives the explanation of the method ofsolution which he promises in the preface. That he hada definite method like that used in the Geometry of Heronis proved by clear verbal explanations in different propositions.As he requires the equation to be in the form of two positiveterms being equal to one positive term, the possible forms forDiophantus are(a) mx 2 + 'px=)m,/r = px + q, (c)mjp + q= px.It does not appear that Diophantus divided by in in order tomake the first term a square rather he; multiplied by m forthis purpose. It is clear that he stilted the roots in the abovecapes in a form equivalent tommThe explanations which show this arc to be found in VI. 6,in IV. 39 and 31, and in V. 10 and VI. 22 respectively. Forexample in V. 10 he has the equation 17ar+ 17 < f2x, and hesays 'Multiply half the coefficient of x into itself and we have1296; subtract the product of the coefficient of x 2 and theterm in units, or 289. The remainder is 1007, the square rootAdd half the coefficient of xof which is not greater than 31.and the result is not greater than 67. Divide by the coefficientof a; 2 ,and x is not greater than f.' In IV. 39 he has the

464 DIOPHANTUS OF ALEXANDRIAequation > 6x+ 2x2 18 and says, 'To solve this, take the squareof half the coefficient of x yi.e. 9, and the product of the unittermand the coefficient of x 2 i.e. 36., Adding, we have 45,the square root of which is not less than 7. Add half thecoefficient of x [and divide by the coefficient of #2 ]whence x;is not less than 5.' In these cases it will be observed that 3 1and 7 are not accurate limits, but are the nearest integrallimits which will serve his purpose.Diophantus always uses the positive sign with the radical,and there has been much discussion as to whether he knewthat a quadratic equation has two roots. The evidence of thetext is inconclusive because his only object, in every case, is toget one solution; in some cases the other root would benegative, and would therefore naturally be ignored as 'absurd'or ' impossible In '. yet other cases where the second root ispossible it can be shown to be useless from Diophantus's pointof view. For my part, I find it difficult or impossible tobelieve that Diophantus was unaware of the existence of tworeal roots 1131 such cases. It is so obvious from the geometricalform of solution based on Eucl. II. 5, 6 and that contained inEucl. VI. 27-9; the construction of VI. 28, too, correspondsin fact to the negative sign before the radical in the case of theparticular equation there solved, while a quite obvious andslight variation of the construction would give the solutioncorresponding to the positive sign.The following particular cases of quadratics occurring inthe Arithmetics may be quoted, with the results stated byDiophantus.X* = 4o; 4;therefore x = 2. (IV. 22)2i/; = 3x+ 18; x = Jfa or -aV (Iv 31 )7o;= 7; x = -J. (VI. 6)a? a -7^= 7;sc =.(VI. 7)630ar-73;/; = 6; x = 3\. (VI. 9)630# 2 + 73x = 6 ;# is rational. (VI. 8)5x < a; 2 r-60 < 8x; x not < 11 and not > 12. (V. 30)17a 2 +17 < 72#fandnot

In the firstDETERMINATE EQUATIONS 465and third of the last three cases the limits are notaccurate, but are integral limits which are a fortiori safe.In the second ff should have been ff and it would have been,more correct to say that, if x is not greater than f% and notless than y|, the given conditions are a fortiori satisfied.For comparison with Diophantus's solutions of quadraticequations we may refer to a few of his solutions of(3) Simultaneous equations involving quadratics.In I. 27, 28, and 30 we have the following pairs of equations.(a) + 7;= 2a(ft) + ,= 2 77).rj= r*,-;*;;It follows, by addition and subtraction, that = a + x,therefore?; = (a + *i) (a .*')= a 2 ./*2=^,and & is found from the pure quadratic equation.In (/3) similarly he assumes g rj = 2x, and the resultingequation is f 2 -f t)~= (a + ,

466 DIOPHANTUS OF ALEXANDRIA(B)Indeterminate equations.Diophantns says nothing of indeterminate equations of thefirst degree. The reason is perhaps that it is a principle withhim to admit rational fractional as well as integral solutions,whereas the whole point of indeterminate equations of thefirst degree is to obtain a solution in integral numbers.Without this limitation (foreign to Diophantus) such equationshave no significance.(a) Indeterminate equations of tJie second deyree.The form in which these equations occur is invariably this :one or two (but never more) functions of x of the formAx* + Bx + C or simpler forms are to be made rational squarenumbers by finding a suitable value for x. That is, we haveto solve, in the most general case, one or two equations of theform Ax* + Bx + C =2y.(1) Single equation.The solutions take different forms according to the particularvalues of the coefficients. Specialmore of them vanish or they satisfy certain conditions.cases arise when one or1. When A or C or both vanish, the equation can alwaysbe solved rationally.Form Bx =zy.Form Bx + C =2y.Diophantus puts for y 2 any determinate square m 2 and x is,immediately found.Form Ax* + Bx y 2 .Diophantus puts for y any multiple of x, as x.2. The equation Ax 2 + C = y2can be rationally solved accordingto Diophantus :(a) when A is positive and a square, say a2 ;in this case we put a 2 x 2 + C = (ax m)2,whencetf-m 2x = +- 2ma(m and the sign being so chosen as to give x a positive value) ;

INDETERMINATE EQUATIONS 467()8) when C is positive and a square, say c 2 ;in this case Diophantus puts Ax 2 + c 2 = (w# + c)2,and obtains2 m cx= 4-. r/r(y) When one solution is known, any number of othersolutions can be found. This is stated in the Lemma toVI. 15. It would be true not only of the cases + Ax 2 + C = 2y ,but of the general case Ax 2 + Bx + C = y\ Diophantus, however,only states it of the case Ax* C = y 2 .His method of finding other (greater) values of x satisfyingthe equation when one (x )is known is as follows. If2A x QC = q~,he substitutes in the original equation (XQfor x and (qkx) for y, where k is some integer.itThen, since A (a* + x)2C = (7 ,)*,while 4aj 2 C =follows by subtraction thatwhenceand the new value of x is ;c +Form Ax 2 c 2 =?/2.1C" ./lDiophantus says (VI. 14) that a rational solution of thiscase is only possible when A is the sum of two squares.[In fact, if x = p/q satisfies the equation, and Ax2 c 2 = /c2,we haveApor A =Form Ax 2 + C = y2Diophantus proves.in the Lemmato VI. 12 that this equationhas an infinite number of solutions when A + C is a square,i.e. in the particular case where x = 1 is a solution. (He doesnot, however, always bear this in mind, for in III. 10 heregards the equation 52x2 +l2 = y* as impossible though52 + 12 = 64 is a square, just as, in III. 11, 266# 2 -10 = y2is regarded as impossible.)Suppose that A + C = q 2 ;the equationis then solved by

468 DIOPHANTUS OF ALEXANDRIAsubstituting in the original equation 1 4- x for x and (qfor y, where k is some integer.kx)3. Form Ax* + Ex +0 = y*.This can be reduced to the form in which the second term iswanting by replacing x by z -.Diophantus, however, treats this case separately and lessfully. According to him, a rational solution of the equationAx* + x + C=y* is only possible(a) when A is positive and a square, say a 2 ;() when G is positive and a square, say c2 ;(y) when ^ J92 AC is positive and a square.In case (a) y is put equal to (ax m), and in case (ft) y is putequal to (mxc).Case (y)is not expressly enunciated, but occurs, as itwere, accidentally (IV. 31). The equation to be solved is4 ,//-,3 x + 1 8 x* = y2.Diophantus first assumes 3 s + 1 8 ./:2=which gives the quadratic 3&+ 18 = 5jj 2 but this 'is;notrational '. Therefore the assumption of 4 x2 for y 2 will not do,caud we must find a square [to replace 4] such that 1 8 times(this square 4- 21) + (|) may be a square'. The auxiliaryequation is therefore 18(m' 2 + l) + f= ?/2,or 72m 2 + 81= asquare, and Diophantus assumes 72 ?// 2 + 8 1 = (8 m + 9)2m= 18. Then, assuming 3;/;+ 18 x = 1 2(!8)equation 32 5 #2 3i/: 18 = 0, whence x = 372\, that is, 2V(2) Double equation,.,whence,/;2,he obtains theThe Greek term is &7rAoar6r??y, 8i7rXfj lo-orr}? or SnrXfj fraxrtr.Two different functions of the unknown have to be madesimultaneously squares. The general case is to solve inrational numbers the equationsmrc 2 + a x + a = u 2b =The necessary preliminary condition isthat each of the twoexpressions can be made a square. This is always possiblewhen the first term (in x z ) is wanting. We take this simplestcase first.

INDETERMINATE EQUATIONS 4691. Double equation of the first degree.The equations arca x -f a = u 2 ,Diophantus lias one general method taking slightly differentforms according to the nature of the coefficients.(a)First method of solution.This depends upon the identity1C the difference between the two expressions in x can heseparated into two factors ^>,

470 DIOPHANTUS OF ALEXANDRIAIn order that this equation may reduce to a simple equation,either(1) the coefficient of x* must' vanish, or a j8= 0,or (2) the absolute term must vanish, that is,orp*-2p 2 (a + b) + (a-6)2= 0,{p 2 -(a + ft)}2= 4ci&,so that ab must be a square number.As regards condition (1)we observe that it is really sufficientif an 2 =2)8m , since, if a x + a is a square, (a x + a)?i 2 is equallya square, and, if ftx + b is a square, so is (/8# + 6)m 2 and,vice versa.That is, (1) we can solve any pair of equations of the form/v * /YtTt i' I ft if * \LA i/v w |^ vt/ iv Ia/i' 2 a; + 6= w-JMultiply by ri2 ,m~ respectively, and we have to solve tiltequationsputa m 2Separate the difference, cm 2 6m 2 ,into two factors^;, 5 andtherefore u /2 =and a m 2 7i 2^ + an 2 = i ( ^ -f g) 2 ,and from either of these equations we get- (a^2+ 6m 2 ),^ =since2>

INDETERMINATE EQUATIONS 471Ex. from Diophantus:65- 60:=r %- 32 )'therefore 260 - 24 x = u /2 1'G5-24#=iC' /2 jThe difference = 195 = 15.13, Sciy ;therefore (15 1 3)* = 05 - 24 # ;that is, 24# = 64, and x =.Taking now the condition (2) that ab is a square, we seethat the equations can be solved in the cases where eithera and b are both squares, or the ratio of a to 6 is the ratio ofa square to a square.If the equations areOLX+ (T = U 2 ,and factors are taken of the difference between the expressionsas they stand, then, since one factor p, as we saw, satisfies the2equation [ />(

472 DIOPHANTUS OF ALEXANDRIAor, since pq = ad 2 )8c 2 ,2p*x*-2x(ocd* + pc*) + q*-*c*d = 0.In order that this may reduce to a simple equation, asDiophantus = requires, the absolute term must vanish, so thatq 2cd. The method therefore only gives one solution, sinceqis restricted to the value 2 cd.Ex. from Diophantus:6a + 4="(IV. 39)Difference 2o?; q necessarily taken to be 2\/4 or 4; factorstherefore \x, 4. Therefore 8# + 4 = (|x + 4)2,and a; =112.(j9)Second method of solution of a double equation of thefirst degree.There is only one case of this in Diophantus, the equationsbeing of the form Jix + n 2 = u 1 }Suppose hx + ?i2 = (y + n}1and (A +/ )o; + n 2 = (y + n)2+ t (y *;therefore Jut = yl-f 2 wy,It only remains to make the latter expression a square,which is done by equating it to (py n)*.The case in Diophantus is the same as that last mentioned(IV. 39).Where I have used y, Diophantus as usual contrivesto use his one unknown a second time.2. Double equations of the second degree.The general form isL+Bx +0 =but only three types appear in Diophantus, namelyp 2 /#2 + (xx + a = u 2 }(1) -7K where, except in one case, a = b./rar + pas-f 6 = w )

INDETERMINATE EQUATIONS 473,* 2 + a + a = t~(The case where the absolute terms are in the ratio of a squareto a square reduces to this.)In all examplesof these cases the usual method of solutionapplies.The usual method does not lierc serve, and a specialartificeis required.Diophantus assumes u2 = m 2 # 2 .Then x = a/(m2equation, we havea) and, bysubstitution in the second)S (---,- ) +-Vw,- a/.,m j a3 which must l)e made a square,Aor a- (3 + Ixifyn? 01)must be a square.Wu have therefore to solve the equation(a/3

474 DIOPHANTUS OF ALEXANDRIAThe species of the firstas follows.class found in the Arithmetica are1 .Equation Ax* + Bx2 + Ox + c/ 2 = if.As the absolute term is a square, we can assumeor we might assume y = m 2 x* + nx + d and determine m, n sothat the coefficients of x, x* in the resulting equation bothvanish.Diophantus has only one case, & 3 3x 2 + 3x + I = y2(VI. 18),and uses the first method.2.Equation Ax* + Bx3 + Cx* + Dx + E = y\ where either A orE is a square.If A is a square (= a 2 ), we may assume y = ax* + - x + n,&(.(>determining n so that the term in x 2 in the resulting equationmay vanish. If E is a square (== we e2 ), may assumey = mx 2 + - x + e, determining m so that the term in x 1 in theA 6resulting equation may vanish.obtain a simple equation in x.We shall then, in either case,3.Equation Ax* + Cx 2 + E = 7/2,but in special cases only whereall the coefficients are squares.4.Equation Ax* + E = 2/2.The case occurring in Diophantus is #4 + 97 = y* (V. 29).Diophantus tries one assumption, y = x2 1 0, and finds thatthis gives= a?2 -jfcwhich leads to no rational result. He,therefore goes back and alters his assumptionsso that heis able to replace the refractory equation by x* + 337 = y\and at the same time to find a suitable value for y, namelyy = #2 26, which produces a rational result, x = \ 2 -.5.Equation of sixth degree in the special formPutting y = x3 + c, we have Ax 2 + B = 2c# 2 ,andi>- iswhich, gives a rational solution if5 - A + 2c

INDETERMINATE EQUATIONS 475a square. Where this does not hold (in IV. 18) Diophantusharks back and replaces the equationa? 16# :) + oj + 64 = y/2by another, &- 128# 3 + x + 4096 = y\Of expressions which have to be made cubes, we have thefollowing cases.There are only two cases of this. First, in VI. 1,cr,24x + 4has to be made a cube, being already a square.2 a cube.naturally makes #DiophantusSecondly, a peculiar case occurs in VI. 17, where a cube hasto be found exceeding a square by2.Diophantus assumes(a? I) 3 for the cube and (x + I) 2 for the square. This givesa; 3 - 3 a; 2 + 3,0 - 1 = ;r2 + 2x + 3,or ,*:' + = 4 2+ ,*: 4. We divide out by #2 + 1, and x = 4. Itseems evident that the assumptions were made with knowledgeand intention. That is, Diophantus knew 'of the solution 27and 25 and deliberately led up to it. It is unlikely that he wasaware of the fact, observed by Fermat, that 27 and 25 are theonly integral numbers satisfying the condition.2. Ax* + Bx 2 + Cx + D = y/3,where either A or D is a cubenumber, or both are cube numbers. Where A is a cube (a3),we have only to assume y = cc+^,3d-(

476 DIOPHANTUS OF ALEXANDRIAMore complicatedis the case in VI. 21 :Diophantus assumes y = mx, whence x = 2/(m22), and/ 2 \ 3 _// 2 \a 2I __ 14-91 ___W- ^ 1"" "2/ Vm - 2 2/ m - 2 '22m 4or -=2(m*-2) 3We have only to make 2m 4 ,or 2m, a cube.II.Method of Limits.As Diophantus often has to find a series of numbers inorder of magnitude, and as he does not admit negativesolutions, it is often necessary for him to reject a solutionfound in the usual course because it does not satisfy thenecessary conditions; he is then obliged, in many cases, tofind solutions lying within certain limits in placeof thoserejected. For example:1 . It is required to find a value of x such that some power ofit, x n ,shall lie between two given numbers, saya and b.Diophantus multiplies both a and b by 2 W 3 W , ,and so on,successively, until some 7^th poweris seen which lies betweenthe two products. Suppose that c n lies between ap u andu bp ;then we can put x = c/p, for n (c/p) lies between a and b.Ex. To find a square between l and 2.Diophantusmultiplies by a square 64; this gives 80 and 128, betweenwhich lies 100. Therefore (^) 2 or f| solves the problem(IV. 31 (2)).To find a sixth power between 8 and 16. The sixth powersof 1, 2, 3, 4 are 1, 64, 729, 4096. Multiply 8 and 16 by 64and we have 512 and 1024, between which 729- 7^9 lies; istherefore a solution (VI. 21).2. Sometimes a value of x has to be found which will give

METHOD OF LIMITS 477some function of x a value intermediate between the valuesof two other functions of x.Ex. 1. In IV. 25 a value of x is required such that 8/(# 2 + x)shall lie between x and x+ 1.One part of the condition gives 8 > x* + x 2 .Diophantusaccordingly assumes 8 :i }= (# + )= X' + x 2 + ^x + -f, which is> a? 3 + x 2 . Thus x + = 2 or 05 = f satisfies one part ofthe condition. Incidentally it satisfies the other, namely8/(x 2 + x) < x+l. This is a piece of luck, and Diophantusis satisfied with it, saying nothing more.Ex. 2.We have seen how Diophantus concludes that, ifthen x is not less than 11 and not greater than 12 (V. 30).The problem further requires that x 2Assuming #2 60 = (as m) 2 we find x =, (m2 + 60)/2m.Since x > 11 and < 12, says Diophantus,it follows that60 shall be a square.24m > m 2 + 60 > 22 TO ;from which lie concludes that m lies between 19 and 21.Putting m = 20, he finds x = 11^.III. Method of approximationto Limits.Here we have a very distinctive method called by DiophantusTTapiyrj. The objectis to solve suchproblems as that of finding two or three square numbers thesum of which is a given number, while each of them eitherapproximates to one and the same number, or is subject tolimits which may be the same or different.Two examples will best show the method.Ex. 1. Divide 13 into two squares each of which > 6 (V. 9).Take half of 13, i.e. 6^, and find what small fraction 1 /x 1added to it will give a square ;thus6 1 H55 or 26 + ^ > must be a square.x~ y

478 DIOPHANTUS OF ALEXANDRIADiophantus assumeswhenceT/= 10, and 1/7/= T fo fi.e.[The assumption of 5 + as the side is not haphazard : 5 ischosen because it is the most suitable as giving the largestrational value for yJ]We have now, says Diophantus, to divide 13 into twosquares each of which is as nearly as possible equal to 2.(f )Now 13 =r 3 2 -f 2 2 [it is necessary that the original numbershall be capable of being expressed as the sum of two squares];and 3 >fby^,while 2 < f by J.But if we took 3-^, 2+-| as the sides of two squares,their sum would be 2(f J)2= - 5,%%2-, which is > 13.Accordingly we assume 3 9#,2 + 11 x as the sides of therequired squares (so that x is not exactly ^ but near it).Thus (3-9#) 2 + (2 + llx)2= 13,and we find x = T T.The sides of the required squares arc f-, f ^f .Ex. 2. Divide 10 into three squares each of which > 3(V.ll).xx[The original number, here 1 0, must of course be expressibleas the sum of three squares.]Take one-third of 10, i.e. 3^, and find what small fraction1/aj 2 added to it will make a square;i.e. we have to make33+ - a square,i.e. 30-h 73-must be a square, or 30+ ^y19 1= a square, where 3/x = 1 /y.Diophantus assumesthe coefficient of y> i.e. 5, being so chosen as to make 1 /y assmall as possible ;

METHOD OF APPROXIMATION TO LIMITS 479therefore y = 2, and 1 /x* = ^; and 3^ + sV = -J^1-, a square.We have now, says Diophantus,squares with sides as near as may be to - 1/.Now 10 = 9 + 1 = 3 2 + (|-)2+ (f)2.Bringing 3, f90 ,18 24 ana orul 55tf* "Sis So >to divide 10 into threef and ~y- to a common denominator, we haveand 3 >|gby,3 ^ 55 lv 7 37T < 3^ by 50 >fs 55 Vvr r< w31y 3o-If now we took 3 f , f + f , + f^ as the sides of squares,the sum of .the squares would be 3 (V") 2 or ^V~> which is > 10.Accordingly we assume as the sides 3 35 x, f + 37 x, f + 3 1 x,where x must therefore be not exactly ^ û^ near ^.Solving (3-35j-) a + (| + 37ar)2+ (| + Slaj)s = 10,or 10-116^-f 3555./; 2 = 10,we find x = ^5% 5thus the sides of the required squares are VrVs -VW"> VT^!thf* QniiaroQ fliarnaoKrê nva 1745041 1651225 1658944tne squares tiiemseives aie -^^23;-, T^"S^^T" "Ô^F^T"'Other instances of the application of the method will befound in V. 10, 12, 13, 14.Porisms and propositions in the Theoryof Numbers.I. Three propositions are quoted as occurring in the PorismsWe have(' it in the Porisms that ...');and some other propositionsassumed without proof may very likely have comefrom the same collection. The three propositions from thePorisms are to the followingeffect.1 . If a is a given number and x, ynumbers such thatx + a = m 2 , y 4- a = ?i2 , then, if xy 4- a is also a square, m and ndiffer by unity (V. 3).[From the first two equations we obtain easilyxy + a = w 2 7i 2 - a (m 2 + n 2 1) + a 2 ,and this isobviously a square if m2 + ?i 2 1 = 2mn, orm n =1.]

480 DIOPHANTUS OF ALEXANDRIA2. If m 2 , (m + I) 2 be consecutive squares and a third numberbe taken equal to 2{w 2 + (m + I)2} + 2, or 4(m 2 + m+ 1),thethree numbers have the property that the product of any twothe sum of those two or the remaining numberplus eithergives a square (V. 5).[In fact, if X, Y, Z denote the numbers respectively,XY+% = (3) 2 , YZ+X,ZX= (2m2 + 3m+2) 2 ,+ Y = (2ma + m+ I)2.]3. The difference of anycubes,(V. 16).two cubes is also the sum of twoi.e. can be transformed into the sum of two cubes[Diophantus merely states this without provingit or showinghow to make the transformation. The subject of thetransformation of sums and differences of cubes was investigatedby Vieta, Bachet and Fermat.]II. Of the many other propositions assumed or implied byDiophantus which are not referred to the Porisms we maydistinguish two classes.1. The first class are of two sorts; some are more or lessof the nature of identical formulae, e.g. the facts that theexpressions {^(a + ft)}2a6 and a 2 (a-h l) 2 + a 2 -f (&+ I) 2 arerespectively squares, that a (a 2 a) + a + (a2a) is always acube, and that 8 times a triangular number plus1gives2a square,i.e.8.%x(x+l) + 1 = (2x+ I). Others are of thesame kind as the first two propositions quoted from thePorisms, e.g.(1)If Z=a 2 # + 2tt, 7=(c6+l) 2 0+2(a+l) or, in otherwords, if xX+ 1 = (ax+ I)2and xY+ 1 = {(a+ !)#+!)*,then XY+ 1 is a square (IV. 20).In fact(2) If JTa = m 2 ,Ya=X7+1 = [a(m+l) 2 ,and Z =then YZa, ZXa, XYa are all squares (V. 3, 4).

PORISMS AND PROPOSITIONS ASSUMED 4&1In fact YZa = {(m + 1) (2m + l) + 2a}2,ZX+a, = {m(2m+ l) + 2a}2,(3) IfX = m* + 2,Fthen the six expressionsYZ-X, ZX-Y, XY-Zare all squares (V. 6).In fact2,&c.2. The second class is much more important, consisting ofpropositions in the Theory of Numbers which we find firststated or assumed in the Arithmetica. It was in explanationor extension of these that Fermat's most famous noteswere written. How far Diophantus possessed scientific proofsof the theorems which he assumes must remain largely amatter of speculation.(a)Theorems on the composition ofof two squares.numbers as the(1) Any square number can be resolved into two squares inany number of ways (II. 8).(2) Any number which is the sum of two squares can beresolved into two other squares in any number of ways (II. 9).(It is implied throughout that the squares may be fractionalas well as integral.)(3) If there are two whole numbers each of which is thesum of two squares, the product of the numbers can beresolved into the sum of two squares in two ways.In fact (a 2 + t 2 ) (c 2 + d 2 ) == (ac bdf + (ad T 6c)2.This proposition is used in III. 19, where the problem isto find four rational right-angled triangles with the same

482 DIOPHANTUS OF ALEXANDRIAhypotenuse. The method is this. Form two right-angledtriangles from (a, b) and (c, d) respectively, by which Diophantusmeans, form the right-angled triangles(a 2 + i 2 ,a 2 -& 2 , 2ab) and (c 2 + cZ 2 ,c a -d 2 , 2cd).Multiply all the sides in each triangle by the hypotenuse ofthe other; we have then two rational right-angled triangleswith the same hypotenuse (a2 + 2 2ft) (c + cZ 2 ).Two others are furnished by the formula above; for wehave only to ''form two right-angled triangles from (ac + Id,ad be)and from (ac bd, ad + bc) respectively. The methodfails if certain relations hold between a, 6, c, d. They mustnot be such that one number of either pair vanishes, i.e.suchthat ad = be or ac = bd, or such that the numbers in eitherpair are equal to one another, for then the triangles areillusory.In the case taken by Diophantus a'2 + 6 2 = 2 2+ 1 2 = 5,c 2 + d 2 = 3 2 + 2 2 = 1 3,and the four right-angled triangles are(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33).On this proposition Format has a long and interesting noteas to the number of ways in which a prime number of theform 4n+l and its powers can be (a) the hypotenuse ofa rational right-angled triangle, (6)the sum of two squares.'He also extends theorem (3)above : If a prime number whichis the sum of two squares be multiplied by another primenumber which is also the sum of two squares, the productwill be the sum of two squares in two ifways the first prime;be multiplied by the square of the second, the product will bethe sum of two squares in three ways the ;product of the firstand the cube of the second will be the sum of two squaresin four ways, and so on ad iufinitum. 9Although the hypotenuses selected by Diophaiitus, 5 and 1 3,are prime numbers of the form 4?i+ 1, it is unlikely that hewas aware that prime numbers of the form 4 n + 1 andnumbers arising from the multiplicationof such numbers arethe only clesses of numbers which are always the sum of twosquares this was first proved by Euler.;(4) More remarkable is a condition of possibility of solutionprefixed to V. 9, 'To divide 1 into two parts such that, if

NUMBERS AS THE SUMS OF SQUARES 483a given number is added to either part, the result will be asquare/ The condition is in two parts. There is no doubt asto the first, 'The given number must not be odd' [i.e.nonumber of the form 4n + 3 or 4^1 can be the sum of twosquares]the text of the second part is corrupt, but the words;actually found in the text make it quite likely that correctionsmade by Hankel and Tannery give the real meaning of theoriginal, ' nor must the double of the given number plus1 bemeasured by any prime number which is less by 1 than amultiple of 4 This '. is tolerably near the true conditionstated by Fermat, The 'given number must not be odd, andthe double of it increased by when 1 divided , by the greatestsquare which measures it, must not be divisible by a primenumber of the form 4 n I/(/8)On numbers which are tJie sum of three squares.In V. 11 the number 3ti + l has to be divisible into threesquares. Diophantus says that a 'must not be 2 or anymultiple of 8 increased by 2 That ''. is, a number of theAs a matterform 24n+7 cannot be the sum of three squares '.of fact, the factor 3 in the 24 is irrelevant here, and Diophantusmight have said that a number of the form 8 n + 1 cannot bethe sum of three squares. The latter condition is true, butdoes not include all the numbers which cannot be the sum ofFermat gives' the conditions to which a must l>ethree squares.subject, proving that 3 a + 1 cannot be of the form 4" (24 k + 7)or 4 w (8fc+ 7), where k = or any integer.(y)Composition of numbers as the sum offour squares.There arc three problems, IV. 29, 30 and V. 14, in which itis required to divide a number into four squares. Diophantusstates no necessary condition in this case, as he does whenit is a question of dividing a number into three or luv squares.Now every number is eitfier a square or the sum of two, threeor four squares (a theorem enunciated by Fermat and provedby Lagrange who followed up results obtained by Euler), andthis shows that any number can be divided into four squares(admitting fractional as well as integral. squares), since anysquare number can be divided into two other squares, integral

484 DIOPHANTUS OF ALEXANDRIAor fractional. It is possible, therefore, that Diophantus wasempirically aware of the truth of the theorem of Fermat, butwe cannot be sure of this.Conspectus of the Aritlimetica,with typical solutions.There seems to be no means of conveying an idea of theextent of the problems solved by Diophantus except by givinga conspectus of the whole of the six Books. Fortunately thiscan be done by the help of modern notation without occupyingtoo many pages.It will be best to classify the propositions according to theircharacter rather than to give them in Diophantus's order.should be premised that x, y, z . . .indicating the first, secondand third . . . numbers required do not mean that Diophantushe gives his un-indicates, any of them by his unknown (?) ;known in each case the signification which is most convenient,numbers at once inhis object being to express all his requiredterms of the one unknown (where possible), thereby avoiding thenecessity for eliminations. Where I have occasion to specifyDiophantus's unknown, I shall as a rule call it,except whena problem includes a subsidiary problem and it is convenientto use different letters for the unknown in the original andsubsidiary problems respectively, in order to mark clearly thedistinction between them. When in the equations expressionsare said to be = u 2 ,*/ 2 , ic; 2 ,t* ... this means simply that theyare to be made squares. Given numbers will be indicated bya, 6, c . . . m, and will take the -ii . place of the numbers used. .by Diophantus, which are always specific numbers.Where the solutions, or particular devices employed, arcspecially ingenious or interesting, the methods of solution willbe shortly indicated. The character of the book will bo bestappreciated by means of such illustrations.[The problems marked with an asterisk are probablyspurious.]It(i)Equations of the first degreewith one unknown.I. 7. # a = m(x b).I. 8. x + a = m(x + b).

DETERMINATE EQUATIONS 485I. 9. a x = m (&&).I. 10. x + b = m(a #).I. 11. x + b = m(# a).I. 39. (a + x)b + (h + x}a = 2(a + &)#,\or (a + 6)a; + (6 + #)a= 2 (a + x) 6,L (a > 6)or (a + 6)x+(a + a;)6= 2(6 + #)a.)Diophantus states this problem in this form, 'Giventwo numbers (a, 6), to find a third number (x)such thatthe numbers(a + x)b, (b + x)a, (a + b)xare in arithmetical progression/The result is of course different according to the orderof magnitude of the three expressions. If a>b (5 and 3are the numbers in Diophantus), then (a + x)b < (b + x)a\there are consequently three alternatives, since (a + x)bmust be either the least or the middle, and (b + x)a eitherthe middle or the greatest of the three products.have(( / -f x) b < (d + b)x < (b + ,r) a,or (a + b)x < (a + x) b < (b + x) a,We mayor(a + x)b < (b -f x)a < (a -f b)x,and the corresponding equations are as set out above.(ii)Determinate systems of equations of the first degree.I. 1. x + y = a, x y = b.I. 2. aj-f y = a, x = my,I. 4. ^ y= a, x = my.1. 3. # + 2/= a, x = my + 6.I. 5. a; + y *= a, # + - y= &, subject to necessary condition.m n1 1 .I. 6. oj + y = a, x -- 7/=6,

kCor.486 DIOPHANTUS OF ALEXANDRIA(I. 12. x l + x%i. 13. Xi + xîI. 15. # + a = m(y a), y + b = n(# 6).[Diophantus puts y = + a, where is his unknown.]I. 16. y + = a, s + # = 6, # + y = r. [Dioph. puts =a: + y + 3.]I. 17. y + z + tv = a, 2 + 10 -f # = 6, w + x + y = c, x + y + z = ri.1. 18. T/ + = a,I. 19.[Dioph. puts 2 = ^ # + y 4- z.1.20. aj + y + z = a, x + y= mz, y + z = TIOJ.I. 21. x = y+ 0, y = s+ ~#, 3 = a + - y (where 05 >y>0),??7/ *y t/^f)with necessary condition.II. 18*. a?-( \m/[Solution wanting.](iii)Determinate systems of equationsfirst degree.reducible to theI. 26. ax = a2^fa = a.I. 29. x + y= a, a? 2 y 2 = 6. [Dioph. puts 2 = #-y.JI. 31. x = my,I. 32. x = my,I. 33. x =I. 34. x =I. 34. Cor. 1. x = my, #y = n(x + y).2. aj = my, #y = 71(03 y).

(I.DETERMINATE EQUATIONS 48735. x = 2my, y = nx.I. 36. x = my, y2= My.I. 37. a; =2my, y = n(x + y).I. 38. x =2my, y = 7i(x y).I. 38. Cor. a; = my, x = 2 tiy.a; = my, # = 2 nx..! =my, x* = n(x + y).a = my, x 1= î(x~y).II. G*. -y = a, x 2 y 2 = x y + b.IV. 36. z =[Solved by means of Lemma : see under (vi) Indeterminateequations of the first degree.](iv)Determinate systems rfeducible to equations ofsecond degree.I. 27. x + y = a, xy = I.[Dioph. states the necessary condition, namely thatci 2 6 must be a square, with the words are so chosen that (a ^6 3 )/36 isa square.]IV. 2. a 3 -y'*= a, xy = ft.

488 DIOPHANTUS OF ALEXANDRIAIV. 15. (y + z)x= a, (z + x)y= 6, (x + y)z= c.[Dioph. takes the third number z as his unknown ;thus # + 2/= rt/2.Assume x = p/z, y =

DETERMINATE EQUATIONS 489whenceand it is only necessary that (a + 1) (jS + 1) (y + 1)shouldbe a square, not that each of the expressions a + 1, + 1,y -f 1 should be a square.Dioph. finds in a Lemma (see under (vi) below) a solutionkv dopto-TM (indeterminately) of acy + (which practically means finding y in terms of #.]IV. 35. z-[The remarks on the last proposition apply mutatismutandis. The lemma in this case is the indeterminatesolution of xy(x + y)= &.]IV. 37. yz = a(v + y + s),cs = b(.r,+y + z), xy = c([Another interesting case of c false hypothesis '. Dioph.first gives x + y + z an arbitrary value, then finds thatnot rational, and proceeds to solve the newthe result isproblem of finding a value of x + y + zto take the placeofthe first value.If w = x +y + z, we have x = civ/y> z = = bw by hypothesis; therefore y = 2 -^-w.For a rational solution this last expression:isquare. Suppose, therefore, that w =etc-y-ac . t>ac . .2must be,and we haveEliminating a?, jy, 0, we obtain = (bc + ca + ab)/ac,and# = (/>c + ca + ab)/a, y = (fa + ca -fab)/b,z = (bc + ca + ab)/e.]Lemma to V. 8. yz = a 2 ,c.r = /> 2 , xy = c 2 .

490 DIOPHANTUS OF ALEXANDRIA(v) Systems of equations apparently indeterminate butreally reduced, by arbitrary assumptions, to determinateequations of the first degree.I. 14.xy = m(x + y). [Value of y arbitrarily assumed.]II. 3*. xy = m(x + y),and acy=II. 1*. (cf. I. 31). x* + y2=II. 2*. (cf.I. 34).x 2 2=7/ m(x y). > [x assumed = 2 y.]II. 4*. (cf. I. 32). x* + y* =m(x-y).\II. 5*. (cf. I. 33).aj 2 ~2/ 2 = m(# + 2/).jII. 7*. x 2 y 2 = m(# 2/)+ ^-[Diopb. assumes # ?/= 2.]T1. 22. #-- cc+-0 = 2/-- 2/H-- a? = --m z+ -y. up n m 'Yip[Value of y assumed.]T A1 1 11111.23. x --- x 4-- w = y-- y H --- x = -- H ?/r^ g7i 7}ip n J1.24. x+ (y + z)= y+-(s +7/6 IvI. 25. X+[Value of y + z assumed.]= w -- w+ -0. [Value ofl yJ assumed.] J

\INDETERMINATE ANALYSIS 491IV. 33. x+ -y m (y -y),y + - x = n(x -x)Z'[Dioph. assumes ^y =1.]Z * *^ \ / /(vi)Indeterminate equations of the first degree.Lemma to IV. 34. xy + (x + y)=a.^ [Solutions >^ ,r

492 DIOPHANTUS OF ALEXANDRIAII. 16. x = mj/, a? + x = u 2 ,a> 2 -}-2/ = ^2 .II. 19. # 2 2=7/ mfv2s 2 ).[Assume ?/ == 2mo; + m 2 ,and one condition is satisfied.]i[Assume x = + m, y = 2m + m 2 ,and one conditionis satisfied.]/II. 22. x 2 + (x + y)-1[Put# + ?/=II. 23. x* (x + y)= u 2 ,II. 24. (# + 7/)2+ # = u 2 ,[Assume = 02 (m 2 1)2II. 25. (^ + 2/)# = ^2;2,y = (>//(^ + ?y)2~~2/ == v "-21)| 2 ,x + y =.]II. 26. xy + x = tt2,xy + y = v*, u + v = c/.[Put y = ^2 aj-l.]IIITT rtiw V7 o ^/"J/ >Of* "

INDETERMINATE ANALYSIS 49311.34. x*2[Since {-|(m n) } + mn is a square, take any numberseparable into two factors (m, n) in three ways. Thisgives three values, say, p, q,r for ^(m n). Putx = P&> y= d?> z = r >ancl * + V + 2 =2 w&7i therefore>(2> +

494 DIOPHANTUS OF ALEXANDRIAThen, since the sum of the first three expressionsisitself equal to x + y + z, we have# = (&2 + c 2 ), yIII. 6. x + y + z = 2 ,III. 7. x-y = y-z,(III.8. x + y + z + a = 2 , 2/ + 2 + C& = w 2 >III. 9. x + y + z a = 2 , y + z a - ii2III. 10. 7/^ + tt = U?, zx + a = v 2 , ay-f a = ^2 .[Suppose 1/2; + a = m 2 ,and let ?/= (m22also let z# + a = iiWe have therefore to make;therefore a; = (/t2(m 2 a) (n 2 a) 2 + a a square.,z + x a = v2,# -f 2/a = w 2 .a)^.a)^, z = l/:Diopliantus takes m 2 = 25, a = 12, n 2 =16, andarrives at 52^2 +12, which is to be made a square.Although 52 . l f-+ 12 is a square, and it Follows that anynumber of other solutions giving a square are possibleby substituting 1 + for in the77 expression, and so on,be solvedDiopliantus says that the equation could easilyif 52 was a square, and proceeds to solve the problem offinding two squares such that each increased bygive a square, in which case their product12 willalso will bea square. In other words, we have to find m 2 and u 2such that m 2 a, n 2 a are both squares, which, as hesays, is easy. We have to find two pairs of squaresdiffering by a.Ifandlet, then, m 2 =III. 11. yz a = u 2 3#,a = v2,xy a = it;2.[The solution is like that of III. 1 mutatis mutwridis.](III. 12. yz + x = u 2 ,0# + ?/= ^2j y + z = î2 -(HI. 13. yzx= u 2 ,zx y = v 2 ,xy z = w 2 .

INDETERMINATE ANALYSIS 495111. 14. yz + x* = u 2 ,zx + y*= v2, xy + z* = w 2 .[Lemma. If a, a+1 be two consecutive numbers,a 2 (a + 1 )2+ a 2 + (a + 1 )2 is a square. Let2/ = w a ,s = (m+1)2Therefore (m 2 + 2m + 2) aj -f (m + 1 )2and (m' 2 + l)# + m 2have to be made squares. This is solved as a doubleequation;in Diophantus's problem m = 2.Second solution. Let x be the first number, m thesecond; then (m+l)# + m is a square= u 2 ,say; thereforex =2(M = w. We have thenm)/ (771+ 1), while 2/(m + \)z + m = a square?i 2 -m2 + . /and(1\,)+ = a squareV m -f 1 / m+1jDiophantus has m = 3, H = 5, so that the expressionsto be made squares are with him.42 + 3 )This isnot possible because, of the correspondingcoefficients,neither pair are in the ratio of squares.In order tosubstitute, for 6, 4, coefficients which are in the ratioof a square to a square he then finds two numbers, say,^>, q to replace 5^, 3 such that 2 )( 1+P + ( 1 = a square, and= (/>+!)/(

496 DIOPHANTUS OF ALEXANDRIAafter multiplying by 25 and 100 respectively, makingexpressions130.T+ 30In the above equations we should only have to make>i 2 + 1 a square, and then multiply the firstby 7i2 + 1 andthe second by (m+ I) 2 .Diophantus, with his notation, was hardly in a positionto solve, as we should, by writingwhich gives x+1 =-/{(&*+ 1) (c- + l)/(aa + 1) }, &c.]III. 1 6.yz (y + z)= u\ zx(z + x)= v 2 , xy-(x + y)= ?'/2.[The method is the same mutatis mutandis as thesecond of the above solutions.]III. 17. xy + (x + y)= u' 2 ,aIII. 18. x-III. 19. (^ 1+ a: a + ii; S8a+ aj 4 )+ aj 1=^fu 24 )- a a=-, ~I/ 8f+^= /2 j[Diophantus finds, in the way we have seen (p. 482),four different rational right-angled triangles with thesame hypotenuse, namely (65, 52, 39), (65, 60, 25), (65,56, 33), (65, 63, 16), or, what is the same thing, a squarewhich is divisible into two squares in four different ways ;this will solve the problem, since, if h, p, b be the threesides of a right-angled triangle, h 22pb are both squares.

INDETERMINATE ANALYSIS 497Put therefore x l + x 2 + x% + x = 6 5 .IV. 4.|lIV. 5.and #!= 2. 39. 52 f 2 , o? 2= 2. 25. 60 2 ,# 3= 2 . 33 . 56 2 ,# 4= 2.16.632this gives 127G8f2= G5, and =IV. 13. x+ 1 =2 i ,y+ = u 1 2 #, + ?/+1 = t;2, y-x + I = w 2 .[Put x =2(m^ + I)21 = 7?if 2 4- 2m;;the second andthird conditions require us to find two squares with x asdifference. The difference m' 2 2 + 2m is separate^! intothe factors w' 2 -f 2m, ; the squareof half the difference= ((m 2 l) + mj-. Put this equal to ?/+!, sothat y = J(m--l)a *+ m(m a -l) + m a -l, and thefirst three conditions are satisfied. The fourth gives4(??i6 m~ + 1 ) | 2 + (m 3 3m) + ?7i' 2 = a square, whichwe can equate to (n m) 2 .]IV. 1 4.x 2 + y* + z~ =(,(;-- ya)+ (//*- 2 ) + (x2 - s2). (^ > y > z.)IV. 16.(4m+l) 2 ;therefore s = 8in+l. By[Put 47?i^ for y, and l>y means of the factors 2mg, 2we can satisfy the second condition by making x equalto half the difference, or m 1. The third conditionis satisfied by subtracting (4m) 2 from some square, saythe first condition13 mg must be a 2square. Let it be 169ij the;22numbers are therefore 1 3 1 ?/ ,5 2 1 ?/04 2, ij + 1 ,andthe last condition gives 10816?/4 4- 22 1?/2= a square,i.e. 10816?; 2 -f 221 = a square= (104ij + I)2, say. Thisgives the value of 17,. and solves the problem.]IV. 17. x + + z = J2IV. 19. ?/0+ 1 = u 2,x*-= u\ tf-z = v 2 ,z 2 -x = w 2 .[We are asked to solve this indeterminately (i> ro>dopbmp). Put for y0 some square minus 1, say m 2 ^2+ 2m; one condition is now satisfied. Put z =,that y = ?7i2^ + 2m.1523.2 K kso

498 DIOPHANTUS OF ALEXANDRIASimilarly we satisfy the second condition by assumingzx = ?i2 2+ 2 Tig ;therefore x = 7i2+ 2n. To satisfy thethird condition, we must have(m 2 n 2 2 + 2mn .m + ng + 4 WTI) + 1 a square.We must therefore have 4mn+ 1 a square and alsomn (m + n)= mw \/ (4mri + 1 ). The first condition issatisfied by n = m + 1 ,which incidentally satisfies the2second condition also. We put therefore yz= (m + I)2and 0#= { (m+ l) + 1 }1, and assume that =,1so that2/ = m 2 +2m, # = (m+ l)2+ 2(m+l), and we haveshown that the third condition is also satisfied.Thus wehave a solution in terms of the undetermined unknown .The above is only slightly generalized from Diophantus.]IV. 20. xx+l = r 2 ,ffO + 1 = s 2 , a+1= f 2 ,= u 2 ,[This proposition depends on the last, x},2 , aJ 3 beingdetermined as in that proposition. If x 3 corresponds to zin that proposition, we satisfy the condition # 3# 4 -f 1 = w-by putting # # = 23 4 |(m + 2) + I} 1, and so find x 4interms of ,after which we have only two conditions moreto satisfy. The condition x^+1 = square is automaticallysatisfied, sinceis a square, and it only remains to satisfy x 2 x^ + 1 = squareThat is,t= m 2 (m+ 2) 2 ^ 2 + 2m(m + 2) (2m + 2) + 4m(m + 2) + 1has to be made a square, which is easy,ofaj;j=2 is a square.since the coefficientWith Diophantus m = 1, so that x l= 4 + 4, # 2= + 2^ ^4= 9^ 6 and 9 2 + 24^+13a square. He equates this to (3 4)2IV. 21. xz = y2has to be mady>z|IV. 22. xyz + x = u 2 ,xyz + y = v 2 , xyz + z = w 2 .lIV. 23. xyz x = u 2 ,xyzy= v 2 , xyz z = w 2 .

'\TINDETERMINATE ANALYSIS 499IV. 29. x 2 + y2 + z 2 + iv 2 + x + y + z + iv a.[Since + x2 x + %is a square,(x 2 + x) + (y2 + y) + (z2 + z) + (w2 + w) + 1is the sum of four squares, and we only have to separatea + 1into four squares.]IV. 31. x + y = 1, (x + a) (y + b)= u 2 .IV. 32. x + y + z = a,IV. 39.x-y = m(y-IV. 40. x~ y 2 = m(y1 _ 2 i . 2__ 2 ___ *'V. 2. #2 = ?/2V_l> O . ,1I// /Ji< /}/ I., // o- r* I// /^* V . tV j^ v /r I, ^^ \M O ,

500 DIOPHANTUS OF ALEXANDRIAV. 6. x-2 = r2,T/-2 = s 2 ,z-2= J2,yzyz = V?, 2# z x = ^2>x y x y = w' 2 >yz x = u'2,zxy= v' 2 ,xy z = it/2.[Solved by means of the proposition numbered (3) onp. 481.]Lemma*! to V. 7.V. 7.(u[Solved by means of the subsidiary problem (Lemma 2)of finding three rational right-angled triangles withequal area. If m, n satisfy the condition in Lemma 1,i.e. if mn + m 2 -f n 2 = p*, the triangles are ' formed 'fromthe pairs of numbers (p, m), (p, ri), (p>m + 7i) respectively.Diophantus assumes this, but it is easy to prove.In his case m = 3, n = 5, so that p = 7. Now, ina right-angled triangle,(hypotenuse)2+ four times areais a square. We equate, therefore, x + y + z to four2times the common area multiplied by and the several,numbers x> y, z to the three ,hypotenuses multiplied byand equate the two values. In Diophantus's case thetriangles are (40, 42, 58), (24, 70, 74) and (15, 112, 113),and 245 = 3360 2 .]V. 8. yz(x + y + z)=[Solved by means of the same three rational rightangledtriangles found in the Lemma to V. 7, togetherwith the Lemma that we can solve the equations 7/0= a 2 ,zx = 6 2 ,xy = c 2 .]11.9. (Of.[These are the problems of

INDETERMINATE ANALYSIS 501described above (pp. 477-9).The problemis ' to divideunity into two (or three) parts such that, if one and thesame given number be added to each part, the results areall squares '.] %(V. 10. x + y = 1, x + a = u zlV. 12. x+y+z= 1, x + a[These problems are like the preceding except thatdifferent given numbers are added. The second of thetwo problemsis not worked out, but the first is worthreproducing. We must take the particular figures usedWe have then toby Diophantus, namely a = 2, b = 6.divide 9 into two squares such that one of them liesbetween 2 and 3. Take two squares lying between 22and 3, say fff ff. We have then to find a square ,lying between them if we can do; this, we can make9 J 2 a square, and so solve the problem.Put 9- = 2 2(3-m), say, so that = 6m/(m2 + 1);and m has to be determined so that lies between17 o,,/| 19 "1 217 6m 19Thereforet ,

502 DIOPHANTUS OF ALEXANDRIAV. 24. 2/V+ 1 = u\ z*x*+l = v*}x*y*+l = w*.V. 25. 2/2 s2 - 1 = u 2 ,z*x 2 - 1 = v2, a?V- 1 = i(;2 .V. 26. 1 -y2 * 2 = u 2 ,1 -'0 2 x 2 = v 2 ,1 -a 2 2=2 it; .?/[These reduce to the preceding set of three problems.]V. 27. 2/2 + s 2 + a = i6 2 ,s 2 + a> 2 + tt = v2,a 2 + ?/ + a = 2 .V. 28. 2/2 + s 2 -a = u 2 ,V. 30. mx + ny = u2 ,[This problem is enunciated thus. 'A rnan buys acertain number of measures of wine, some at 8 drachmas,some at 5 drachmas each. He pays for them a squarenumber of drachmas ;and if 60 is added to this number,the result is a square, the side of which is equal to thewhole number of measures. Find the number bought ateach price.'Let = the whole number of measures ;therefore260 was the number of drachmas paid, and 2 60= a square, say ( m) 2 hence =; (m2 + 60)/2m.Now % of the price of the five-drachma measures +of that of the eight-drachma measures = therefore;260, the total price, has to be divided into two partssuch that -|of one -f | of the other = .We cannot have a real solution of this unless*>i-(f-60) and

INDETERMINATE ANALYSIS 503Let the first part = 62; therefore J (second part)= 11^ 0, or second part= 92 Sz.Therefore 5z + 92 - 8z = 72%, and z = f 5therefore the number of five-drachma measures is ff andthe number of eight-drachma measures ff .]Lemma 2 to VI. 1 2. ax 2 + b = u 2 (where a + b = c2Lemma to VI. 1 5. o* - &=u 2 (where ad2 - 6 = c 2 ).)\f [III. 15]. xy+'x + y = u*, x+l =i\ v[III. 16]. iry-(x + y)= u*, 0-1 =). \ ( see p 467above.)(viii)Indeterminate analysis of the third degree.IV. 3.x*y = u, xy = u 3 .T \r n O O .> .>I IV. 7. or + 2/2= u 2 ,z 2 -I- j/2 = v\8.|1V.IV. 9.!1V. IV. 12. x + :i(really reducible10. oj 3 + 2/3= x + y. \ to the secondIV A f . 11 JL J. . T*-,fl- ,(/ f7 v -,n ^~~ i)[We may give as examples the solutions of IV. 7,,IV. 8, IV. 11.IV. 7. Since 3 2 + 7/~= a cube, supposeTo make x* + y* a square, put x 3 = a 2which also satisfies # 3 2jy=2. We2 a& a square. Let a =,6=2^;have then to maketherefore a2 -f 6 = 2 5 2^,2ab = 4^2,y = 2, z = g,and we have only to make5 2 a cube. = 5, and # 3 = 125, 2/2= 100, e 2 = 25.

504 DIOPHANTUS OF ALEXANDRIAIV. 8.Suppose x = ,must be the side of the cube m 3 3 + ,andy* =m3 3 ;therefore u= (m + 1 )To solve this, we must have 3m 2 + 3m + 1between consecutive cubes) a square.Put(the difference3m 2 + 3m +1 =2(l-?irn) and m =, (3 + 2n)/(n*-3).IV. 11.Assume x = (m-f 1), y = m, and we haveto make (3m 3 +3m 2 + 1)2only to make 3m2 + 3rn+ 1IV. 18. x* + y - u*, y 2 + x = v\IV. 24. x + y = a, xy*~ u? u.equal to 1, i.e. we havea*square.]= [y a x m therefore ax x 2 has to be made a cube,3minus its side, say (mx I)(mx 1).Therefore ax x 2 = m 3 z* 3 m 2 ,/j2-f 2 m#.To reduce this to a simple equation, we have only toput m = %aJ\IV. 25. x + y + z = a, ,/:?/= { (x-y) + (x-z) + (?y-^)}:i-(,/ > T/> s)[The cube = B(x zf. Let cc = (m+ 1), = m, sothat 2/= 8/ (m 2 + m), and we have only to contrive that8/(m 2 + m) lies between m and m/ + 1.Dioph. takes the'Jfirst limit 8 > mf+ m 2 and, puts8 = (m-f ):5orwhence m/ = f ;therefore # =,/= !, 2 = |. Or,multiplying by 15, we have x = 40 ,?/ = 27 ,s = 25 .The first equation then gives .](TV. 26. CCT/ + CC = u 3 ,xy + y-= v*.UV. 27. aôj = u 3 ,xyy = v 3 .tIV. 28. xy + (x + y)= u 3 ,xy~(x + y)= v 3 .[05+ j/= ^ (u3v 3 ), xy = ^ (u 3 + v 3 ) ; thereforewhich latter expression has to be made a square.

INDETERMINATE ANALYSIS 505Diophantus assumes u = + 1, v =1, whencemust be a* square, ortherefore 32 3 = 36 2 ,and = f. Thus u, v are found,and then x, y.The second (alternative) solution uses the formula thatt(?-t) + (?-& + = a cube. Put x = y = 2 -and one condition is satisfied. We then only have tomake (*_) --(^-) or 3-2g 2 a cube (less thanIV. 38. (( + 2/ + s) 5 = w 3 ,[# + y + s =[Suppose a? + 2/+ z = 525^-^, ,--;therefore4= -J-u(i6 + 1) + v~ + w:l.Diopliantus puts 8 for up, but we may take any cube, asm 3 ;and he assumes t/ 2 = (^2substitute ('2w22)'. We2for whichI)',we mightthen have the triangularnumber %u(u+ 1) = 2?i2 2 u* m 3 . Since 8 times atriangular number plus 1gives a square,16/t 2 2 8?i 4 -8m 3 + l = a square= (4?? -A;)'2and the problem is solved.], say,V. 15.[Let x + y + z = ^,u 3 = m 3 n â = n 3 ,^3 ,w = 3 3^3ptherefore = { (m 3 -1) + (7i3 - 1) + (y>3 - 1) }3

506 DIOPHANTUS OF ALEXANDRIAV. 16.9k+ 14 = a square= (3& Z)2,say ; and k is found.Retracing our steps, we find andtherefore x, y, z.]V. 17. x-V. 18.):J-f z =V. 19.[Put5 =2(r -l)g fl 2that p*1 + q,whence 2 = (p2 -! +r/ 2 -l +r a -l), so1 + T* 1 must be made a fourthpower. Diophantus assumes 2 2= ^> (mr = 2 2(7>i I)1 )2, q 2 = (m + 1 )2,,since m 4 2m 2 + m 2 + 2m + m 2 2m=m 4 .]V. I9a. a; + 2/+ = i2,^-V. 19. 1), c. x + y + z = a, (a + y + $f # =(./+ ?y+ 0)3+ y = w 2 , (,TV. 20.[IV. 8]. a;~?y = 1, a :i -?/ 3 =[IV. 9, 10]. B 8 +7/(^2[IV. 11].>_ = ^>-2/).[V. 15]. cc3 + 2/3 + s 3 -3 =u 2 .[V. 1 6]. 3 - (*3+ y 3 + z 3 ) = u2 .[V. 17].

INDETERMINATE ANALYSIS 507(ix)Indeterminate analysis of the fourth degree.V. 29. o[' Why ',seek tivosays Format, ' did not Diophantus fourth powers such that their sum is a square. Thisproblem is, in fact, impossible, as by my method I amable to prove with all rigour.' No doubt Diophantusknew this truth empirically. Let x* = 2 ,2/2 = p*,z* = q*.Therefore 4 + p* + (f= a square= (2r) 2 , say ;therefore g 2 = (r2this expression a square.p* q*)/Zr,and we have to makeDiophantus puts r = p2 + = 4, q* 4, so that the expressionreduces to Sp^/(2jr + or 8) *p*/(fP + 4).To make2tliis a square, let .//- + 4 = (jj + I) , say therefore p = l,;2and p*= 2i, g= 4, r=6; or (multiplying by 4)jp2= 9, j2= 16, r = 25, which solves the problem.][V. 18].(See above under V. 18.)(x)Problems of constructing right-angled triangles witlisides in rational numbers and satisfying variousother conditions.[I shall in all cases call the hypotenuse z,and theother two sides a 1 , y y so that the condition a? 2 + y*= 2appliesspecified.]in all cases,[Lemma to V. 7]. xy = ayj^ = # 2 ?/2 .VI. 1. z-x = u*> z-y = v\I. 2.in addition to the other conditions[Form a right-angled triangle from g, m, so thatz = 2 + w 2 ,x = 2m, ?/= 2 w 2 ;thus z y = 2m2 ,and, as this must be a cube, we put in = 2 ;therefore# = * 4 + 4 must be a cube, or 2=a cube,say /t:J ,and = W + 2.]

508 DIOPHANTUS OF ALEXANDRIAVI. 3.[Suppose the required triangle to be hg,pg,bg; therefore%pbg* a = 4- a = square 2 2 , say,and the ratio of ato ?i 2To find n, p, b so as to satisfy this condition, form%pb must be the ratio of a square to a square.a right-angled triangle from m,>VI. 4.1 / 2a\-therefore -J6 = m 2 5 Assume n- =(m+ -f ))l,~ \therefore n% %vb = 4 a + .,m 2; and (4 < t +')U - \aor 4a 2 + - o -9 has to be made a square. Put4a 2 'm 2 + a(4a2 + 1)= (2am + fc)2,and we have a solution.Diophantus has a = 5, leading to 100m2 + 505 = a square= (10m + 5)2,say, which gives m = 2 /- and n = ^ 3-./*, y, 6 are thus determined in such a way that%pbg* + a = ^2 2 gives a rational solution.]VI. 5. tt-i#2/ = , />^,so that= a >and for a rational solution of this equationwe must Rave (%p)2+ a(%pty a square.Diophantusassumes p = 1, ?> = m, whence âm + or 2am-fl= a square.But, since the triangle is rational, tow 2 + I= a square.That is, we have a double equation. Difference= m 2 2am = m(m 2a). Put2am + 1 =2{i(m-m~2a) }= a 2 , and>i = -(a 2 l)/2a.The sides of the auxiliary triangle are thus determinedin such a way that the original equation in is solvedrationally.]VI. 7. \m-x = a.

INDETERMINATE ANALYSIS 509rVI. 8.IVI. 9.%xy-(x + y)-a.[With the same assumptions we have in these casesto make {%('p + fy}* + a(%pb) a square. Diophantusassumes as before 1 ,m for the values of p, b, and obtainsthe double equationi (m +2 1 )+ Jam = square] r ?m 2 + l= square){VI.11.m 2 + (2 a + 2)m + 1= square)orfm 92 + 1 = square)solving in the usual way.]10.[In these cases the auxiliary right-angled triangle hasto be found such that= a square.Diophantus assumes it formed from 1, m + 1 ;thusJ(fc+p)= 2 i [ma + 2m + 2 + m 2 + 2m} 2 &) = a (m + 1 ) (?>i2 + 2 ??i).thereforew 4 + (a + 4)m3 + (See + 6)m2 + (2a + 4)m + 1= a squareand 7?iis found.]Lemma 1 to VI. 12. x =r VI. 12. %xy + x = u*I VI. 13. %xyx = tt2,%xyy= v2 .[These problems and the two following are interesting,but their solutions run to some length; therefore onlyone case can here be given. We will take VI. 1 2 withits Lemma 1 .

510 DIOPHANTUS OF ALEXANDRIALemma 1.If a rational right-angled triangle be formedfrom m, n, the perpendicular sides are 2mu, m 2 n 2 .of the two to be 2mn.We will suppose the greaterThe first two relations are satisfied by making m = 2 n.Form, therefore, a triangle from,2.The third conditionthen gives 6 4 + 3 2 = a square or 6 2 + 3 = athere are an infinitesquare. One solution is = 1(andnumber of others to be found by means of If =it). 1,the triangle is formed from 1,2.VI. 12. Suppose the triangle to be (&, b,p). Then(%pb)+p = & square = 2(A; ), say, and g=p/(l

INDETERMINATE ANALYSIS 511[The auxiliary right-angled trianglein this case mustbe such thatm2/ijp %pb.p(h p) is a square.If, says Diophantus (VI. 14), we form a triangle fromthe numbers X 19 X% and suppose that p = 2 X lX 2,and ifwe then divide out 2by (X Xl 2)which is ,equal to h p,such thatwe must find a square/o 2 [= m*/(Xl X 2 )2]Whp %pb . p is a square.The problem, says Diophantus, can be solved if X ltX 2m'2are ' similar plane numbers ' (numbers such as aft,-aft).This is stated without proof, but it can easily be verifiedthat, if A;2 = XlX< 2,the expressionis a square. Dioph.takes 4, 1 as the numbers, so that & 2 = 4. The equationfor m becomes8 . 1 7m 2 4 . 1 5 . 8 . 9 = a square,or136m 2 -4 320 = a square.The solution m = 2 36 (derived from the fact thatP - m 2 /(î- Ar 2

512 DIOPHANTUS OF ALEXANDRIAso that AC = 4 (n-).Therefore (Eucl. I. 47)so that = 7 ft 2 / 32 u = ^\?i. [Dioph. has n = 1.]VI. 17.[Let be the area and =#2/, let & 2 . SinceTherefore 2 + 2 fc musta'^/ = 2, supposea? = 2, T/= .be a cube. As we have seen (p. 475), Diophantustakes (m 3 I) for the cube and (m-h 2 I) for & 2 , givingm 8 3m 2 + 3m l=m 2 + 2m + 3, whence m=4. Thereforek = 5, and we assume %xy = ^, s; = 25,with,r = 2, ?/= I as before. Then we have to make. VI. 18. %wj + z = w\ x + y + z = v*.VI. 19. %xy + x = u 2 ,[Here a right-angled triangle is formed from one oddnumber, say 2 + l, according to the Pythagorean formulam 2 2+ {|(7>iwhere m is an,=I)} 2 2{^(wi2 + l)}odd number. The sides are therefore 2+l, 2 2 +2,2 2 + 2 + 1 . Since the perimeter= a cube,%.l= a cube.Or, if we divide the sides by +1, 4 + 2 has to bemade a cube.Again \xy + x = -^2^s + -f= a sijuarc,which reduces to 2 + 1 = a square.But 4 + 2 is a cube. We therefore put 8 for the cube,and =1*.]VI. 20. %xy + x u s , x+y+ z = v2.VI. 21.[Form a right-angled triangle from , 1, i.e. (2, 2 1,+ 1).Then 2^2+ 2 must be a square, and 3 + 2

INDETERMINATE ANALYSIS 513a cube. Put 2 2 + 2 = m 2 2 ,so that = 2/(m2~2),and we have to make8 8 2 2m, 4Make 2m a cube =3 ?&,so that 2m 4 = m 3 7, 3 ,andgmzr^w 3 -therefore = --; * and must be maden 8greater than 1, in order that 2 1may be positive.Therefore 8 < < 16 ;this is satisfied by M = 3ffi or w j=%7 -, and m = |.]VI. 22.[(1) First seek a rational right-angled triangle suchthat its perimeter and its area are given numbers,say p, m.Let the perpendiculars be --, 2???^; therefore the hypotenuse= p-3mf, and (Eucl. I. 47)or- - = + 4mj2+ 4m = 4mpg + -~that is, ( />2 -f 4 771) ^ = 4 r>^> ^ 2 + 2^.(2) In order that this may have a rational solution,( \ (if + 4m) ]28p 2 7>i must be a square,i.e. 4m 2 6 > 2 m + jp4= a square,or m 2 f p' 2 m + T^p 4 = a squareÂlso, by the second condition, m+p = a square)To solve this, we must take for p some number whichis both a square and a cube (in order that itmay bepossible, by multiplying the second equation by somesquare, to make the constant term equal to the constant1583.2 L 1

514 DIOPHANTUS OF ALEXANDRIAterm in thq first). Diophantus takes p = 64, makingthe equationsm 26 144m + 1048576 = a square]m + 64 = a square)Multiplying the second by 16384, and subtracting the twoexpressions, we have as the difference m 2 22528m.Diophantus observes that, if we take m, m 22528 asthe factors, we obtain m = 7680, an impossible value forthe area of a right-angled triangle of perimeter p = 64.We therefore take as factors llm, -j^m 2048, and,equating the square of half the difference ( = ffm + 1024)to 16384m + 1048576, we have ?>i = a |j|4 .(3) Returning to the original problem,substitute this value for m inwe have toand we obtain= 0,VI. 23.the solution of which is rational, namely= -fife (orDiophantus naturally takes the first value, though thesecond gives the same triangle.]2=VI. 24. z = u* + u, x v* v, y = iu\[VI. 6, 7]. ($x[VI. 8, 9]. {$([VI. 10, 11]. {[VI. 12.] y + (a-y).%xy= u2 ,x = v*. (x > y.)[VI. 14, 16]. u*zx-%ay .x(z-x) = v 2 .(u 2 < or >^ treatise on Polygonal Numbers.The subject of Polygonal Numbers on which Diophantusalso wrote is, as we have seen, an old one, going back to the

THE TREATISE ON POLYGONAL NUMBERS 515Pythagoreans, while Philippus of Opus and Speusippus carriedon the tradition. Hypsicles (about 170 B.C.) is twice mentionedby Diophantus as the author of a ' definition ' ofa polygonal number which, althoughit does not in termsmention any polygonal number beyond the pentagonal,amounts to saying that the nth a-gon (1 counting as thetirst) is M {2 + (n-l)(a-.2)j.Theon of Smyrna, Nicomachus and lamblichus all devotesome space to polygonal numbers. Nicomachus in particulargives various rules for transforming triangles into squares,squares into pentagons, &c.1. If we put two consecutive triangles together, we get a square.In fact2. A pentagonis obtained from a square by adding to ita triangle the side of which is 1 less than that of the square ;similarly a hexagon from a pentagon by adding a trianglethe side of which is 1 loss than that of the pentagon, and so on.In fact3. Nicomachus sets out the first triangles, squares, pentagons,hexagons and heptagons in a diagram thus :Triangles13 6 10 15 21 28 36 15 55,SquaresI 4 1) 16 25 36 19 64 81 100,Pentagons1 5 12 22 35 51 70 92 117 145,Hexagons1 G 15 28 45 66 91 120 153 190,Heptagons1 7 18 34 55 81 112 148 189 235,and observes that :Each polygon is equal to the polygon immediately above itin the diagram plus the triangle with 1 less in its side, i. e. thetriangle in the preceding column.Ll2

516 DIOPHANTUS OF ALEXANDRIA4. The vertical columns are in arithmetical progression, thecommon difference being the triangle in the preceding column.Plutarch, a contemporary of Nicomachus, mentions anothermethod of transforming triangles into squares. Every triangularnumber taken eight times and then increased by 1gives a square.In fact, 8.%n(n+l) + l = (2w+l)a.Only a fragment of Diophantus's treatise On PolygonalNumbers survives. Its character is entirely different fromthat of the Arithmetica. The method of proof is strictlygeometrical, and has the disadvantage, therefore, of being longand involved. He begins with some preliminary propositionsof which two may be mentioned. Prop. 3 proves that, if a bethe first and I the last term in an arithmetical progressionof n terms, and if s is the sum of the terms, 2s = n(l + a).Prop. 4 proves that, if 1, 1+6, 1 + 26, ... 1+(71 1)6 be anA. P., and s the sum of the terms,

THE TREATISE ON POLYGONAL NUMBERS 517The last proposition, which breaks off in the middle, is :Given a number, to find in how many wayspolygonal.it can beThe proposition begins in a way which suggests thatDiophantus first proved geometrically that, ifthen-4) 2 = {2 + (2n- 1) (a-2)] 2 ,2P = n{2+ (n -l)(a-2)}.Wertheim (in his edition of Diophantus) has suggested arestoration of the complete proof of this proposition, andI have shown (in my edition) how the proof can be madeshorter. Wertheim adds an investigation of the main problem,but no doubt opinions will continue to differ as towhether Diophantus actually solved it.

XXICOMMENTATORS AND BYZANTINESWE have come to the last stage of Greek mathematics it;only remains to include in a last chapter references to commentatorsof more or less note who contributed nothingoriginal but have preserved, among observations and explanationsobvious or trivial from a mathematical point of view,valuable extracts from works which have perished, orhistorical allusions which, in the absence of original documents,are precious in proportion to their rarity. Nor mustit be forgotten that in several cases we probably owe to thecommentators the fact that the masterpieces of the greatmathematicians havet survived, wholly or partly, in theoriginal Greek or at all. This may have been the case evenwith the works of Archimedes on which Eutocius wrote commentaries.It was no doubt these commentaries whicharoused in the school of Isidorus of Miletus (the colleagueof Anthemius as architect of Saint Sophia at Constantinople)a new interest in the works of Archimedes and caused themto be sought out in the various libraries or wherever they hadlain hid. This revived interest apparently had the effect ofevoking new versions of the famous works commented uponin a form more convenient for the student, with the Doricdialect of the original eliminated; this translation of theDoric into] the more familiar dialect was systematicallycarried out in those books only which Eutocius commentedon, and it is these versions which alone survive. Again,Eutocius's commentary on Apollonius's Conies is extant forthe first four Books, and it is probably owing to their havingbeen commented on by Eutocius, as well as to their beingmore elementary than the rest, that these four Books alone

SERENUS 519survive in Greek. Tannery, as we have seen, conjecturedthat, in like manner, the first six of the thirteen Books ofDiophantus's Arithmetica survive because Hypatia wrotecommentaries on these Books only and did not reach theothers.The first writer who calls for notice in this chapteris onewho was rather more than a commentator in so far as hewrote a couple of treatises to supplement the Conies ofI mean SERENUS. Serenus came from AntinoeiaApollonius,or Antinoupolis, a city in Egypt founded by Hadrian (A. D.117-38). His date is uncertain, but he most probably belongedto the fourth century A.D., and came between Pappusand Theon of Alexandria. He tells us himself that he wrotea commentary on the Conies of Apollonius. 1 This hasperished and, apart from a certain proposition 'of Serenusthe philosopher, from the Lemmas 'preserved in certain manuscriptsof Theon of Smyrna (to the effect that, if a number ofrectilineal angles be subtended at a point on a diameter of acircle which is not the centre, by equal arcs of that circle, theangle nearer to the centre is always less than the angle moreremote), we have only the two small treatises by him entitledOn the Section of a Cylinder and On the Section of a Cone.These works came to be connected, from the seventh centuryonwards, with the Conies of Apollonius, on account of theaffinity of the subjects,and this no doubt accounts for theirsurvival.They were translated into Latin by Commandinusin 1566 ;the first Greek text was brought'out by Halley alongwith his Apollonius (Oxford 1710), and we now have thedefinitive text edited by Heiberg (Teubner 1896).(a)On the Section of a Cylinder.The occasion and the object of the tract On the Section ofa Cylinder are stated in the preface. Serenus observes thatwere under themany persons who were students of geometrysrroneous impression that the oblique section of a cylinderwas different from the oblique section of a cone known as anwhereas it is of course the same curve. Hence heellipse,thinks it necessary to establish, by a regular geometrical1Serenus, Opitscula, ed. Heiberg, p. 52. 25-6.

is520 COMMENTATORS AND BYZANTINESproof, that the said oblique sections cutting all the generatorsare equally ellipseswhether they are sections of a cylinder orof a cone. He begins with ' a more general definition ' of a'If in twocylinder to include any oblique circular cylinder.equal and parallel circles which remain fixed the diameters,while remaining parallel to one another throughout, are movedround in the planes of the circles about the centres, whichremain fixed, and if they carry round with them the straight linejoining their extremities on the same side until they bring itback again to the same place, let the surface described by thestraight line so carried round be called a cylindrical surface!The cylinderis the figure contained by the parallel circles andthe cylindrical surface intercepted by them; the parallelcircles are the bases, the axis . the straight line drawnthrough their centres; the generating straight line in anyposition is a side. Thirty-three propositions follow. Of theseProp. 6 proves the existence in an oblique cylinder of theparallel circular sections subcontrary to the series of whichthe bases are two, Prop. 9 that the section by any plane notparallel to that of the bases or of one of the subcontrarysections but cutting all the generatorsis not a circle ;thenext propositions lead up to the main results, namely those inProps. 14 and 16, where the said section is proved to have theproperty of the ellipse which we write in the formand in Prop. 17, where the property is put in the Apollonianform involving the latus rectum, QV = PV 2 . VR (see figureon p.137 above), which is expressed by saying that the squareon the semi-ordinate is equal to the rectangle applied to thelatus rectum PL, having the abscissa PFas breadth and fallingshort by a rectangle similar to the rectangle contained by thediameter PP' and the latus rectum PL (whichis determinedby the condition PL . PP'= DIY* and is drawn at right anglesto PF). Prop. 18 proves the corresponding property withreference to the conjugate diameter DD' and the correspondinglatus rectum, and Prop. 19 gives the main property in theform QF 2 : PF . P / F =2Q'F': PF 7 . P*V. Then comes theproposition that 'it is possible to exhibit a cone and a cylinder'which are alike cut in one and the same ellipse (Prop. 20).

SERENUS 521Serenus then solves such problems as these: Given a cone(or cylinder) and an ellipse on it, to find the cylinder (cone)which is cut in the same ellipse as the cone (cylinder)(Props. 21, 22); given a cone (cylinder), to find a cylinder(cone) and to cut both by one and the same plane so that thesections thus made shall be similar ellipses (Props. 23, 24).Props. 27, 28 deal with similar elliptic sections of a scalenecylinder and cone there are two;pairs of infinite sets of thesesimilar to any one given section, the first pair being thosewhich are parallel and subcontrary respectively to the givensection, the other pair subcontrary to one another but not toeither of the other sets and having the conjugate diameteroccupying the corresponding placeother sets, and vice versa.In the propositions (29-33) from this pointto the transverse in theto the end ofthe book Serenus deals with what is really an optical problem.It is introduced by a remark about a certain geometer,Peithon by name, who wrote a tract on the subject ofparallels. Peithon, not being satisfied with Euclid's treatmentof parallels, thought to define parallels by means of anillustration, observing that parallels are such lines as areshown on a wall or a roof by the shadow of a pillar witha light behind it. This definition, it appears, was generallyridiculed ;and Serenus seeks to rehabilitate Peithon, whowas his friend, by showing that his statement is after allmathematically sound. He therefore proves, with regard tothe cylinder, that, if any number of rays from a point outsidethe cylinder are drawn touchingit on both sides, all the rayspass through the sides of a parallelogram (a section of thecylinder parallel to the axis) Prop. 29 and if they areproduced farther to meet any other plane parallel to thatof the parallelogram the points in which they meet the planewill lie on two parallel lines (Prop. 30) ;he adds that the lineswill not seem parallel (vide Euclid's Optics, Prop. 6).Theproblem about the rays touching the surface of a cylindersuggests the similar one about any number of rays from anexternal point touching the surface of a cone ;these meet thesurface in points on a triangular section of the cone (Prop. 32)and, if produced to meet a plane parallel to that of thetriangle, meet that plane in points forming a similar triangle

522 COMMENTATORS AND BYZANTINES(Prop. 33). Prop. 31 preceding these propositions is a particularcase of the constancy of the anharmonic ratio of apencil of four rays.If two sides AB, AC of a triangle meeta transversal through D, an external point, in E, .Fantl anotherray AG between AB and AC cuts DEF in a point G suchthat ED : DF = EG :GF, then any other transversal through1) meeting AB, AG, AC in K, L, M is also divided harmoni-i.e. KD DM = KL : LM.:cally, To prove the succeeding propositions,32 and 33, Serenus uses this proposition and areciprocal of it combined with the harmonic property of thepole and polar with reference to an ellipse.(/3)On the Section of a Gone.The treatise On the Section of a Cone is even less important,although Serenus claims originality for it. It deals mainlywith the areas of triangular sections of right or scalene conesmade by planes passing through the vertex and either throughthe axis or not through the axis, showing when the area ofa certain triangle of a particular class is a maximum, underwhat conditions two triangles of a class may be equal in area,and so on, and solving in some easy cases the problem offinding triangular sections of given area.This sort of investigationoccupies Props. 1-57 of the work, these propositionsincluding various lemmas required for the proofs of thesubstantive theorems. Props. 58-69 constitute a separatesection of the book dealing with the volumes of right conesin relation to their heights, their bases and the areas of thetriangular sections through the axis.The essence of the first portion of the book up to Prop. 57is best shown bymeans of modern notation. We will call hthe height of a right cone, r the radius of the base ;in thecase of an oblique cone, let p be the perpendicular from thevertex to the plane of the base, d the distance of the foot ofthis perpendicular from the centre of the base, r the radiusof the base.Consider firstthe right cone, and let 2x be the base of anytriangular section through the vertex, while of course 2r isthe base of the triangular section through the axis. Then, ifA be the area of the triangular section with base 2x,

SERENUS 523Observing that the sum of x* and ra x 2 + IP is constant, wesee that A 2 and,therefore A, is a maximum when= r*-a; + A*, or a* = (r8 + A 8 ) ;and, since a? is not greater than r, it follows that, for a realvalue of x (other than r),h is less than r, or the cone is obtuseangled.When h is not less than r, the maximum triangle isthe triangle through the axis and vice versa (Props. 5, 8) ;when h = r, the maximum triangle is also right-angled(Prop. 13).If the triangle with base 2 c is equal to the triangle throughthe axis, A a r* = c2(ra -c8 + A a ),or (ra -c 2 ) (c 2 -/* 2 ) = 0, and,since c A 2 r 2 ,and the triangle with base 2x is greater than either of theequal triangles with bases 2r, 2c, or 2h (Prop. 11).In the case of the scalene cone Sereiius compares individualtriangular sections belonging to one of three classes with othersections of the same class as regards their area. The classesare :(1) axial triangles, including all sections through the axis;(2) isosceles sections,perpendicular to the projectionplane of the base ;i.e. the sections the bases of which areof the axis of the cone on the(3) a set of triangular sections the bases of which are (a) thediameter of the circular base which passes through the foot ofthe perpendicular from the vertex to the plane of the base, and(b)the chords of the circular base parallel to that diameter.After two preliminary propositions (15, 16) and somelemmas, Serenus comparesthe areas of the first class oftriangles through the axis. If, as we said, p is the perpendicularfrom the vertex to the plane of the base, d the distanceof the foot of this perpendicular from the centre of the base,with areaand the angle which the base of any axial triangleA makes with the base of the axial triangle passing throughp the perpendicular,A =This area is a minimum when 6 = 0, and increases with

ases524 COMMENTATORS AND BYZANTINESuntil 6 = f TT when it is a maximum, the triangle being thenisosceles (Prop. 24).In Prop. 29 Serenus takes up the third class of sections withbases parallel to d. If the base of such a section is 2x,A = aVr 2 -and, as in the case ofmaximum valuethe right cone, we must have for a realoj 2 = ^ (r2+p*), while x < r,so that, for a real value of x other than r, p must be less thanr, and, ifp is not less than r, the maximum triangle is thatwhich isperpendicular to the base of the cone and has 2r forits base (Prop. 29). If p

SERENUS 525the base of the cone to the projection of the vertex on itsplane ;the areas of the axial triangles are therefore proportionalto the generators of the cone with the said circle asbase and the same vertex as the original cone. Prop. 50 is tothe effect that, if the axis of the cone is equal to the radius ofthe base, the least axial triangle is a mean proportionalbetween the greatest axial triangle and the isosceles triangularsection perpendicular to the base that;is, with the above notation,if r = V(p* + tV), then r V(f + d 2 ) :rp = rp:p b ^ c> rf,then a + d< b + c (a, d are the sides other than the base of oneaxial triangle,and 6, c those of the other axialtriangle comparedwith it; and if ABC, ADEbe two axial triangles andthe centre of the base, BA* + AC*= DA 2 + AE* because eachof these sums is equal to 2 A O 2 + 2 BO 2 ,Prop. 1 7).This propositionagain depends on the lemma (Props. 52, 53) that, ifstraight lines bo 'inflected' from the ends of the base ofa segment of a circle to the curve (i. e. if we join the endsthe sum ofof the base to any point on the curve) the lino (i.e.the chords) is greatest when the point taken is the middlepoint of the arc, and diminishes as the point is taken fartherand farther from that point.Let B be the middle point of thearc of the segment ABC, 7), E anyother points on the curve towards(7; I say thatBC>AD+I)C>AE+EC.With B as centre and BA as radiusdescribe a circle, and produce AB,AD, AE to meet this circle in F, G,H. JoinFC,GC,HC.Since AB = EG = BF, we have AF = AB + BC.Also theangles BFC, BCF arc equal, and each of them is half ofthe angle ABC.

526 COMMENTATORS AND BYZANTINESAgain LAGG = LAFG = \LAEG = \LADG\therefore the angles DOC, DOG are equal and DG =thereforeSimilarlyAG = AD + DC.EH = KG and ^7/ = AE+ EC.But, by Eucl. III. 7 or 15, AF>AG>AIf, and so on ;thereforeAB + BC > A D + DC > AE + EC, and so on.In the particular case where the segment ABC isa semicircleProp.AB* + BC 2 = AG* = AD* + DC*, &c., and the result of57 follows.Props. 58-69 are propositions of this sort : In equal rightcones the triangular sections through th$ axis are reciprocallyproportional to their bases and conversely (Props. 58, 59) ;right cones of equal height have to one another the ratioduplicate of that of their axial triangles (Prop. 62); rightcones which are reciprocally proportional to their bases haveaxial triangles which are to one another reciprocally in thetriplicate ratio of their bases and conversely (Props. 66, 67);and so on.THEON OF ALEXANDRIA lived towards the end of the fourthcentury A.D.Suidas places him in the reign of Theodosius I(379-95); he tells us himself that he observed a solar eclipseat Alexandria in the year 365, and his notes on the chronologicaltables of Ptolemy extend down to 372.Commentary on the Syntaxis.We have already seen him as the author of a commentaryon Ptolemy's Syntaxis in eleven Books. This commentary isnot calculated to give us a very high opinion of Theon'smathematical calibre, but it is valuable for several historicaland we are indebted to it for a usefulnotices that it gives,account of the Greek method of operating with sexagesimalfractions, which is illustrated by examples of multiplication,division, .and the extraction of the square root of a non-squarenumber by way of approximation. These illustrations ofnumerical calculation have already been given above (vol. i,

THEONOF ALEXANDRIA 527pp. 58-63). Of the historical notices we may mention thefollowing. (1) Theon mentions the treatise of Menelaus OnChords in a Circle, i.e. Menelau&'s Table of Chords, which camebetween the similar Tables of Hipparchus and Ptolemy. (2) Aquotation from Diophantus furiiishes incidentally a lower limitfor the date of the Arithmetica. (3) It is in the commentaryon Ptolemy that Thcon tells UH that the second part of EuclidVI. 33 relating to sectors in equal circles was inserted by himselfin his edition of the Elements, a notice which is of capitalimportance in that it enables the Theonine manuscripts ofEuclid to be distinguished from the ante-Theonine, and istherefore the key to the question how far the genuine textof Euclid was altered in Theon's edition. (4) As we haveseen (pp. 207 sq.), Theon, & propos of an allusion of Ptolemyto the theory of isoperimetric figures, has preserved for usseveral propositions from the treatise by Zenodorus on thatsubject.Theon's edition of Euclid's Elements.Wo are able to judge of the character of Theon's edition ofEuclid by a comparison between the Theonine manuscriptsand the famous Vatican MS. 190, which contains an earlieredition than Theon's, together with certain fragments ofancient papyri. It appears that, while Theon took sometrouble to follow older manuscripts,it was not so much histhe most authoritative text as to make what heobject to get considered improvements of one sort or other, (l)He madealterations where he found, or thought he found, mistakes inthe original; while he tried to remove some real blots, healtered other passages too hastily when a little more considerationwould have shown that Euclid's words are right or couldbe excused, and offer no difficulty to an intelligent reader.(2) He made emendations intended to improve the form orin general they were prompted by a desirediction of Euclid ;to eliminate anything which was out of the common in expressionor in form, in order to reduce the language to one and thesame standard or norm. (3)He bestowed, however, mostattention upon additions designed to supplement or explainthe original ; (a) he interpolated whole propositions where hethought them necessary or useful, e.g. the addition to VI. 33

528 COMMENTATORS AND BYZANTINESalready referred to, a second case to VI. 27, a porism or corollaryto II. 4, a second porism to III. 16, the proposition VII. 22,a lemma after X. 12, besides alternative proofs here and there ;(6) he added words for the purpose of making smoother andclearer, or more precise, things which Euclid had expressedwith unusual brevity, harshness, or carelessness he ; (c) suppliedintermediate steps where Euclid's argument seemed toodifficult to follow. In short, while making only inconsiderableadditions to the content of the Elements, he endeavouredto remove difficulties that might be felt by learners in studyingthe book, as a modern editor might do in editing a classicaltext-book for use in schools ;and there is no doubt that hisedition was approved by his pupils at Alexandria for whom itwas written, as well as by later Greeks, who used it almostexclusively, with the result that the more ancient text is onlypreserved complete in one manuscript.Edition of the Optics of Euclid.In addition to the Elements, Theon edited the Optics ofEuclid; Theon's recension as well as the genuine work isincluded by Heiberg in his edition. It is possible that theCatoptrica included by Heiberg in the same volume is also byTheon.4Next to Theon should be mentioned his daughter HYPATIA,who is mentioned by Theon himself as having assisted in therevision of the commentary on Ptolemy. This learned ladyissaid to have been mistress of the whole of pagan science,especially of philosophy and medicine, and by her eloquenceand authority to have attained such influence that Christianityconsidered itself threatened, and she was put to death bya fanatical mob in March 415. According to Suidas she wrotecommentaries on Diopharitus, on the Astronomical Canon (ofPtolemy) and on the Conies of 'Apollonius. These workshave not survived, but it has been conjectured (by Tannery)that the remarks of Psellus (eleventh century) at the beginningof his letter about Diophantus, Anatolius, and theEgyptian method of arithmetical reckoning were taken bodilyfrom some manuscript of Diophantus containing an ancientand systematic commentary which may very well have beenthat of Hypatia. Possibly her commentary may have extended

HYPATIA. PORPHYRY 529only to the first six Books, in which case the fact that Hypatiawrote a commentary on them may account for the survival ofthese Books while the rest of the thirteen were first forgottenand .then lost.It will be convenient to take next the series of Neo-Platonist commentators. It does not appear that AmmoniusSaccas (about A.D. 175-250), the founder of Neo-Platonism, orhis pupil Plotinus (A.D. 204-69), who first expounded thedoctrines in systematic form, had any special connexion withmathematics, but PORPHYRY (about 232-304), the disciple ofPlotinus and the reviser and editor of his works, appears tohave written a commentary on the Elements. This we gatherfrom Proclus, who quotes from Porphyry comments on Eucl.I. 14 and 26 and alternative proofs of I. 18, 20. It is possiblethat Porphyry's work may have been used later by Pappus inwriting his own commentary, and Proclus may have got hisreferences from Pappus, but the form of these references sug-that he had direct access to the original commentary ofgestsPorphyry.IAMBLICHUS (died about A.D. 330) was the author of a commentaryon the Introductio arithmetica of Nicomachus, andof other works which have already been mentioned. He wasa pupil of Porphyry as well as of Anatolius, also a disciple ofPorphyry.But the most important of the Neo-Platonists to the historianof mathematics is PROCLUS (A.D. 410-85). Proclusreceived his early training at Alexandria, where Olympiodoruswas his instructor in the works of Aristotle, andmathematics was taught him by one Heron (of course adifferent Heron from the 'mechauicus Hero' of the Metrica,&c.).He afterwards went to Athens, where he learnt theNeo-Platonic philosophyfrom Plutarch, the grandson of Nestorius,and from his pupil Syrianus, and became one of itsmost prominent exponents.He speaks everywhere with thehighest respect of his masters, and was in turn regarded withextravagant veneration by his contemporaries, as we learnfrom Marinus, his pupil and biographer.On the death ofSyrianus he was put at the head of the Neo-Platonic school.He was a man of untiring industry, as is shown by the

530 COMMENTATORS AND BYZANTINESnumber of books which he wrote, including a largenumber ofcommentaries, mostly on the dialogues of Plato (e.g. theTimaeus, the Republic, the Parme aides, the Gratylus). Hewas an acute dialectician and pre-eminent among his contemporariesin the range of his learning; he was a competentmathematician ;he was even a poet. At the same time hewas a believer in all sorts of myths and mysteries, anda devout worshipper of .divinities both Greek and Oriental.He was much more a philosopher than a mathematician. Inhis commentary on the Timaeus, when referring to the questionwhether the sun occupies a middle place among theplanets, he speaks as no real mathematician could havespoken, rejecting the view of Hipparchus and Ptolemy because6 Qtovpyos (sc. the Chaldean, says Zeller) thinks otherwise,cwhom it is not lawful to disbelieve '. Martin observes too,rather neatly, that ' for Proclus the Elements of Euclid hadthe good fortune not to be contradicted either by the ChaldeanOracles or by the speculations of Pythagoreans old and new '.Commentary on Euclid, Book I.For us the most important work of Proclus is hiscommentaryon Euclid, Book I, because it is one of the main sourcesof our information as to the history of elementary geometry.Its great value arises mainly from the fact that Proclus hadaccess to a number of historical and critical works which arenow lost except for fragments preserved by Proclus andothers.(a) Sources of the Commentary.The historical work the loss of which is most deeply to bedeplored is the History of Geometry by Eudemus. Thereappears to be no reason to doubt that the work of Eudemuswas accessible to Proclus at first hand. For the later writersSimplicius and Eutocius refer to it in terms such as leave nodoubt that they had it before them. Simplicius, quotingEudemus as the best authority on Hippocrates's quadratures'of lunes, says he will set out what Eudemus says word forword ',adding only a little explanation in the shape of referencesto Euclid's Elements 'owing to the memorandum-likestyle of Eudemus, who sets out his explanations in the abbre-

PROCLUS 531viated form usual with ancient writers. Now in the secondbook of the history of geometry he writes as follows '.* Inlike manner Eutocius speaks of the paralogisms handed downin connexion with the attempts of Hippocrates and Antiphonto square the circle, 'with which I imagine that all personsare accurately acquainted who have examined (7reovc/i/*j>ouy)the geometrical history of Eudemus and know the GeriaAristotelica '. 2The references by Proclus to Eudemus by name are notindeed numerous ; they are five in number but on the other;hand he gives at least as many other historical data which canwith great probability be attributed to Eudemus.Proclus was even more indebted to Geminus, from whomhe borrows long extracts, often mentioning him by namethere are some eighteen such referencesbut often omittingto do so. We are able to form a tolerably certain judgementas to the origin of the latter class of passages on thestrength of the similarity of the subjects treated and the viewsexpressed to those found in the acknowledged extracts. Aswe have seen, the work of Geminus mainly cited seems tohave borne the title The Doctrine or Theory of the Mathematics,which was a very comprehensive work dealing, in a portion ofit, with the * classification of mathematics '.We have already discussed the question of the authorshipof the famous historical summary given by Proclus. It isdivided, as every one knows, into two distinct parts betweenwhich comes the remark, ' Those who compiled historiesbring the development of this science up to this point. Notmuch younger than these is Euclid, who ',&c. The ultimatesource at any rate of the early part of the summary mustpresumably have been the great work of Eudemus abovementioned.It is evident that Proclus had before him the original worksof Plato, Aristotle, Archimedes and Plotinus, the SvfipiKTa ofPorphyry and the works of his master Syrianus, as well as agroup of works representing the Pythagorean tradition on itsmystic, as distinct from its mathematical, side, from Philolausdownwards, and comprising the more or less apocryphal12Simplicius on Arist. Phys., p. 60. 28, Diels.Archimedes, ed. Heib., vol. iii, p. 228. 17-19.

532 COMMENTATORS AND BYZANTINESXoyoy of Pythagoras, the Oracles (\6yia) and Orphicverses.The following will be a convenient summary of the otherworks used by Proclus, and will at the same time give anindication of the historical value of his commentary onEuclid, Book I :Eudemus :Geminus :History of Geometry.The Theory of the Mathematical Sciences.Heron: Commentary on the Elements of Euclid.APorphyry:Pappus:Apollonius of Perga :geometry.work relating to elementaryPtolemy :Posidonius :Carpus: Astronomy.On the 'parallel-postulate.A book controverting Zeno of Sidon.Syrianus: A discussion on the angle.Character (ft) of the Commentary.We know that in the Neo-Platonic school the pupils learntmathematics ;and it is clear that Proclus taught this subject,and that this was the origin of his commentary. Manypassages show him as a master speaking to scholars in one;place he speaks of my ' hearers *.1 Further, the pupils whomhe was addressing were beginners in mathematics ;thus in onepassage he ''says that he omits for the present to speak of thediscoveries of those who employed the curves of Nicomedesand Hippias for trisecting an angle, and of those who used theArchimedean spiral for dividing an angle in a given ratio,because these things would be ' too difficult for 2beginners '.But there are signs that the cpmmentary was revised andre-edited for a larger public; he speaks for instance in oneplace of ' those who will corne across his work \ 3 There arealso passages, e.g. passages about the cylindrical helix, conchoidsand cissoids, which would not have been understood bythe beginners to whom he lectured.1Proclus on Eucl. I, p. 210. 19.3Ib., p. 84. 9.2/&., p. 272. 12.

PROCLUS 533The commentary opens with two Prologues. The first ison mathematics in general and its relation to, and use in,philosophy, from which Proclus passes to the classification ofmathematics. Prologue II deals with geometry generally andits subject-matter according to Plato, Aristotle and others.After this section comes the famous summary (pp. 64-8)ending with a eulogium of Euclid, witli particular referenceto the admirable discretion shown in the selection of the propositionswhich should constitute the Elements of geometry,the ordering of the whole subject-matter, the exactness andthe collusiveness of the demonstrations, and the power withwhich every questionis handled. Generalities follow, such asthe discussion of the nature of elements, the distinction betweentheorems and problems according to different authorities, andfinally a division of Book I into three main sections, (1) theconstruction and properties of triangles and their parts andthe comparison between triangles in respect of their anglesand sides, (2) the properties of parallels and parallelogramsand their construction from certain data, and (3)the bringingof triangles and parallelograms into relation as regards area.Coming to the Book itself, Proclus deals historically andcritically with all the definitions, postulates and axioms inorder. The notes on the postulates and axioms are precededby a general discussion of the- principles of geometry, hypotheses,postulates and axioms, and their relation to oneanother ;here as usual Proclus quotes the opinions of all theimportant authorities.discusses once more the difference between theorems andAgain, when he comes to Prop. 1, heproblems, then sets out and explains the formal divisions ofa proposition, the enunciation (Trporatny), the settiny-out(cAcflecny), the definition or specification (Siopurpos), the consti*uction(KaraaKtvri), the proof (a7ro#e*ty), the conclusionand finally a number of other technical terms,(crvfjLTTtpao-jjLa),e.g. things said to be given, in the various senses of this term,the lemma, the case, the porism in its two senses, the objection(ci/

534 COMMENTATORS AND BYZANTINEScases, mainly for the sake of practice, and thirdly he addresseshimself to refuting objections which cavillers had taken ormight take to particular propositions or arguments. He doesnot seem to have had any notion of correcting or improvingEuclid; only in one place does he propose anything of hisown to get over a difficulty which he finds in Euclid ;this iswhere he tries to prove the parallel-postulate, after givingPtolemy's attempt to prove it and pointing out objections toPtolemy's proof.The book is evidently almost entirely a compilation, though'a compilation in the better sense of the term '. The onusin it isprobandi is on any one who shall assert that anythingProclus's own ; very few things can with certainty be said tobe so. Instances are (1)remarks on certain things which hequotes from Pappus, since Pappus was the last of the commentatorswhose works he seems to have used, (2)a defenceof Geminus against Carpus, who criticized Geminus's view ofthe difference between theorems and problems, and perhaps(3) criticisms of certain attempts by Apollonius to improve onEuclid's proofs and constructions ;but the only substantialexample is (4) the attempted proof of the parallel-postulate,based on an *axiom ' to the effect that, * if from one point twostraight lines forming an angle be produced ad injinitum, thedistance between them when so produced ad infiuitum exceedsany finite magnitude (i. e. length) an ', assumption whichpurports to be the 1equivalent of a statement in Aristotle.Philoponus says that Proclus as well as Ptolemy wrote a whole2book on the parallel-postulate.It is still not quite certain whether Proclus continued hiscommentaries beyond Book I. He certainly intended to do so,for, speaking of the trisection of an angle by means of certainwe may perhaps more appropriately examineccurves, he says,these things on the third Book, where the writer of theElements bisects a given circumference', and again, aftersaying that of all parallelograms which have the same perimeterthe square'is the greatest and the rhomboid least offor it'all ',he adds, But this we will prove in another place,is more appropriate to the discussion of the hypotheses of the1De caelo, i. 5, 271 b 28-30.2Philoponus on Anal. Post. i.10, p. 214 a 9-12, Brandis.

PttOCLUS 535second Book 3 . But at the time when the commentary onBook I was written he was evidently uncertain whether he'would be able to continue it, for at the end he says, For mypart, if I should be able to discuss the other Books in the'same way, I should give thanks to the gods; but, if othercares should draw me away, 1 beg those who are attracted byof the other Books asthis subject to complete the expositionwell, following the same method and addressing themselvesthroughout to the deeper and more sharply defined questionsinvolved '. lWachsmuth, finding a Vatican manuscript containinga collection of scholia on Books I, II, V, VI, X, headed Eh raTOi\Ta Trpo\afjif$av6p.va K T&V HpoK\ov criTopdSvjvand seeing that the scholia on Book I wereKar* tTriTOfjLrjis,extracts from the extant commentary of Proclus, concludedthat those on the other Books were also from Proclus; butthe TT/DOin7r/>oXa/i/Jai>6/zj>a rather suggests that only thescholia to Book I are from Proclus. Heiberg found andpublished in 1903 a scholium to X. 9, in which Proclus isexpressly quoted as the authority, but he does not regardthis circumstance as conclusive. On the other hand, Heiberghas noted two facts which go against the view that Procluswrote on the later Books: (1) the scholiast's copy ofProclus was not much better than our manuscripts ;inparticular, it had the same lacunae in the notes to I. 36,37, and I. 41-3; this makes it improbable that the scholiasthad further commentaries of Proclus which have vanishedfor us ; (2) there is no trace in the scholia of the noteswhich Proclus promised in the passages already referred to.All, therefore, that we can say is that, while the Wachsmuthscholia may be extracts from Proclus, it is on the wholeimprobable.Hypotyposis of Astronomical HypothesesAnother extant work of Proclus which should be referredto is his Hypotyposis of Astronomical Hypotheses, a sort ofreadable and easy introduction to the astronomical systemof Hipparchus and Ptolemy. It has been well edited byMunitius (Teubner, 1909). Three things may be noted as1Proclus on Eucl. I, p. 432. 9-15.

536 COMMENTATORS AND BYZANTINESregards this work. It contains 1 a description of the methodof measuring the sun's apparent diameter by means ofHeron's water-clock, which, by comparison with the correspondingdescription in Theon's commentary to the Syntaxisof Ptolemy,is seen to have a common source with it. Thatsource is Pappus, and, inasmuch as Proclus has a figure (reproducedby Manitius in his text from one set of manuscripts)corresponding to the description, while the text of Theon hasno figure, it is clear that Proclus drew directly on Pappus,who doubtless gave, in his account of the procedure, a figuretaken from Heron's own work on water-clocks. A simpleproof of the equivalence of the epicycle and eccentric hypothesesis 2quoted by Proclus from one Hilarius of Antioch.An interesting passageis that in chap. 4 (p. 130, 18) whereSosigenes the Peripatetic is said to have recorded in his work'on reacting spheres' that an annular eclipse of the sun issometimes observed at times of perigee; this is, so far asI know, the only allusion in ancient times to annular eclipses,and Proclus himself questions the correctness of Sosigenes'sstatement.Commentary on the Republic.The commentary of Proclus on the Republic contains somepassages of great interest to the historian of mathematics.The most importantis that 3 in which Proclus indicates thatProps. 9, 10 of Euclid, Book II, are Pythagorean propositionsinvented for the purpose of proving geometrically thefundamental property of the series of side-' and ' diameter-'numbers, giving successive approximations to the value of\/2 (see vol. i, p. 93). The explanation 4 of the passage inPlato about the Geometrical Number is defective and disappointing,but it contains an interesting reference to onePaterius, of date presumably intermediate betwefen Nestoriusand Proclus. Paterius is said to have made a calculation, inunits and submultiples, of the lengths of different segments of1Proclus, Hypotyposis, c. 4, pp. 120-22.8Ib., c. 3, pp. 76, 17 sq.8Prodi Diadochi in Platonis EempuUicam Commentarii, ed. Kroll,vol. ii, p. 27.Ib., vol. ii, pp. 36-42.

PKOCLUS. MARINUS 537straight lines in a figure formed by taking a triangle withsides 3, 4, 6 as ABC, then drawingBD from the right angle B perpendicularto AC, and lastly drawingperpendiculars DE, DF to AB, BC.A diagram in the text with thelengths of the segments shown alongsidethem in the usual numericalnotation shows that Paterius obtained from the data AB = 3,BC = 4, CA = 5 the following:= -ye'= 3>This is an example of the Egyptian method of stating fractionspreceding by some three or four centuries the expositionof the same method in the papyrus of Akhmlm.MARINUS of Neapolis, the pupil and biographer of Proclus,wrote a commentary or rather introduction to the Data ofEuclid. 1 It ismainly taken up with a discussion of thequestion ri TO 5c5o/ici/o^, what is meant by given 1 Therewere apparently many different definitions of the term givenby earlier and later authorities. Of those who tried to defineit in the simplest way by means of a single differentia, threeare mentioned by name. Apollonius in his work on i/eu

538 COMMENTATORS AND BYZANTINEScombined two of these ideas and called it assigned OY fixedand procurable or capable of being found (rr6pip.oi>)\ otherscfixed and known ',and a third class * known and procurable'These various views are then discussed at length.DOMNINUS of Larissa, a pupil of Syrianusat the same timeas. Proclus, wrote a Manual of Introductory Arithmetic eyxeipi'StovdpiQftrjTiKfjs /o-aya>y^y, which was edited by Boissonade 1and is the subject of two articles by Tannery, 2 who also lefta translation of it, with prolegomena, which has since beenpublished. 3 It is a sketch of the elements of the theory ofnumbers, very concise and well arranged, and -is interestingbecause it indicates a serious attempt at a reaction against theIntroductio arithmetica of Nicomachus and $ return to thedoctrine of Euclid. Besides Euclid, Nicomachus and Theonof Smyrna, Domninus seems to have used another sourco,now lost, which was also drawn upon by lamblichus. At theend of this work Domninus foreshadows a more completetreatise on the theory of numbers under the. title Elements ofArithmetic (apiQ^riK^ oTcuxôxny), but whether this wasever written or not we do not know. Another tractattributed to Domninus 7ro>9 GCTTI \6yov K \6yov a0eAea>(how a ratio can be taken out of a ratio) has been publishedwith a translation by Ruelle 4 ;if it is not by Domninus, itprobably belongs to the same period.A most honourable place in our history must be reservedfor SIMPLICIUS, who has been rightly called 'the excellentSiinplicius, the Aristotle-commentator, to whom the world cannever be grateful enough for the preservation of the fragmentsof Parmenides, Empedocles, Anaxagoras, Melissus,Theophrastus and others' (v. Wilamowitz-Mollendortt). Helived in the first half of the sixth century and was a pupil,first of Ammonius of Alexandria, and then of Damascius,the last head of the Platonic school at Athens. When in theyear 529 the Emperor Justinian, in his zeal to eradicatepaganism, issued an edict forbidding the teaching of philo-1Anecdota Graeca, vol. iv, pp. 413-29.2Mtmoires scientifiques, vol. ii, nos. 35, 40.8 Revue des Mudes yrecques, 1906, pp. 359- 82 ; Mtmoiresvol. iii, pp. 256-81.4Revue de Philoloyie, 1883, p. 83 sq.

DOMNINUS. SIMPLICIUS 539sophy at Athens, the last members of the school, includingDamascius and Simplicius, migrated to Persia, but returnedabout 533 to Athens, where Simplicius continued to teach forsome time though the school remained closed.Extracts from Eudemus.To Simplicius we owe two long extracts of capital importancefor the history of mathematics and astronomy. Thefirst is his account, based upon and to a large extent quotedtextually from Eudemus's History of Geometry, of the attemptby Antiphon to square. the circle and of the quadratures oflunes by Hippocrates of Chios. It is contained in Simplicity'scommentary on Aristotle's Physics, 1and has been the subjectof a considerable literature extending from 1870, the datewhen Bretsclmeider first called attention to it, to the latestcriticaledition with translation and notes by Rudio (Teubner,1907). It has already been discussed (vol.i,pp. 183-99).The second, and not less important, of the two passages isthat containing the elaborate and detailed account of thesystem of concentric spheres, as first invented by Eudoxus forexplaining the apparent motion of the sun, moon, and planets,and of the modifications made by Callippus and Aristotle. Itis contained in the commentary on Aristotle's De caelo*\Simplicius quotes largely from Sosigenes the Peripatetic(second century A.D.), observing that he in his turn drewfrom Eudemus, who dealt with the subject in the secondbook of his History of Astronomy. It is this passage ofSimplicius which, along with a passagein Aristotle's Metaphysics,'*enabled Schiaparelli to reconstruct Eudoxus's system(bee vol. i, pp. 329-34). Nor must it be forgotten that it is inSimplicius's commentary on the Physics* that the extractfrom Geminus's summary of the Meteorologica of Posidoniusoccurs which was used by Schiaparelli to support his viewthat it was Heraclides of Pontus, not Aristarchus of Samos,who firstpropounded the heliocentric hypothesis.Simplicius also wrote a commentary on Euclid's Elements,Book I, from which an-Nairizi, the Arabian commentator,1Simpl. in Phys., pp. 54-69, ed. Diels.2Simpl. on Arist. De ceio, p. 488. 18-24 and pp. 493-506, ed. Heiberg.3 JUetaph. A. 8, 1073 b 17-1074 a 14.4Simpl. in Phys., pp. 291-2, ed. Diels.

540 COMMENTATOKS AND BYZANTINESmade valuable extracts, including the account of the attempt of'Aganis' to prove the parallel-postulate (see pp. 228- 30 above).Contemporary with Simplicius, or somewhat earlier, wasEUTOCIUS, the commentator on Archimedes and Apollonius,As he dedicated the commentary on Book I On the Sphereand Cylinder to Ammonius (a pupil of Proclus and teacherof Simplicius), who can hardly have been alive after A.D. 510,Eutocius was probably born about A.D. 480. His date usedto be put some fifty years later because, at the end of the commentarieson Book II On the Sphere and Cylinder and onthe Measurement of a Circle, there is a note to the eft'ect*thatthe edition was revised by Isidorus of Miletus, the mechanicalengineer, our teacher 9 .But,in view of the relation to Ammonius,it is impossible that Eutocius can have been a pupil ofIsidorus, who was younger than Anthemius of Tralles, thearchitect of Saint Sophia at Constantinople in 532, whosework was continued by Isidorus after Anthemius's deathabout A.D. 534. Moreover, it was to Anthemius that Eutociusdedicated, separately, the commentaries on the first fourBooks of Apollonius's Conies, addressing Anthemius as my'dear friend '. Hence we conclude that Eutocius was an eldercontemporary of Anthemius, and that the reference to Isidorasisby an editor of Eutocius's commentaries who was a pupil ofIsidorus.For a like reason, the reference in the commentaryon Book II OH the Sphere and Cylinder 1 to a SiaprjTrjsinvented by Isidorus ' our teacher ' for drawing a parabolamust be considered to be an interpolation by the same editor.Eutocius's commentaries on Archimedes apparently extendedonly to the three works, On theSpttere and Cylinder,Measurement of a Circle and Plane Equilibriums, and thoseon the Conies of Apollonius to the first four Books only.We are indebted to these commentaries for many valuablehistorical notes. Those deserving special mention here are(1) the account of the solutions of the problem of the duplicationof the cube, or the finding of two mean proportionals,by 'Plato', Heron, Philon, Apollonius, Diocles, Pappus,Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes, (2)the fragment discovered by Eutocius himself containing the1Archimedes, ed. Heiberg, vol. iii, p. 84. 8-11.

EUTOCIUS. ANTHEMIUS 541missing solution, promised by Archimedes in On the Sphereand Cylinder, II. 4, of the auxiliary problem amountingto the solution by means of conies of the cubic equation(ax)x 2 = fcc2, (3) the solutions (a) by Diocles of the originalproblem of II. 4 without bringing in the cubic, (b) by Dionysodorusof the auxiliary cubic equation.ANTHEMIUS of Tralles, the architect, mentioned above, washimself an able mathematician, as is seen from a fragment ofa work of his, On Burning-mirrors.This is a document ofconsiderable importance for the history of conic sections.Originally edited by L. Dupuy in 1777, it was reprinted inWestermann's IlapaSogoypdQoi (Scriptures rerum mirabiliumGraeci), 1839, pp. 149-58. The first and third portions ofthe fragment are those which interest us. 1 The first givesa solution of the problem, To contrive that a ray of the sun(admitted through a small hole or window) shall fall in agiven spot, without moving away at any hour and season.This is contrived by constructing an elliptical mirror one focusof which is at the point where the ray of the sun is admittedwhile the other is at the point to which the ray is requiredto be reflected at all times. Let B be the hole, A the pointto which reflection must always take place,BA being in themeridian and parallel to the horizon. Let BG be at rightangles to BA, so that OB is an equinoctial ray and let BD be;the ray at the summer solstice, BE a winter ray.Take F at a convenient distance on BE and measure FQequal to FA. Draw HFG through F bisecting the angleA FQ, and let BG be the straight line bisecting the angle EBCbetween the winter and the equinoctial rays. Then clearly,since FO bisects the angle QFA, if we have a plane mirror inthe position HFG, the ray BFE entering at B will be reflectedto A.To get the equinoctial ray similarly reflected to A, join GAtand with G as centre and GA as radius draw a circle meetingBO in K. Bisect the angle KGA by the straight line GLMmeeting BK in L and terminated at M, a point on the bisectorof the angle OBJ}. Then LM bisects the angle KLA also, andKL = LA, and KM = MA.the ray BL will be reflected to A.1If then GLM is a plane mirror,See Bibliotheca matliematica, vii 3 , 1907, pp. 225-33.

542 COMMENTATORS AND BYZANTINESBy taking the point N on BD such that MN = MA, andbisecting the angle NMA by the straight line MOP meetingBD in 0, we find that, if MOP is a plane mirror, the ray BOis reflected to A.Similarly, by continually bisecting angles and making moremirrors, we can get any number of other points of impact. Makingthe mirrors so short as to form a continuous curve, we getthe curve containing all points such that the sum of the distancesof each of them from A and B is constant and equal to BQ, BK,or BN. ' If then ', says Anthemius, c we stretch a string passedHround the points A, B, and through the first point taken on therays which are to be reflected, the said curve will be described,which is part of the so-called " ellipse ", with reference towhich (i.e. by the revolution of which round BA) the surfaceof impact of the said mirror has to be constructed/We have here apparently the first mention of the constructionof an ellipse by means of a string stretched tight roundthe foci. Anthemius's construction depends upon two propositionsproved by Apollonius (1) that the sum of the focaldistances of any point on the ellipse is constant, (2) that thefocal distances of any point make equal angles with thetangent at that point, and also (3) upon a proposition notfound in Apollonius, namely that the straight line joining

ANTHEMIUS 543the focus to the intersection of two tangents bisects the anglebetween the straight lines joining the focus to the two pointsof contact respectively.In the third portion of the fragment Anthemius proves thatparallel rays can be reflected to one single point from a parabolicmirror of which the pointis the focus. The directrix isused in the construction, which follows, mutatis mutandis, thesame course as the above construction in the case of the ellipse."As to the supposition of Heiberg that Anthemius may alsobe the author of the Fraymentum mathematicum Bobiense, seeabove (p. 203). The Papyrus of Akhmim.Next in chronological order must apparently be placedthe Papyrus of Akhmim, a manual of calculation writtenin Greek, which was found in the metropolis of Akhmim,the ancient Panopolis, and is now in the Musê duGizeh. It was edited by J. Baillet 1 in 1892. Accord-it was written between the sixth anding to the editor,ninth centuries by a Christian. It is interesting becauseit preserves the Egyptian method of reckoning, with properfractions written as the sum of primary fractions or submultiples,a method which survived alongside the Greek andwas employed, and even exclusively taught, in the East. Theadvantage of this papyrus, as compared with Ahmes's, is thatwe can gather the formulae used for the decomposition ofordinary proper fractions into sums of submultiples. Theformulae for decomposing a proper fraction into the sum oftwo submultiples may be shown thus :,-. u 1 11Memoires publics par les membres de la Mission archto1ogiquefran$aiseau Caire, vol. ix, part 1, pp. 1-89.

544 COMMENTATORS AND BYZANTINES1 l lFx -L - ,176 /16,+ 3.1K+ /16 + -3. 3.1K 11\ 1 77^112"7 /3and again-- = + T R -.y"j-\=\ L & /lo -r 7\ /lOTfl.f\l UoU7 16 ( \ o S ) / (" V 3"~)2u / &3(3)= +cdf cd~'+C 9dfaJT. cd + dfaExample.O

THE PAPYRUS OF AKHMIM. PSELLUS 545& n TG Tis- KG e* TS rr TO ? where w is succesunsively 11, 12, ... and 20; the results are all arranged as thesums of integers and submultiples.The Geodaesia of a Byzantine author formerly called, withoutany authority, ' Heron the Younger ' was translated intoLatin by Barocius in 1572, and the Greek text was publishedwith a French translation by Vincent. 1 The place of theauthor's observations was the hippodrome at Constantinople,and the date apparently about 938. The treatise was modelledon Heron of Alexandria, especially the Dioptra, while somemeasurements of areas and volumes are taken from theMetrica.MICHAEL PSELLUS lived in the latter partof the eleventhcentury, since his latest work bears the date 1092. Thoughhe was called ' first of philosophers', it cannot be said thatwhat survives of his mathematics suits this title. Xy landeredited in 1556 the Greek text, with a Latin translation, ofa book purporting to be by Psellus on the four mathematicalsciences, arithmetic, music, geometry and astronomy, but it isevident that it cannot be entirely Psellus's own work, sincethe astronomical portionis dated 1008. The arithmetic containsno more than the names and classification of numbersand ratios. The geometry has the extraordinary remark that,while opinions differedas to how to find the area of a circle,the method which found most favour was to take the area asthe geometric mean between the inscribed and circumscribedsquares; this gives TT = VS = 2-8284271 ! Theonly thing ofPsellus which has any value for us is the letter published byTannery in his edition of Diophantus. 2 In this letter Psellussays that both Diophantus and Anatolius (Bishop of Laodiceaabout A. p. 280) wrote on the Egyptian method of reckoning,1Notices et extmits, xix, pt. 2, Paris, 1858.aDiophantus, vol. ii, pp. 37-42.1528.2 N n

546 COMMENTATORS AND BYZANTINESand that Anatolius's account, which was differentand moresuccinct, was dedicated to Diophantus (this enables us todetermine Diophantus's date approximately). He also notesthe difference between the Diophantine and Egyptian namesfor the successive powers of dpidftSs: the next power afterthe fourth (SwapoSfyapif = #4 ),'i.e. x 5 , the Egyptians calledthe first undescribed ' ((JfAoyoy 7rpv Ttvvdpw fiadr)fidr(i)vor TtTpdf$ip\ov). The arithmetical portion contains, besidesexcerpts from Nicomachus and Euclid, a paraphrase of Diophantus,Book I, which Tannery published in his edition ofDiophantus ] the musical section with;part of the preface waspublished by Vincent, 2 and some fragments from Book IV byMartin in his edition of the Astronomy of Theon of Smyrna.MAXIMUS PLANUDES, a monk from Nicomedia, was theenvoy of the Emperor Andronicus II at Venice in the year1297, and lived probably from about 1260 to 1310. Hewrote scholia on the first two Books of Diophantus, whichare extant and are included in Tannery's edition of Diophantus.8They contain nothing of particular interest excepta number of conspectuses of the working-out of problems ofDiophantus written in Diophantus's own notation but withsteps in separate lines, and with abbreviations on the left ofwords indicating the operations (e.g. ?K0. = e/c0

PSELLUS. PACHYMERES. PLANUDES 547Rechenbuch des Maximus Planudes in Greek by Gerhardt(Halle, 18(55) and in a German translation by H. Waeschke(Halle, 1878). There was, however, an earlier book under thesimilar title 'Apxy rtjy /zeyaAijr KOL 'IvSiKrjs ^(/uôpmy (sic),written in 1252, which is extant in the Paris MS. Suppl. Gr.387 ;and Planudes seems to have raided this work. Hebegins with an account of the symbols which, he says, were'invented by certain distinguished astronomers for the mostconvenient and accurate expression of numbers. There arenine of these symbols (our 1, 2, 3, 4, 5,6, 7, 8, 9), to which isadded another called Tzifra, (cypher), written and denotingaero. The nine signs as well as this one are Indian/But this is, of course, not the first occurrence of the Indiannumerals; they were known, except the zero, to Gerbert(Pope Sylvester II) in the tenth century, and were used byLeonardo of Pisa in his Liber abaci (written in 1202 andrewritten in 1228). Planudes used the Persian form of thenumerals, differing in this from the writer of the treatise of1252 referred to, who used the form then current in Italy.It scarcely belongs to Greek mathematics to give an accountof Planudes's methods of subtraction, multiplication, &c.Extraction oftliesquare root.As regards the extraction of the square root, he claims tohave invented a method different from the Indian methodand from that of Theon. It docs not appear, however, thatthere was anything new about it. Let us try to see in whatthe supposed new method consisted.Planudes describes fully the method of extracting thesquare root of a number with several digits,a method whichis essentially the same as ours. This appears to be what herefers to later on as ' the Indian method '. Then he tells ushow to find a first approximation to the root when the numberis not a complete square.'Take the square root of the next lower actual squarenumber, and double it :then, from the number the square rootof whicli is required, subtract the next lower square numberso found, and to the remainder (as numerator) give as denominatorthe double of the square root already found.'N n 2

548,COMMENTATORSAND BYZANTINES*The example givenis

PLANUDES. MOSCHOPOULOS 549the scholia to Eucl., Book X, the same method is applied.Examples have been given above (vol. i, p. 63). The supposednew method was therefore not only already known to thescholiast, but goes back, in all probability, to Hipparchus.Two problems.Two problems given at the end of the Manual of Planudesare worth mention. The first is stated thus :'A certain manfinding himself at the point of death had his desk or safebrought to him and divided his money among his sons withthe " following words, I wish to divide my money equallybetween my sons the first shall have one :piece and ^th of therest, the second 2 and th of the remainder, the third 3 and?th of the remainder/' At this point the father died withoutgetting to the end either of his money or the enumeration ofhis sons. I wish to know how many sons he had and howmuch money.' The solution isgiven as (n I) 2 for the numberof coins to be divided and (n1) for the number of his sons;or rather this is how itmight be stated, for Planudes takesn = 7 arbitrarily. Comparing the shares of the first two wemust clearly havel + -(a-l) = 2n' + i{a-(l + +n 2)},l N 'nwhich gives x = (n1 )2;therefore each of (/i 1) sons received(71-1).The other problem is one which we have already met with,that of finding two rectangles of equal perimeter such thatthe area of one of them is a given multiple of the area ofthe other. If n is the given multiple, the rectangles are2(>*1, n s n, 2 ) andl('ftl, 7&3 n) respectively. Planudesstates the solution correctly, but how he obtained it is not clear.We find also in the Manual of Planudes the 'proof by nine '(i.e. by casting out nines), with a statement that it was discoveredby the Indians and transmitted to us through theArabs.MANUEL MOSCHOPOULOS, a pupil and friend of MaximusPlanudes, lived apparently under the Emperor Andronicus IIMichael VIII(1282-1328) and perhaps under his predecessor(1261-82) also. A man of wide learning, he wrote (at the

550 COMMENTATORS AND BYZANTINESinstance of Nicolas Rhabdas, presently to be mentioned) atreatise on magic squares ;he showed, that is, how the numbers1, 2, 3 ... 7i 2 could be placed in the n 2 compartments ofa square, divided like a chess-board into n 2 small squares, in sucha way that the sum of the numbers in each horizontal andas well as in the rowseach vertical row of compartments,forming the diagonals, is always the same, namely %n (n 2 + 1).for the cases in whichMoschopoulos gives rules of procedure11 = 2 m + 1 and n = 4 m respectively, and these only, in thetreatise as we have it ;he promises to give the case wheren = 4ra+2 also, but does not seem to have done so, as thetwo manuscripts used by Tannery have after the first two casesthe words reAoy rov avTov. The treatise was translated byDe la Hire, 1 edited by S. Giinther, 2 and finally edited in animproved text with translation by Tannery. 3The work of Moschopoulos was dedicated to Nicolas Artavasdus,called RHABDAS, a person of some importance in thehistory of Greek arithmetic. He edited, with some additionsof his own, the Manual of Planudes; this edition exists inthe Paris MS. 2428. But he is also the author of two letterswhich have been edited by Tannery in the Greek text withFrench translation. 4 The date of Rhabdas is roughly fixedby means of a calculation of the date of Easter 'in the currentyear ' contained in one of the letters, which shows that itsdate was 1341. It is remarkable that each of the two lettershas a preface which (except for the words T^V 8j\a>

MOSCHOPOULOS. RHABDAS 551of Smyrna, arithmetician and geometer, TOV 'PafiSa,at theinstance of the most revered Master of Requests, GeorgiusChatzyces, and most easy for those who desire to study it/A long passage, called Kpa

552 COMMENTATORS AND BYZANTINESfor 20, stretch out the forefinger straight and vertical,keep fingers 3, 4, 5 together but separate from itand inclined slightly to the palm in this;positiontouch the forefinger with the thumb ;30, join the tips of the forefinger and thumb ;40, place the thumb on the knuckle of the forefingerbehind, making a figure like the letter f ;50, make a like figure with the thumb on the knuckleof the forefinger inside ;as for 50 and60, place the thumb inside the forefingerbring the forefinger down over the thumb, touchingthe ball of it ;70, rest the forefinger round the tip of the thumb,making a curve like a spiral ;80, fingers 3, 4, 5 being held togetherand inclinedat an angle to the pdflm, put the thumb across thepalm to touch the third phalanx of the middlelinger (3) and in this position bend the forefingerabove the first joint of the thumb ;90, close the forefinger only as completely as possible.(d) The same operations on the right hand give the hundreds,from 100 to 900.The first letter also contains tables for addition and subtractionand for multiplication and division ;as these are saidto be the 'invention of Palamedes', we must suppose thatsuch tables were in use from a remote antiquity. Lastly, thefirst letter contains a statement which, though applied toparticular numbers, expresses a theorem to the effect that(a + 10^ + ... + 10 r 'XJ (&o+ 10^ + ... + 10 w ftn )is not > 10+"+-,where,Oj ... 6 , ^ ... are any numbers from to 9.In the second letter of Rhabdas we find simple algebraicalproblems of the same sort as those of the Anthologia Graecaand the Papyrus of Akhmim. Thus there are five problemsleading to equations of the typexfmx+ - + ... =a.n

RHABDAS 553Rhabdas solves the equation + - = a, practically as we^m n r *should, by multiplying up to get rid of fractions, whence heobtains x = mna /(m + n).equations x + y = a, ma? = ny ;Again he solves the simultaneousalso the pair of equationsxm y n,xy+ 2- = y+-- = a.Of course, m, n, a... have particularnumerical values inallcases.Rhabdas's Rule for approximatingto the square root ofa non-square number.We find in Rhabdas the equivalent of the Heronian formulafor the approximation to the square root of a non-squarenumber A = a* + 6, namelyba = a + ;2ahe further observes that, if a be an approximation by excess,then o^ = A/OLis an approximation by defect, and ^(a + cXj)is an approximation nearer than either. This last form is ofcourse exactly Heron's formula a = } (a+ \ The formulawas also known to Barlaam (presently to be mentioned), whoalso indicates that the procedure can be continued indefinitely.It should here be added that there is interesting evidenceof the Greek methods of approximating to square roots in twodocuments published by Heiberg in 1899. 1 The first ofthese documents (from a manuscript of the fifteenth centuryat Vienna) gives the approximate square root of certain nonsquarenumbers from 2 to 147 in integers and proper fractions.The numerals are the Greek alphabetic numerals, but they aregiven place-value like our numerals : thus OLTJ= 18, #= 147,~ = 9 and so on: is indicated by 4 or, sometimes, by .PTJ 28All these square roots, such as \/(21)= 4, \/(35)= 5^J,= '/(1 12) 10||, and so on, can be obtained (either exactly or,in a few cases, by neglecting or adding a small fraction in the1 (Byzantinische Analekten' in Abh. zurGesch. d. Math. ix. Heft, 1899,pp. 163 sqq.

554 COMMENTATORS AND BYZANTINESnumerator of the fractional part of the root)in one or otherof the following ways:(1) by taking the nearest square to the given number A,say a2 and, using the Heronian formulae(2) by using one or other of the following approximations,wherea 2 < A < (tt+1)2,and A = a* + b =namely, b b7a+ _, f b ,2 a 2a+ --2 aor a combination of two of these with(JL(3) the formula that, if y < -,oCct> thena ma + nc cT) mb 4- nd dIt is clear that it is impossible to denyknowledge of these simple formulae.to the Greeks theThree more names and we have done.IOANNES PKDIASIMUS, also called Galenus, was Keeper of theSeal to the Patriarch of Constantinople in the reign ofAndronicus III (1328-41). Besides literary works of his,some notes on difficult points in arithmetic and a treatise onthe duplication of the cube by him are said to exist in manuscripts.His Geometry, which was edited by Friedlein in 1866,follows very closely the mensuration of Heron.BAKLAAM, a monk of Calabria, was abbot at Constantinopleand later Bishop of Geraci in the neighbourhood of Naples;he died in 1348. He wrote, in Greek, arithmetical demonstrationsof propositions in Euclid, Book1II, and a Logistic insix Books, a laborious manual of calculation in whole numbers,1Edited with Latin translationby Dasypodius in 1564, and includedin Heiberg and Menge's Euclid, vol. v, ad fin.

PEDIASIMUS. BARLAAM. ARGYRUS 555ordinary fractions and sexagesimal fractions (printed atStrassburg in 1592 and at Paris in 1600). Barlaam, as wehave seen, knew the Heronian formulae for finding successiveapproximations to square roots, and was aware that they couldbe indefinitely continued.ISAAC ARGYRUS, a monk, who lived before 1368, was one ofa number of Byzantine translators of Persian astronomicalworks. In mathematics he wrote a Geodaesia and scholia tothe first six Books of Euclid's Elements. The former is containedin the Paris MS. 2428 and is called 'a method ofgeodesy or the measurement of surfaces, exact and shortened ' ;the introductory letter addressed to one Colybosis followedby a compilation of extracts from the Geometrica and Stereometricaof Heron. He isapparently the author of somefurther additions to Rhabdas's revision of the Manual ofPlanudes contained in the same manuscript. A short tractof his On ' the discovery of the squarei*oots of non-rationalsquare numbers Jis mentioned as contained in two other manuscriptsat Venice and Rome respectively (Codd. Marcianus Gr.333 and Vaticanus Gr. 1058), where it is followed by a tableof the squareroots of all numbers from 1 to 102 in sexagesimalfractions (e.^. ^2 = 1 24' 51" 48'", ^3=1 43' rÔ'"). 11Heiherg, ' Byxantinische Analekten ',in Abh. zur Cesch. d. Math, ix,pp. 169-70.

APPENDIXOn Archimedes's proof of the subtangent-property ofa spiral.THE section of the treatise On Spirals from Prop. 3 toProp. 20 is an elaborate series of propositions leading upto the proof of the fundamental property of the subtangentcorresponding to the tangent at any point on any turn of thespiral. Libri, doubtless with this series of propositions inmind, remarks (Histoire des sciences math&matiques en Italie,i, p. 31) that'Apr&s vingt si&cles de travaux et de d^couvertes,les intelligences les plus puissantes viennent encore(Schouer contre la sy&th&se difficile du Traite des Spiralesd'Arehimfcde/ There is no foundation for this statement,which seems to be a too hasty generalization from a dictum,apparently of Fontenelle, in the Histoire de VAcadtmie desSciences pour Vann&e 1704 (p. 42 of the edition of 1722),who says of the proofs of Archimedes in the work OnSpirals: 'Elles sont si longues, et si difficiles a embrasser,que, comme on l'a pft voir dans la Preface de 1'Analyse desInfiniment petits, M. Bouillaud a avou qu'il ne les avoitjamais bien entendues, et queVi&te les ainjustement soup-ônnês de paralogisme, parce qu'il n'avoit pA non plusparvenir & les bien entendre. Mais Routes les preuves qu'onpeut donner de leur difficult^ et de leur obscurit^* tournenta la gloire d'Archimfede; car quelle vigueur d'esprit, quellequantit^ de vftes diff

APPENDIX 557obscurity ' of Archimedes ; while, as regards Vieta, he has'shown that the statement quoted is based on an entire misapprehension,and that, so far from suspecting a fallacy inArchimedes's proofs, Vieta made a special study of the treatiseOn Spirals and had the greatest admiration for that work.But, as in many cases in Greek geometry where the analy-omitted or even (as Wallis was tempted to suppose) ofsis isset purpose hidden, the reading of the completed syntheticalproof leaves a certain impression of mystery; for there isnothing in it to show why Archimedes should have takenprecisely this line of argument, or how he evolved it. It isa fact that, as Pappus said, the subtangent-property can beestablished by purely 'plane 1 methods, without recourse toa 'solid* vtvorts (whether actually solved or merely assumedIf, then, Archimedes chose the morecapable of being solved).difficult method which we actually find him employing,it isscarcely possible to assign any reason except his definitepredilection for the form of proof by reductio ad absurdumbased ultimately on his famous ''or Axiom.LemmaIt seerns worth while to re-examine the whole question ofthe discovery and proof of the property, and to see howArchimedes's argument compares with an easier f plane ' proofsuggested by the figures of some of the very propositionsproved by Archimedes in the treatise.In the first place, we may be sure that the property wasnot discovered by the steps leading to the avSproofit stands.I cannot but think that Archimedes divined the result by anargument corresponding to our use of the differential calculusfor determining tangents. He must have considered theinstantaneous direction of the motion of the point P describingthe spiral, using for this purpose the parallelogram ofvelocities. The motion of P iscompounded of two motions,one along OP and the other at right angles to it.Comparingthe distances traversed in an instant of time in the two directions,we see that, corresponding to a small increase in theradius vector r, wo have a small distance traversed perpendicularlyto it, a tiny arc of a circle of radius r subtended bythe angle representing the simultaneous small increase of theNow r has a constant ratio to 6 which we callangle 6 (AOP).a (when 6 is the circular measure of the angle 6).Consequently

558 APPENDIXthe small increases of r and 6 are in that same ratio.Thereforewhat we call the tangent of the angle OPT isr/a,i.e. OT/r = r/a; and OT = r 2 /a, or rd, that is, the arc of acircle of radius r subtended by the angle 0.To i>rove this result Archimedes would doubtless begin byan analysis of the following sort. Having drawn OT perpendicularto OP and of length equal to the arc ASP, he had toprove that the straight lino joining P to T is the tangentat P. He would evidently take the line of trying to showthat, if any radius vector to the spiral is drawn, as OQ', oneither side of OP, Q' is always on the side of TP towards 0,or, if OQ' meets TP in F, OQ' is alwaysless than OF. Supposethat in the above figure OR' isany radius vector between OPand OS on the backward''side of OP, and that OR meets thecircle with radius OP in R, the tangent to it at P in (r, thespiral in JB', and TP in F'.We have to prove that R, R' lioon oppositesides of F', i.e. that RR' > RF' ;and again, supposingthat any radius vector OQ' on the ' forward ' side ofOP meets the circle with radius OP in Q, the spiral in Q' andTP produced in F, we have to prove that QQ' < QF.Archimedes then had to prove thatRO < RR': RO, and(1)F'M :(2) FQ:QO>Qff:QQ.Now (1) is equivalent toF'R : RO < Yarc RP\ : Tare A8P\. since RO = PO.

APPENDIX 559But (arc A HP) = OT, by hypothesis ;therefore it was necessary to prove, alternando, that(3) F'R : (arc RP) < RO :i.e.OT, or PO :OT,< 'PM: MO, where OM is perpendicular to SP.Similarly, in order to satisfy (2), it was necessary toprove that(4) / PM: MO.Now, as a matter of fact, (3) is a fortioriF'R :(chord RP) < PM:MO;satisfied ifwe cannot substitute the chord PQ forbut in the case of (4)the arc PQ, and we have to substitute PG', where G' is thepoint in which the tangent at P tothe circle meets OQ produced for;of course PG' > (arc PQ), so that (4)is a fortiori satisfied ifFIG. 1.It is remarkable that Archimedesuses for his proof of the'two cases Prop.8 and Prop. 7 respectively, and makesno use of Props 6 and 9, whereasthe above argument points precisely to the use of the figuresof the two latter propositions only.For in the figure of Prop. 6 (Fig. 1),if OFP isany radiuscutting AB inF, and if PB produced cuts OT, the parallel toAB through 0, in H, it is obvious, by parallels, thatPF: (chord PB) = OP :Pll.Also PH becomes greater the farther P moves from Btowards A, so that the ratio PF :P.B diminishes continually,while it is always less than OB : BT (where BT is the tangentat B and meets Oil in T), i.e.alwaysless than BM: MO.Hence the relation (3)isalways satisfied for any point R' ofthe spiral on the ' backward ' side of P.is equivalent to (1),from which it follows that F'RBut (3)isalways less than RR', so that R' always lies on the sideof TP towards 0.

560 APPENDIXNext, for the point Q' on the ' forward * side of the spiralfrom P, suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2)any radius OP of the circle meets AB produced in F, andthe tangent at B in G;PIG. 2.parallel through to AB, in H, T.and draw BPH, BGT meeting OT, theThen PF:BG>FG: BG, since PF > FG,> OG :GT, by parallels,> OB: BT, a fortiori,>BM:MO\and obviously, as P moves away from B towards OT, i.e.moves away from B along BT, the ratio OG:GT increasescontinually, while, as shown, PF:BG isalways > BM:MO,and, a fortiori,PJP:(arcPJ5) >BM:MO.That is,as G(4)isalways satisfied for any point Q' of the spiral''forward of P, so that (2)is also satisfied, and QQ' isalwaysless than QF.It will be observed that no i/eflcm, and nothing beyond'plane ' methods, is required in the above proof, and Pappus'scriticism of Archimedes's proofis therefore justified.Let us now consider for a moment what Archimedes actuallydoes. In Prop. 8, which he uses to prove our proposition inthe 'backward' case (-R', JR, F'), he shows that,isany ratio whatever less than PO : OT or PM:MO, we canfind points F', G corresponding to any ratio PO : 0V whereOT < 0V < OF, i.e. we can find a point F fcorresponding toa ratio still nearer to PO : OT than PO:OV is. This provesthat the ratio RF' :if PO:OVPG, while it isalways less than PM:MO,

APPENDIX 561approaches that ratio without limit as R approaches P. Butthe proof does not enable us to say that RF' -.(chord PR),which is > RF' :PG, is also always less than PM : MO. Atfirst sight, therefore, it would seern that the proof must fail.Not so, however; Archimedes is nevertheless able to provethat, if P V and not PT is the tangent at P to the spiral, anabsurdity follows. For his proof establishes that, if PFis thetangent and OF' is drawn as in the proposition, thenF'O : RO < OR: OP,'or F'O < OR', wîch is impossible Why this '. is impossibledoes not appear in Props. 18 and 20, bat it follows from theargument in Prop. 13, which proves that a tangeijt to the spiralcannot meet the curve again, and in fact that the spiral iseverywhere concave towards the origin.Similar remarks apply to the proof by Archimedes of theimpossibility of the other alternative supposition (that the tan-than T is).gent at P meets OT at a point U nearer toArchimedes's proof is therefore in both parts perfectly valid,in spite of any appearances to the contrary. The only drawbackthat can be urged seems to be that, if we assume thetangent to cut OT at a point very near to T on either side,Archimedes's construction brings us perilously near to infinitesimals,and the proof may appear to hang, as it were, ona thread, albeit a thread strong enough to carry it.But it isremarkable that he should have elaborated such a difficultproof by means of Props. 7, 8 (including the solid* vsvvis ofProp. 8),when the figures of Props. 6 and 7 (or 9) themselvessuggest the direct proof above given, which is independent ofany i/et/cny.P. Tannery, 1 in a paper on Pappus's criticism of the proof asunnecessarily involving ' solid ' methods, has given anotherproof of the subtangent-property based on ' plane ' methodsonly ; but I prefer the method which I have given abovebecause it corresponds more closely to the preliminary propositionsactually given by Archimedes.1Tannery, Memoires stientijiquesi i,1912, pp. HOO-16.1523.2 O O

INDEX OF UKEEK. WORDS[The pages are those of the first volume except where otherwise stated.]47.-o/ :cicriTc* (Plato)iii. 355.rjTos, -op, that cannot be gonethrough, i.e. infinite 343.vvciTos, -ov ii. 462 :anay&yt] elsu&vvarov, &C. 372.postulate 373.l 11.y, -ov, irrational 84, DO :nfp\

sensesj| SlOTTTplKr)564 INDEX OF GREEK WOKDSy,*little altar 1 ,properly awedge-shaped solid ii. 319, ii. 333 :measurement of (Heron),ii. 332-3 :(= a-fyvio-Kos)of a certain kindof solid number 107, ii.240, ii.315.ycco&iKri'ci = mensuration 1 6.T(G);jiTpovu.(va of Heion ii. 318, ii.453.y\w\is (arrow-head), Pythagoreanname for angle 166.yvtopw, ynomon, q.v.= :perpendicular 78, 175.yvatpifjios, -ov, known :Kara yvot>p.ovayi/a>/n/ioi>,analternative term for 6e8o/ueW,yivenii. 537.'in theyvoopifjL^s, recognized manner'ii. 79. !ypupua, * figure 1 or proposition, oftheorem of Eucl. 1. 47, 144. !!ypnfjififf I dia or fK TO>V ypap,p.u>v oftheoretical proof ii. 257, 258.ypap,piKus, -?, -di/, linear: used ofprime numbers 73 : ypappiKtil'7riard(7is, Considerations onCurves ', by Demetrius ii. 359 :ypappiKus, graphically 93.ypufatv, to draw or write on 159,173 : also to prove 203 H., 339.ii. 537-8.S) -rj y -oi/, yiven'.of,to iy prove 328.di? 371.secondary of composite:numbers 72 =:dcvrepa p.vpuis (10,000 2 40. )aprjrrjs, compasses 308, ii. 540.:8iXWi, separando or dividendo(in transformation of ratios)386.8iaip

INDEX OF GREEK WORDS 565a class of locus of unknown, or term, in an equationoy, ii. 185,*prime 72.(Diophantus) ii. 4GO.ii. 193.(If 9 pta, V, one : ef TrXttco, 'several (f>68tov, Method ii. 246.ones' (definition of * number')70.V, lever or balance :n-fp!cl

566 INDEX OF GREEK WORDSKnT(t(TTfpurpoi work9 by Eratosthenesii. 108.Pythagorean school, opp. to/, (>!/ :parjuLariKo nVararopff Kavovos, Sectio canon is, aKovapariKoi 11 :Matfi/iifmici)attributed to Euclid 17, 444.rais of ii.Ptolemy 273-4:KaTovnpais rail' dptOp&v, naming of partita, ra (Plato) 288.numbers (Archimedes)ii. 23. peBoptov, boundary ii, 449.xaroTjrpiKiJ, theory of mirrors 18. pciovpov irpoca-Kapifavpwov (Heron),Ki/Tpo#apiK, problems on centre of curtailed and pared in front (cf.gravity ii. 24, ii. 350.scarify), of a long, narrow, triangularprism (Heib.) ii. 319.KcWpoi/, centre : CK roCf) Kt'vrpov=radius 381.pos: /nf'p/;, parts (= proper fraction)dist. from ps'pos (aliquotof Sporus 234.part) 42 (cf. p. 294).K(paTQi8i]s (yttivla) 178, 882.v, inflect : jccitXdcrdm 337.pccro\dfiot>, mean-Jimler (of Eratosthenes)ii:i04, ii. 359.pappy, conchoid 238.Kviov ii. 211./X6T*'o>pO, -Ol/ I 7Tpl pfTf da/HttP, WOl'kKoXovpos, -ov, truncated ii. 333: (of by Posidonius ii. 219, ii. 231-2.pyramidal number) 107.pfTeWpOOKOTTtKT) 18.KOO-KII/OV, siere (of Eratosthenes) 16, Mfrp//0tir, Mensurae (Heronian) ii.100, ii. 105.319.Ko^XoeiS/}? ypappy, coMoid 238./nr>off, lengthKvftoKvftos, cube-cube, = used by Plato of side:sixth power of square containing a squareof unknown (Diophantus)ii. 458. number of units of area 204.Ki;|3oKv/3o0Toi>, reciprocal of Kv/3-/ufc), term for problemsKV&OS ii. 458.about numbers of apples (e.g.) 14,Ki'poe, cube :ii.*(>$v avfrj (Plato)442.297: cube of unknown (Diophantus)ii. 458 vofioupcvos (TOTTOS), LittleKV&OS cftXrjero? Astronomy : ii. 273.=* ninth power of unknown a, mina (= 1000 drachmae) M:(Egyptian)ii. 546.stands for, 31.KvK\tKfj 0to>pui,De motu circular! i /io?pn, fraction l/360th of circumferenceor a degree 45, 61 ::by Cleomedes ii. 235.poipajeucXifcof, -*}, -ov, circular, used of TOTTIKI}, xp viK n *l (i Hypsicles) ii.square numbers ending in 5 or 6,108.povas, monad or unit 43 : definitionsof, 69 povd8av :(Tvcrrrjpn= number,\dnctv : forms used to express minus. 69 1devTfpwdovpevrjand sign for (Diophantus),ii. 459.= = povns 10,TptwSovpfi-rj p. 100, &c. (Iambi.)Xetyir, wanting (Diophantus) :Aetyet 114: pnvas Ofariv Hvovcra = point= minus ii. 459.69, 283.Xeis: Kara Xcîy, word for word /iopioi/, part183.\irrov ya fraction (Heron) 48 = = or fraction : pnpiov orv popiv divided by (Diophantus)44.: aminute (Ptolemy) 45.pvpias (with or without irpvTtj or\fjppa, lemma 373.ctnXrj) myriad (10,000), p.Xoyicr/Aoy, calculation 13.or din\j) 10,0002, &c. 40.Xoyicrri/cv, art of calculation, opp.to dpiOprjriKT] 13-16, 53.(solid ?) 156, 178.Xoyo?, ratio : Xdyoi; drroTopf^ sectioto , verge (towards) 196. 239,ii.rationis, by Apollonius 175.337,ii. 65.W, inclinatio or * verging', asubjects of instruction type of problem 235 41, 260, ii.10-11 : term first appropriated 199, ii. 385: vevoras in Archimedesii 65-8: two books ofto mathematics by Pythagoreans11 :7Tp\ r)t> paQripdrw, a work vtva-fLs by Apolloninsii. 189 92by Protagoras 179. ii. 401, ii. 412-13.

INDEX OF GREEK WORDS 567, goal or end of race-course114.o]3oXoff, obol :sign for, 31, 49, 50.'oXu/iTTiovZ/cm, work by Eratosthenesii. 109.oi/uf, a wedge-shaped figure ii. 319,ii. 333.ôpy avoir OIIKTJ 1R.op&oy, .a, -or, right or perpendicular:np6la TrXevpu, latus rectumii. 139: opffla faapcrpot, *erectdiameter', in double hyperbola,ii. 134ifajf : topurfJLfvos, defined,i.e. determinate94, 340.ifov (KVK\OS), dividing circle :fcon*o(Eucl.)351 f 352., (1) definition 373: (2) limitor boundary 293 :(3) term (in aproportion) 306 n.de/Lu'a or ov&V, sign for (0), 39, 45.ITa /3w Knl Kiva> rav yav, saying ofArchimedes ii. 18.Trap* qv bvvavran (nl Karayofifi/nirfTa'}-pivots), expression for parameterof ordinates ii. 139.7ra/wi]3o\jJ, application : TT. TWI/ xwptWapplication of areas 150 : Ta (yivnptva aijfifca, thefoci of a central conic, ii. 156 :parabola (the conic) 150,ii. 138.ii.Hapadooypa(f)oi 541.7Tfif)atiooc ypuppy, paradoxical curve(of Menelaus) ii. 260-1,ii. 360.177, ii. 234.-orqff,vj to pull awry:nnpf O-TTOii.398.nearness to equality, approximation: TTo/Jtacmjroi" n-ycoy^(Diophantus)ii. 477, ii. 500.r, axe-shaped figureii. 315.to * five ' , (= count) 26.176, ii. 104.ntpnivov(ra TrocruTrjs= unit, 69.Trc'paffylimit or extiemity 293:limiting surface 166 :mpas cru>xXfio*',definition of figureii. 221.mpKrmiiiTloy, odd-even : with Neo-Pythagoreansis of formfia 19,) >72.q.v.tyXucor, -i/, -oi', how great (of magnitude)12.size 384.-a, -oi/, transverse :or TrXcvpa ii. 139.o$-, -oV, (easily) formableii. 487.firaWir, a work by Eratosthenesii. 104.ôr multitude :, TrXôr = ?v unit,69 :irXfjQos fapta-pwov= number,70 :TrXrjQos /Liorddo>f aopio-TO*i/, def.of unknown * quantity ' 94, ii.456.iv6i$, i brick, a solid number ofa certain form 107, ii. 240, ii,315.>XXa7rXo-ie7Ti/u* = p^ff, multiplex super*partien*, ratio of formp+ 1 ^,ni+n103., multiplex superparticularit,= ratio of form18.--nf 103., , -oi/, multiple 101., a compound pulley ii,Tropi/io?, ov (7ro/)iffiv), procurable:one sense of ktopevnsii. 538.= TTopto-jiifl, porisin: (1) corollary,(2) a certain type of proposition372-3, ii. 533.7roo-oj>, quantity, of number, 12.TrofroYqy, quantity 69, 70 : numbeidefined as Troo-oT^ros- xvpa K povtldav crvyKfijjifvov 70.irpofj.i,Kr)f, prolate (= oblong) 203;but distinguished from trtpopfiKrn83, 108.7rpO(r(iyo)yiov 3C9.= TTporno-tf enunciation 370,ii.533.TTpwros, prime 72.71 TWO" If,C/W 372.7iv0prt v, base ==; digit 55-7, 1 15-17eVi'rpiros jriGprjv 306-7.irvpapif, pyramid 126.irvpciw, Truptor, burning mirrorn(pl TTvpc/wp, work by Dioclef264, ii. 200 ; irpl TOV irvpiov, b)Apolloniusii. 194.pqro?, -T;, -(if, rational : used in senseof * given ' ii. 537.:Trepi ponSn a mechanica1 ,work by Ptolemyii. 295.

568 INDEX OF GREEK WORDS103.224.of Archimedes ii. 23, ii.49.9a form of sun-dial ii. 1,ii. 4.ri, scene-painting 18, ii.2o0ia, nickname of Democritus 176.iTTrctpn, spire or tore ii. 117: varietiesOf (facxfc)p,svrj or fTraXXttTTou ra), ii. 204.ord0/i7, plumb-line 78, R09.a-rnrrjpy sign for, 81.T79, -6, the writer of Elements(o-rot^cioj/), used of Euclid357.orpoyyuXos, -or, round or circular293.cru/iTr/pacr/ia, conclusion (of proposition).370, ii. 533.o-vpQecris (Xctyov), composition (of aratio) 385.9collection :MtydXrj po\ov, sign for, 49.309.water-clocksii.iSy forthcoming : positive term ,dist. from negative (Xetyi?) ii.459.vTTfTrt/LtfpiJy, subsuperpartiens, YQCIprocalof irippfi9 102.{/rcTTi/LKipiof, subsuperparticularis, reciprocalof c môpios 101.V7rppo\f], exceeding (in applicationof areas): name given to Jiyper-bola 150,ii. 138.tvjrfpTc\iof 9 vTTfprfX^, over-perfect(number) 74, 100.'YTTotfe'cms T&V 7iXai/ci>/JuVa>, work byii. 293.Ptolemy{'TTOTToXXaTrXttO'lOS', UT

INDEX OF GREEK WORDS569101-3.oreiWii/, subtend 193 n.^&c.-irXrji;, starting-point (of racecourse)114.

iENGLISH INDEX[The pages are those of the first volume except where otherwise stated.]Abacus 46-8.'Abdelmelik al-Shlrazi ii. 128.Abraham Echellensis ii. 127.Abu Bekr Muh. b. al- Hasan al-Karkhi, see al-KarkhT.Abu '1 Fath al-Isfahanl ii. 127.Abu '1 Wafa al-JBuzjaniii. 328, ii.450, ii. 453.Abu Nasr Mansar ii. 262.Achilles of Zeno 275-6, 278-80.Adam, James, 305-7, 313.|Addition in Greek notation 52.Adrastua ii. 241, 243, 244.Aetius 158-9, 163, ii. 2.!'Agania 1 :attempt to prove paral-|lei-postulate 358, ii. 228-30.Agatharchus 174.Ahnies (Papyrus Rhind) 125, 130,jii. 441.iAkhmiin, Papyrus of, ii. 543-5.Albertus Pius ii. 26.Al-Chazin! ii. 260-1.Alexander the 'Aetolian' ii. 242.Alexander Aphrodisiensis 184, 185,186, 222, 223, ii. 223, ii. 231.Alexeieff, ii. 324-5 n.Al-Fakhri, by al-Karkhi 109, ii.449-50.Algebra beginnings in Egypt : ii.440 : /law-calculations ii. 440 -1 :Pythagorean, 91-7 : exanthema ofThymaridas 94-6.Algebra, geometrical, 150 4 : applicationof areas (q.v.) 150-3:scope of geometrical algebra153-4 : method of proportionib.Al-Hajjaj, translator of Euclid,362 : of Ptolemyii. 274.Alhazen, problem of, ii. 294.Al-Kafi of al-Karkhi 111.Al-KarkhT : on sum of109-10, 111, ii. 51,ii. 449.Allman, G. J. 134, 183.Almagest ii. 274.Alphabet, Greek : derived fromPhoenician, 31 -2 :Milesian, 33 -4:^Mast-numerical use of alphabet,35-6 n.Alphabetic numerals 31-40, 42-4.Amasis 4, 129.Auienemhat I 122, TH 122.Ameristus 140, 141, 171.Amyclas (better Amyntas) 320-1.Amyntas 320 1.Analemma of Ptolemyii. 286-92:of Diodorua ii. 287.Analysis already used by Pythagoreans168 supposed invention::by Plato 291-2: absent fromEuclid's Elements 371-2: definedby Pappus ii. 400.Anatolius 11, 14, 97, ii. 448, ii. 545-6.Anaxagoras explanation of :eclipses7, 162, 172 : moon borrows lightfrom sun 138, 172, ii. 244 : centrifugalforce and centripetaltendency 172-3: geometry 170:tried to square circle 173, 220 :on perspective 174 : in Erastae22, 174.Anaximander 67, 177 : introducedgnomon 78,139,140: astronomy139, ii. 244 : distances of sun andmoon 139 : first map of inhabitedearth ib.Anaximenes ii. 244.Anchor-ring, see Tore.Anderson, Alex., ii. 190.Angelo Poliziano ii. 26.Angle * of a segment ' and ' of asemicircle ' '179 :angle of contact' 178-9, ii. 202.Anharmonic property, of arcs ofgreat circles ii. 269-70 : of straightlines ii. 270, ii. 420-1.

jENGLISH INDEX 571Anthemius of Trails 243, ii.194,ii. 200 3, ii.518, ii. 540, ii.541-3.Antiphon 184, 219, 221-2. 224,_271.Apastamba-&ulba-Sntm 145-6.Apelt, E. F. 330.Apelt, 0. 181 n., 182.Apices 47.Apollodovus, author of Chronica,176.Apollotlorus o \oyi(TTiK('>$ : distich of,131.133, 134, 144, 145.Apollonius of Perga ii. 1, ii. 126.Arithmetic : MKVTUKLQV 234, ii.194, ii. 253 (approximation toTT, ?&.), 'tetrads 1 40, continuedmultiplications 54-7.ii.Astronomy 195-6: A. andTycho Brahe 317, ii. 196: onepicycles and eccentrics ii. 195 6,ii. 243 : ii.trigonometry 253.Conies ii. 126-75: text ii. 126-8, Arabic translations ii.127,prefaces ii. 128-32, characteristicsii. 132-3: conies obtainedFrom oblique cone ii. 134 8,prime property equivalent toCartesian equation (oblique axes)ii. 139, now names, parabola, &c.150, 167, ii. 138, transformationof coordinates ii. 141-7, tangentsii. 140-1, asymptotes ii. 148-9,rectangles under segments of intersectingchords ii. 152-3, harmonicpropertiesii. 154-5, focalproperties (centralii.-conies) 1567, normals as maxima and minimaii. 159-67, construction ofnormals ii. 166-7, number ofnormals through pointii. 163-4,propositions giving evolute ii.164-5.On contacts ii. 181-5 (lemmasto, ii. 416-17), three-circle problemii. 182-5.Sectio ration is ii. 175 9 (lemmasto, ii. 404-5).Sectio spatiiii. 179-80, ii. 337,ii. 339.Determinate section ii. 180-1(lemmas to, ii. 405-12).Comparison of dodecahedronand icosahedron 419 20, ii. 192.Duplication of cube 262 3, ii.194.*General treatise ' ii. 192-3, ii.253 : on Book- 1 of Euclid 358.i>fv0-3 method:attributed to Pythagoras 150,equivalent to solution of generalquadratic 150-2, 394-6.Approximations to \/2 (by meansof * side- ' and 'diameter-' numbers)91 3, (Indian) 146 : to v/3(Ptolemy) 45, 62-3, (Archimedes)ii:51-2: to TT232-5, ii. 194, ii.253: to surds (Heron) ii. 323 6,cf. ii. 547-9, ii. 553-4: to cuberoot (Heron)ii. 341-2.Apuleius of Madaura 97, 99.Archibald, R. C. 425 n.Archimedes 3, 52, 54, 180, 199, 202,203 H., 213, 217, 224-5, 229, 234,272, ii. 1.Traditions ii. 16-17, engines ii.17, mechanics ii. 18, generalestimate ii. 19-20.Works : character of, ii. 20 2,works extant ii.22-3, lost ii. 23-5, 103 ; text ii. 25-7, MSS. ii. 26,editions ii. 27 : The Method ii. 20,21, 22, 27-34, ii. 246, ii. 317-18 :On. the Sphere and Cylinderii. 34-50 : Measurement of a circhii. 50-6, ii. 253 : On Conoids and Spheroidsii. 56 -64 : On Spirals 230-1,ii. 64 75 (cf. ii.377-9),ii. 556-61 :Sand- reckoner ii. 81-5 :Quadratureof Parabola ii. 85-91 : mechanicalworks, titles ii. 23-4,Plane, eqtultbrhimsii. 75-81 : OnFloating Bodies ii.91-7, problemof crown ii. 92-4 : Liber assumjttorumii. 101-3: Cattle-problem14, 15, ii. 23, ii. 97-8,ii. 447 :Catoptrics 444, ii. 24.Arithmetic : octads 40-1, fractions42, value of TT 232-3, 234,ii. 50-6 :approximations to \/îi. 51-2.*Ast 10110my ii. 17 18, sphere-

572 ENGLISH INDEXmakingii. 18, on Aristarchus'shypothesisii. 3-4.Conies, propositions in, 438-9,ii. 122-6.Cubic equation solved by coniesii. 45-6.On Democritus 180, 327,equality of angles of incidenceand reflection ii. 353-4, integralcalculus anticipated ii. 41-2, 61,62-3, 74, 89-90: Lemma or Axiomof A. 326-8, ii. 35 i/eiVei? : in, ii.65-8 (Pappus on, ii. 68): on semiregularsolids ii. 98-101 :triangle,area in terms of sides ii. 103:trisection of any angle 240-1.Archytas 2, 170, 212-16, ii. 1: onp.a6f)iJLaTa 11, on logistic 14, on 1as odd-even 71 : on means 85, 86:no mean proportional between nandn-f 1, 90,215: on music 214:mechanics 213 : solution of problemof two mean proportionals214, 219, 245, 246-9, 334, ii. 261.Argyrus, Isaac, 224 .,ii. 555.Aristaeus :comparison of five regularsolids 420 : Solid Loci (conies)438, ii. 116, 118-19Aristaeus of Croton 86.Aristarchus of Samos 43, 139, ii. 1-15, ii. 251 : date ii. 2 : o-Kttyrj of,ii. 1 :anticipated Copernicus ii.2-3: other hypothesesii. 3, 4:treatise On sizes and distances ofSun and Moon ii. 1, 3, 4-15, trigonometricalii.purpose 5 : numbersin, 39 : fractions in, 43.Aristonophus, vase of, 162.Aristophanes 48, 161, 220.Aristotelian treatise on indivisiblelines 157, 346 8.Aristotherus 348.Aristotle 5, 120, 121 011 :origin ofscience 8 : on mathematical subjects16-17 : on first principles, definitions,postulates, axioms 336-8.Arithmetic :reckoning by tens26-7, why 1 is odd-even 71 : 2even and prime 73 : on Pythagoreansand numbers 67 9 : onthe gnomon 77-8, 83.Astronomy Pythagorean :system164-5, on hypothesis of concentricspheres 329, 335, ii. 244,on Plato's view about the earth314 15.On the continuous and infinite342-3 :proof of incommensurabilityof diagonal 91 on :principleof exhaustion 340 : on Zeno'sparadoxes 272, 275-7, 278-9, 282:on Hippocrates 22 : encomium onDemocritus 176.Geometry illustrations from,:335, 336, 338-40, on parallels339, proofs differing from Euclid's338-9, pro positions not in Euclid340, on quadratures 184-5, 221,223, 224 n., 271, on quadratureby lunes (Hippocrates) 184-5,198-9: on Plato and regularsolids 159 : curves and solids inA. 341.Mechanics 344 6,445-6: parallelogramof velocities 346 'Aristotle's:wheerii. 347-8.:Aristoxenus 24 u., 66.Arithmetic (1 j= theory of numbers(opp. to \oyi(JTtKr)) 13 16: early4Elements of A rithmetic '90, 216 rsystematic treatises, NicomachusIntrod. Ar. 97-112, Theon ofSmyrnal 12- 3,Iamblichus,Comm.on Nicomachus 113-15, Domninusii. 538. (2) Practical arithmetic :originated with Phoenicians 1201, in-primary education 19-20.Arithmetic mean, defined 85.Antlimetica of Diophantus 15-16,ii. 449-514.Arithmetical operations: see Addition,Subtraction, &c.Arrow of Zeno 276, 280 1.Aryabhatta 234.Asclepius of Trail es 99.Astronomy in elementary education19 as :secondary subject 20-1.Athelharcl of Bath, first translatorof Euclid 362 4.Atheriaeus 144, 145.Athenacus of Cyzicus 320-1.*Attic 1(or 'Herodianic') numeials30-1.August, E. P. 299,302,361.Autolycus of Pitane 348 : worksOn the moving Sphere 348-52, OnRisings and Settings 352-3 : relationto Euclid ;^5i"2.Auverus, C. ii. 26.Axioms : Aristotle on, 336 : CommonNotions in Euclid 376 : Axiomof Archimedes 326-8, ii. 35.

ENGLISH INDEX 573Babylonians civilization : of, 8, 9 :system of numerals 28 9 sexagesimalfractions 29: *:perfectproportion '86.Bachet, editor of Diophantus ii.454-5, ii. 480.:Bacon, Roger on Euclid 367-8.Baillet, J. ii. 543.Baldi, B. ii. 308.Barlaam ii. 324 n., ii. 554-5.Barocius ii. 545.Barrow, I., edition of Euclid, 369-70 : on Book V 384.Bathycles 142.Baudhayana & S. 146.Baynard, D. ii. 128.Benecke, A. 298, 302-3.Benedetti, G. B. 344, 446.Bert rand, J. ii. 324 n.Bessarion ii. 27.Besthorn, R. 0. 362, ii. 310.Billingsley, Sir H. 369.Bjornbo, A. A. 197 n., 363, ii. 262.Blass, C. 298.Blass, F. 182.Boeckh, A. 50, 78, 315.Boetius 37, 47, 90: translation ofEuclid 359.Boissonade ii. 538.Bumbelli, Rafael,ii. 454.Borchardt, L. 125, 127.Borelli, G. A. ii. 127.Bouillaud (Bullialdus)ii. 238, ii.556.Braunmiibl, A. von, ii. 268-9 ??., ii.288, ii. 291.Breton (de Champ), P. 436, ii. 3(>0.Bretsclmeider, C. A. 149, 183,324 r>,ii. 539.Brochard, V. 276-7, 279 n., 282.Brougham, Lord, 436.Brugsch, H. K. 124.Bryson 219, 223-5.Burnet, J. 203 M., 285, 314-15.Butcher, S.H. 299,300.ii.Buzengeiger 324 n.Cajori, F. 283 N.:Calculation, practical tbc abacus46-8, addition and subtraction52, multiplication (i) Egyptian52-3 (Russian ?53n.), (ii)Greek53-8, division 58-60, extractionof square root 60-3, of cube root63 4, ii. 341 2.Callimachus 141-2.Callippus: Great Year 177: system ofconcentric spheres 329, 335, ii. 244.Cambyses 5.Camerarius, Joachim,ii. 274.Camerer, J. G. ii. 360.Campanus, translator of Euclid363-4.Canonic = theory of musical intervals17.Cantor, G. 279.Cantor, M. 37-8, 123, 127, 131, 135,182, ii. 203, ii. 207.Carpus of Antioch 225, 232, ii.359.Case (TTTWC) 372, ii. 533.Cassini ii. 206.Casting out nines 115 17, ii. 549.Catoptric, theory of mirrors 18.Catoptrica : treatises by Euclid (?)442, by Theon (?) 444, by Archimedes444, and Heron 444, ii. 294,ii. 310, ii. 352-4.Cattle-problem of Archimedes 14,15, ii. 23, ii. 97-8, ii. 447.Cavalieri, B. 180, ii. 20.Censorinus 177.Centre of gravity: definitions ii.302, ii. 350-1, ii. 430.Ceria Arlstotelica ii. 531.Chalcidiua ii. 242, 244.Chaldaeans : measurement of anglesby ells ii 2 1 5- 1 6 . : order of planetsii. 242.Charmandrus ii. 359.Chaslcs, M. ii.19, 20: on Porisms435-7,ii.419.Chords, Tables of, 45, ii. 257, ii.259-60.Chvysippusl79: definitionofunit69.Cicero 144, 359,ii. 17, 19.Circle: division into degrees ii.214-15 :squaring of, 173, 220 35,Antiphon 221-2, Bryson 223-4,by Archimedes's spiral 225, 230-1, Nicomedes, Dinostratus, andquadratrix 225-9, Apollonius225, Carpus 225 ; approximationsto TT 124, 232-5, ii. 194, ii. 253,ii. 545.Cissoid of Diodes 264-6.Clausen, Th. 200.Cleanthes ii. 2.1Cleomedes: 'paradoxical eclipse 6:De motu circulars ii. 235-8, 244.Cleonides 444.Cochlias 232, ii. 193.

574 ENGLISH INDEX//.,1Cochloids 238-40 : sister of cochloid'case in Diopbantus ii. 465, ii. 512. Diophantus of Alexandria : date ii.4Curtze, M. 75 ii. 309.225, 281-2.Cyrus 129.Coins and weights, notation for, 31. Dactylus, l/24th of ell, ii. 216.Columella ii. 303.Damastes of Sigeum 177.Commandinus, F., translator of Damianus ii. 294.Euclid, 365, 425, Apollonius ii. Darius-vase 48-9.127, Analemma of Ptolemy ii. D'Armagnac, G. ii. 26.281,PlanJspherium ii.292, Heron's Dasypodiusii. 554 w.Pneumatica ii.308,Pappus ii. 36Q, De la Hire ii. 550.Serenus ii. 519.De lev! et ponderoso 445-6.Conchoid of Nicomedes 238-40. Decagon inscribed in circle, side of,Conclusion 370, ii. 533.416 : area of, ii. 328.Cone: Democritus on, 179-80, ii. Dee, John, 369, 425.110: volume of, 176, 180, 217, Definitions :Pythagorean 166 : in327, 413, ii. 21, ii. 332 : volume Plato 289, 292-4 : Aristotle on,of frustum ii. 334: division of 337: in Euclid 373: Definitionsfrustum in given ratio ii. 340-3. of Heron, ii. 314 16.Conic sections : discovered by Menaechmus252-3, ii. 110-16: Eu-359.ii.Demetrius of Alexandria ii. 260,clid's Conies and Aristaeus's Solid Democritus of Abdera 12, 119, 121,Aor/438, ii. 116-19: piopositions 182 : date 176, travels 177 :Aris-încluded in Euclid's Conies ii. totle's encomium 176 : list of121-2 (focus-directrix property works (1) astronomical 177, (2)243-4, ii. 119 21), conies in Archimedesmathematical 178 : on irrationalii. 122-6 : names due to lines and solids 156-7, 181 : onApollonius 150, ii. 138 Apollonius'sConies ii. 126-75 : conies cular sections of cone 179 80, ii.angle of contact 178-9: on cir-:in Fmymentum Bobienxe ii. 200- 110 : first discovered volume of203: in Anthemius ii. 541-3.cone and pyramid 176, 180, 217,Conon of Samoa ii. 16, ii. 35 ( J.ii. 21 : atoms mathematically divisibleConstruction 370, ii. 533.ad inf. 181 :'E*C7rT

ENGLISH448 : works and editions ii, 448-55: Arithmeticalb-lb: fractionsin, 42-4 : notation and definitionsii. 455-61 :signs for unknown (x)and powers ii. 456-9, for minusii. 459: methods ii. 462-79 : determinateequations ii. 462-5,484- 90 :indeterminate analysisii.466- 76, 491-514 : 'Poiisms' ii.449, 450, 451, ii. 479- 80 :propositionsin theory of numbers ii.481-4 :conspectus of Arithmeiicaii. 484-514 : On Polygonal NumfowlG,84,ii.514 17: 'Moriastica'ii. 449.DioptralS, ii.256 :" Heron's Dioptraii. 345-6.Division :Egyptian method 53,Greek 58 60 :example with sexagesimalfractions (Theon of Alexandria)59- 60.Divisions (of Fif/ures), On, by Euclid425-30 :similar problems inHeron ii. 3:iG 40.Dodecagon, area of, ii. 328.Dodecahedron :discovery attributedto Pythagoras or Pythagoreans65, 141, 158-60, 162 :early occurrence160: inscribed in sphere(Euclid) 418 19,iPappus)ii. 369 :Apollonius on, 419-20: volumeof, ii. 335.Donminus ii. 538.Dositheus ii. 34.Dufaciu, P. 446.Dupuis, J. ii. 239.Earth :measurement Kii. of, 82,(Eratosthenes) ii. 106-7, (Posidonius)ii. 220.Ecliptic : obliquity discovered byOenopidcs 174, ii. 244: estimateof inclination (Eratosthenes, Ptolemy)ii. 107-8.Eci hantus 317, ii. 2.Edi'u, Temple of Horus 124.Egypt : priest s 4- 5, 8- 9 : i elationswith Greece 8; origin of geometryin, 120-2: orientation of temples122.Êgyptian mathematics : numeralsystem 27-8, fractions 28, multiplication,&c. 14-15, 52-3: geometry(mensuration) 122-8: triangle(3, 4, 5) right-angled 122,147: value of TT124,125: measure-INDEX 575ment oi pyramids 126-8: maps(regional) 139 :algebra in PapyrusRhlnd, &c. ii. 440-1.Eisenlohr, A..123, 126, 127.Eisenmann, H. J. ii. 360.Elements : as known to Pythagoreans166-8: progress in, downto Plato 170-1, 175-6,201-2, 209-13, 216-17 : writers of Elements,Hippocrates of Chios 170-1, 201-2, Leon, Theudius 320-1 : othercontributors to, Leodamas, Archytas170, 212-13, Theaetetus209-12, 354, Hermotimus of Colophon320, Eudoxus 320, 323-9,354: Elements of Euclid 357 419:the so-called 4 'Books XIV, XV419-21.Ell, as nieasuie of angles ii. 215-16.Empedocles on Pythagoras 65.:Enestiom, G. ii. 341-2.Enneagon Heron's measurement:of side ii. 259, of area ii. 328 9.Exanthema of Thymaridas (systemof simple equations) 94 : othertypes reduced to, 94-6.Equations : simple, in PapyrusRhind, &c. ii. 441 : in epcttithentaof Thymaridas and in laniblichus94 6 : in Greek anthologyii.441-3: indeterminate, see Indeterminate:Analysis see /,voQuadratic, Cubic.Eratosthenes ii. 1, 16: date, &c.ii. 104: siere (mMTKivttir) for findingprimes 16, 100, ii. 105 : on duplicationof cube 244- 6, 251, 258- 60 :theP/d/o/f /nut ii. 104-5 : On Mean*ii. 105 6, ii. 359 : Measurement ofearth ii. 106-7, ii.242,ii. 346 :astronomy ii. 107-9 ^chronology:and Gt'oymphicaii. 109: on Octafterisib.Erycinus ii. 359, 365-8.Euclid 2-3,93, 131 :iJate,&e. 354-6: stories of, 25, 354, 357: relationto predecessors 354, 357 :Pappus on, 356-7.Arithmetic: classification anddefinitions of numbers 72-3, 397,* *perfect numbers 74, 402 : formulafor right-angled trianglesin rational numbers 81-2, 405.Conic* 438-9, ii. 121-2, focusdirectrixpropertyii. 119-21 : onii. 125.ellipse 439, ii. Ill,

576 ENGLISH;Data 421-5, Divisions (of Exhaustion, method of, 2, 176, 202,figures) 425-30, ii. 336, 339.217, 222, 326, 327 9 :developmentof, by Archimedes 224,Elements: text 360-1, Theon'sii.edition 358, 360, ii. 527-8, translationby Boetius 359, Arabic35-6.translations 362, ancient commentaries358-9, editio princeps 441 in :False hypothesis: Egyptian use ii.Diophantus ii. 488, 489.of Greek text 360, Greek texts of Fermat, P. 75 n., ii. 20, ii. 185, ii.Gregory, Peyrard, August, Heiberg360-1 Latin :translations, 435.454, ii. 480, ii. 481-4 : on PorismsAthelhard 362-3, Gherard 363, Fontenelle ii. 556.Campanus 363-4, Commanclinus Fractions :Egyptian (submultiples365: first printed editions, Ratdolt364-5, Zamberti 365: first tems 42-4 : Greek notation 16. :except $) 27 8, 41: Greek sys-introduction into England 363 sexagesimal fractions, Babylonian29, in Greek 44-5.:first English editions, Billingsley,*&o. 369-7*0: Euclid in Middle Friendly ' numbers 75.Ages 365-9, at Universities 368 -9: analysis of, 373-419: arrangementof postulates and axioms GeeponicuSi Liber, 124, ii. 309, ii.Galilei 344, 446.361 : 1. 47, how originally proved 318, ii. 344.147-9 :parallel-postulate 358, Geminus 119, ii. 222-34 : on arithmeticand logistic 14 : on divi-375, ii. 227-30, ii. 295-7, ii. 534 :so called 'Books XIV, XV 419- sions of optics, &c. 17-18 : on21.original steps in proof of Eucl. IMechanics 445-6 : Music 444- 32, 135-6: on parallels 358:5, Sectio canon is 17, 90, 215, attempt to prove parallel-postulateii. 227-30 : on original way444-5: 0/>tfc$17-18,441-4: Phaenomena349, 351-2, 440-1, ii. of producing the three conies249: Potisms 431-8, lemmas to,ii. Ill :encyclopaedic work onii. 419-24: Pseudana 480-1 : Surface-Loci243-4, 439-40, lemmas donius's Meteorologicamathematics ii. 223-31 : onPosi-ii. 231-2 :to, ii. 119-21, ii. 425-6.Introduction to Phaenomena ii.Eudeuius 201, 209, 222: History of232-4.Geometry 118, 119, 120, 130, 131, Geodesy = (yeo>8auriu) mensuration135, 150, 171 : on Hippocrates's (as distinct from geometry) 16-17.lunes 173, 182, 183-98: Histonj Geometric mean, defined (Archytas)of Astronomy 174, 329, ii. 244. 85 : one mean between twoEudoxus 24, 118, 119, 121, 320, squares (or similar numbers), two322-4: new theory of propoition between cubes (or similar solid(that of Eucl. V. ii) 2, 153, 216, numbers) 89-90, 112, 201, 297,325-7 : discovered method of exhaustion2, 176, 202, 206, 217, consecutive numbers 90, 215.400 : no rational mean between;222, 326, 327-9 :problem of two Geometrical harmony '(Philolaus'smean proportionals 245, 246, 219 name for cube) 85-6.51 : discovered three new means Geometry origin in Egypt 120-2 : :86: 'general theorems 1 323-4:secondary educationOn speeds, theory of concentric feometryin 0-1.spheres 329-34, ii. 244 : Phaenomenaand Mirror 322.546.Georgius Pachymeres ii. 453, ii.Eugeniue Siculus, Admiral, ii. 293. Gerbert (Pope Sylvester II) 365-7 :Euler, L. 75n., ii.482, ii.483.geometry of, 366 : ii. 547.Euphorbus (= Pythagoras) 142. Gerhardt, C. J. ii. 360,ii. 547.Eurytus 69.Gherard of Cremona, translator ofEutocius 52, 57-8, ii. 25, ii. 45, ii, Euclid and an-Nairm 363, 367,126, ii. 518, ii. 540-1. ii. 309 : of Menelaus ii. 252,ii. 262.

ENGLISH INDEX 577Ghetaldi, Marino, ii. 190.Girard, Albert, 435, ii. 455.Gnomon :history of term 78-9 :gnomons of square numbers 77-8, of oblong numbers 82-3, ofpolygonal numbers 79 in :applicationof areas 151-2 : use byal-Karkhi 109-10 : in Euclid 379 :sun-dial with vertical needle 139.Gomperz, Th. 176.Govi, G. ii. 293 n.Gow, J. 38.Great Year, of Oenopides 174-5,of Callippua and Democritus 177.Gregory, D. 360-1,440, 441, ii. 127.Griffith, F. LI. 125.Giinther, S. ii. 325 w., ii. 550.Guldin's theorem, anticipated byPappusii. 403.Halicarnassus inscriptions 32 - 3,34.llalley, E., editions of Apollonius'sConies ii. 127-8, and Srctio rationixii. 175, 179, of Menelaus ii.252, ii. 262, of extracts fromPappus ii. 360, of Serenns ii. 519.Hal ma, editor of Ptolemyii. 274,275.Hammer-Jen sen, I. ii. 300 n., ii.304 fi.Hankel, II. 145, 149, 288, 369, ii.483.Hardy, G. II. 280.Harmonic mean (originally 'biibcontrary')85.Harpedonaptac, rope-stretchers ''121-2, 178.Hfirun ar-Rashid 362.Han - calculations( Egyptian)ii.440-1.Hecataeus of Miletus 65, 177.Heiben, J. L. 233 w.Heiberg, J. L. 184, 187 n., 188,192 w., 196-7., 315, 361, ii. 203,ii.309, 310, 316, 318, 319, ii. 519,ii. 535, 543, 553, 555 n.llelcephlll.Hendecagon in a circle (Heron) ii.259,ii. 329.Henry, C. ii. 453.Heptagon in a circle, ii. 103 :Heron's measurement of, ii. 328.Heraclides of Pontus 24, ii. 231-2 :discovered rotation of earth aboutaxis816-17,ii.2 3, and that Venusand Mercury revolve about sun312, 317, ii. 2, ii. 244.Heraclitus of Ephesus 65.Heraclitus, mathematician ii. 192,ii. 359,ii. 412.Hermannus Secundus ii. 292.Hermesianax 142 n., 163.Hermodorus ii. 359.Hcrmotimus of Colophon 320-1 :Elements and Loci ib., 354.4Horodianic ' (or ' Attic ')numerals30-1.Herodotus 4, 5,48, 65, 121, 139.Heron of Alexandria 121, ii. 198,ii. 259 : controversies on date ii.298-307 : relation to Ctesibiusand Philon ii. 298-302, to Pappusii. 299-300, to Posidonius andVitruvius ii. 302-3, to ayrimensoresii. 303, to ii.Ptolemy 303-6.Arithmetic: fractions 42-4, multiplications58, approximation tosurds ii. 51, ii. 323-6, approximationto cube root 64, ii. 341 7 2,quadratic equations ii. 344, indeterminateproblems ii. 344,444-7.Character of works ii. 307-8 :list of treatises ii. 308-10.Geometry ii. 31 0-1 4, Definitionsii. 314-16: comni. on Euclid'sElements 358,ii. 310-14 :proof offormula for area of triangle interms of sides ii. 321-3 :duplicationof cube 262-3.Metrica ii. 320-43: (1) mensurationii. 316-35: trianglesii.320-3, quadrilateralsii. 326,regular polygons ii. 326-9, circleand segmentsii. 329-31 volumes:ii. 331-5, fropucTKos ii. 332-3, frustumof cone, sphere and segmentii. 334, tore ii. 334-5, five regularsolids ii. 335. (2) divisions offigures ii. 336-43, of frustum ofcone ii. 342-3.Mechanics ii. 346-52 : on Archimedes'smechanical works ii.23-4, on centl-eof gravityii. 350-1,352.Belopoel'ca 18, ii. 308-9, Catoptrica18, ii. 294, ii. 310, ii. 352-4.Dioptra ii. 345-6, Pneitmaticaand Automata 18, ii. 308, 310.On Water-clocks ii. 429, ii. 536.Heron, teacher of Proclus ii. 529.

578 ENGLISH INPEXHeron the Younger'ii, 545.Heronas 99.Hicetas 317.Hierius 268,ii. 359.BKeronymus 129.Hilal b. Abi Hilal al-Himsi ii. 127.Hiller, E. ii. 239.Hilprecht, H. V. 29.Hipparchus ii. 3, 18, 198, 216, 218 :date, &c. ii. 253 : work ii. 254-6 :on epicycles and eccentrics ii.243, ii. 255 : discovery of precessionii. 254: on mean lunar monthii. 254-5 :catalogue of stars ii.255 :geography ii. 256 : trigonometryii. 257-60, ii. 270.Bippasus 65, 85, 86, 214: constructionof 'twelve pentagons insphere ' 160.Hippias of Elis: taught mathematics23: varied accomplishmentsib.j lectures in Sparta 24 :inventor ofquadratrix 2, 171, 182,219, 225-6.Hippocrates of Chios 2, 182, 211:taught for money 22 first writer:of Elements 119, 170, 171 : elementsas known to, 201 - 2 :assumes fcvcm equivalent to solutionof quadratic equation 88,195-6 : on quadratures of lunes170, 171, 173, 182, 183-99, 220,221: proved theorem of Eucl. XI I2, 187, 328 : reduced duplicationof cube to problem of findingtwo mean proportionals 2, 188,200, 245.Hippolytus: on irvQucvcs (bases) and'rule of nine 'and ' seven 1 115-16.Hippopede of Eudoxus 338-4.Hornet 5.*Horizon ' : use in technical sense byEuclid 352.Horsley, Samuel, ii. 190, ii. 360.Hultsch, F. 204, 230, 349, 350, ii. 51,ii. 308, ii. 318, 319,ii. 361.Hunrath, K. ii. 51.Hunt, A. S. 142.Hypatia ii. 449, ii. 519, ii. 528-9.Hypotenuse, theorem of square on,142, 144-9 : Proclus on discoveryof, 145 :supposed Indian origin145-6.Hypsicles author of so-called BookXIV :of Eucl. 419-20, ii. 192 definitionof 'polygonal number' :84,ii. 213, ii. 515 : 'Ap

Jan, C. 444.Joachim Camerarius ii. 274.Joachim, H. H. 348 n.Johannes de Sacrobosco 368.Jordanus Nemorarius ii. 328.Jourdain, P. E. B. 283 n.Kahun Papyri 125, 126.Kant 173.Keil, B. 34-5.Kepler ii. 20, ii. 99.KCchly, H. A. T. ii. 309.Koppa (9 for 90)= Phoenician QophQOKubitschek, W. 50.Lagrange ii. 483.Laird, A. G. 306 n.Laplace 173.Larfeld,W. 31 n., 33-4.Lawson 486.Leibniz 279, ii. 20.Lemma 373, ii. 533.Leodamas of Thasos 120; 170, 212,291, 319.Leon 319.Leon (of Constantinople)ii. 25.Leonardo of Pisa 367, 426, ii. 547.Lepsius, C. R. 124.Leucippus 181.Libri, G. ii. 556.'Linear ' (of numbers) 73.1Linear* loci and problems 218-19.Lines, classification of, ii. 226.Livyii. 18.Loci : classification of,218-19, plane,solid, linear 218: loci on surfaces219: 'solid loci 'ii. 116-19.Loftus, W. K. 28.Logistic (opp. to 'arithmetic'),science of calculation 13-16, 23,53.Loqistica speciosa and mimerosa(Vieta)ii. 456.Loria, G. iv-v, 350 n., ii. 293 n.Luca Paciuolo 367, ii. 324 n.Lucas, E. 75 n.Lucian 75 H., 77, 99, 161,ii. 18.Lucretius 177.ENGLISH INDEX 579Manus, for number 27.Marinus 444, ii. 192, ii. 537-8.Martianus Capella 359, 365.Martin, T. H. ii. 238, ii. 546.-Maslama b. Ahmad al-MajrlÎii.292.Massalia 8.Mastaba tombs 128.Mathematics: meaning 10-11, classificationof subjects 11-18 :branches of applied mathematics17-18 : mathematics in Greekeducation 18-25.Maurolycusii. 262.Means :arithmetic, geometric, andBubcontrary (harmonic) knownin Pythagoras's time 85 defined:by Archytas ib. fourth, :fifth, andsixth discovered, perhaps by Eudoxus86, four more by Myonideaand Euphranor 86: ten meansin Nicomachus and Pappus 87-9,Pappus's propositions 88-9 no:rational geom. mean between successivenumbers (Archytas) 90,215.Mechanics, divisions of, 18 : writerson, Archytas 213, Aristotle 344-6,445-6, Archimedes ii. 18, ii.23-4,ii. 75-81, Ptolemy ii. 295, Heron ii.346-52, Pappusii. 427-34.Megethionii. 360.Memus, Johannes Baptista,ii. 127.Menaechmus 2, 25, 251-2, 320-1 :discoverer of conic sections 251-3, ii. 110-16 solved problem of:two mean proportionals 245, 246,251-5: on 'problems '318.Menelaus of Alexandria ii. 198, ii.252-3 :date, &c. ii. 260-1: Table ofChords ii. 257: Sphaericaii. 261-73 : Menelaus's theorem ii. 266-8, 270 : anharmonic propertyii.269 :7rapd8oos curve ii. 260-1.Mensa Pythagorea 47.Mensuration in :primary education19 in :Egypt 122-8 : in Heron ii.316-35.Meton 220.Metrodorus ii. 442.Magic squaresii. 550.Minus, sign for. in Diophantus ii.Magnus, Logistica 234-5.459-60.Mamercus or Mamertius 140, 141, Mochus 4.171.Moschopoulos, Manuel, ii. 549-50.al-Ma'mun, Caliph 362.Muhammad Bagdadinus 425.al-Mansur, Caliph 362.Multiplication: Egyptian methodpp 2

530 ENGLISH INDEX52-3, Greek 53-4, 'Russian' 53 n.:examples from Eutocius, Heron,Theon 57-8: Apollonius's continuedmultiplications 54-7.Multiplication Table 53.Jtfumm, an angular measure ii. 215.Musical intervals and numericalratios 69, 75-6, 85, 165.Myriads, ' first ',*second ', &c., notationfor, 39-40.Nagl, A. 50.an-Nairizi : comm. on Euclid 363,ii. 224, ii. 228-30, ii. 309-10.NasTraddin at-Tusi: version of Euclid362, of Apollonius's Coniesii. 127 : of Ptolemyii. 275.Naucratis inscriptions 33.Nemesius 441.Neoclides 319.Ner (Babylonian) (=600) 28,ii.215.Nesselmann, G. H. P. ii. 450-1, ii.455-6.Newton 370,ii. 20,ii. 182.Nicolas Rhabdas 40, ii. 324 n , ii.550-3.Nicomachus of Gerasa 12, 69, 70,72, 73, 74, 76, 83, 85, 86, ii. 238,ii. 515: works of, 97: Introductioarithmetical character of treatise98-9, contents 99-112, classificationof numbers 99-100: on 'perfect1 numbers 74, 100-1 : on tenmeans 87 : on a 'Platonic' theorem297 : sum of series ofnatural cubes 109-10.Nicomedes 225-6,ii. 199: cochloidsor conchoids 238-40: duplicationof cube 260-2.Njloxenus 129.Nine, rule of, 115-16: casting outnines ii. 549.Nipsus, M. Junius, 132.Nix, L. ii.128, 131, ii. 309.Noel, G. 282.Number: defined, by Thales 69, byModeratus, Eudoxus, Nicomachus,Aristotle 70 : classification'of numbers 70-4 :perfect ','over-perfect* and ' defective 1*numbers 74-5, friendly' 75,figured 76-9 :'oblong 1 , 'prolate 182-3, 108, 114, similar plane andsolid numbers 81-2, 90, solid'numbers classified 106-8 : thenumber in the heaven '(Pythagorean)68, 'number' of an object69.Numerals: systems of, decimal, quinary,vigesimal 26 : origin ofdecimal system 26-7 :Egyptian27-8; Babylonian systems (1)decimal 28, (2) sexagesimal 28-9:Greek (1) 'Attic' or 'Herodianic'30-1:(2) alphabetic system,original in Greece 31-7, howevolved 31-2, date of introduction33-5, mode of writing 36-7,comparison of two systems 37-9 :notation for large numbers, Apollonius'stetrads 40, Archimedes'soctads 40-1.Nyuiphodorus 213.'Oblong' numbers 82-3, 108, 114:gnomons of, 82-3.Ocreatus, 111.Octads, of Archimedes 40-1.Octagon, regular, area of, ii. 328.Octahedron 159, 160, 162 : volumeof, ii. 335.'Odd* number defined 70-1: 1called * odd-even ' 71 :'odd-even','odd-times-odd ', &c., numbers71-4.Oenopides of Chios 22, 121 : discoveredobliquity of ecliptic 138,174, ii. 244: Great Year of, 174-5 :called perpendicular gnomon-wise78, 175 : two propositions in elementarygeometry 175.Olympiodorus 444.One, the principle of number 69.ii.Oppermann 324 n.Optics divisions of,17-18 : : of Euclid441-4 : of Ptolemyii. 293-4.Oval of Cassini ii. 206.Oxyrhynchus Papyri 142.Pamphile, 131, 133, 134.Pandrosion ii. 360.Pappus (see also Table of Contents,underChap.XIX)ii.l7-18,ii. 175,180, 181, 182, 183, 185, 186, 187,188, 189, 190, ii.207, 211, 212,213, ii. 262, ii. 337, ii. 355-439 :on Apollonius's tetrads 40, onApollonius's continued multiplications54-7 : on ten means87-9 : on mechanical works ofArchimedes ii. 23-4 : on conies

ENGLISH INDEX 581'of Euclid and Apollonius 438,proof of focus-directrix propertyii. 120-1 :commentary on Euclid358, ii. 356-7, on Book X 154-5,209, 211, ii. 193 : commentary onEuclid's Data 421-2, ii. 357, onDiodorus's Analemma- ii. 287,scholia on Synfaxfsii. 274 : onclassification of problemsandloci (plane, solid, linear) 218-19,ii.117- 18, criticism on Archimedesand Apollonius 288, ii.68, ii.167:on surface-loci 439-40, ii. 425-6 :on Euclid's Porisnts 431-3, 436-7,ii. 270, ii. 419-24 on ' Treasury:1of Analysis 421, 422, 439, ii. 399-427: oncoc/tfo/ 112,

582 ENGLISH INDEX201,297, 400: on * perfect ' proportion86 a : proposition inproportion 294: two geometricalpassages in Meno 297-303 :propositions'on the section* 304,324-5.'Platonic* figures (the regularsolids) 158, 162, 294-5, 296-7.Playfair, John, 436.Pliny 129, ii. 207.Plutarch 84, 96, 128, 129, 130, 133,144, 145, 167, 179, ii. 2, 3, ii. 516:on Archimedes ii. 17-18.Point: defined as a 'unit havingposition ' 69, 166 : Plato on points293.Polybius 48, ii. 17 n. f ii. 207.Polygon: propositions ahout sumof exterior or interior angles- 144:measurement of regular polygonsii. 326-9.Polygonal numbers 15, 76, 79, ii.213, ii. 514-17.Polyhedra, see-Solids. **Porism (1)= corollary 372: (2) acertain type of proposition 373,431-8 : Porisws of Euclid, seeEuclid of :Diophantus, see Diophantus.Porphyry 145: commentary on Euclid'sElements 358, ii. 529.Poselger, F.T. ii.455.Posidonius ii. 219-22 : definitionsii.221, 226 ;on parallels 358, ii.228 : versus Zeno of Sidon ii.221-2 :Meteorologicaii. 219 :measurement of earth ii. 220 : onsize of sunii. 108, ii. 220-1.Postulates : Aristotle on, 336 : inEuclid 336, 374-5 : in Archimedes336,Powers, R. E. 75 n.Prestet, Jean, 75 n.Prime numbers and numbers primeto one another 72-3: defined 73:2 prime with Euclid and Aristotle,not Theon of Smyrna and Neo-Pythagoreans ib.Problems : classification 218-19 :ii. 75.plape and solid ii. 117-18 :problemsand theorems 318, 431, ii.583.Proclus 12, 99, 175, 183, 213, 224 .,ii. 529-37 : Comm. on Eucl. I. ii.'530-5 : sources ii. 530-2 : summary'118-21, 170, object of, 170-1 : on discoveries of Pythagoras84-5, 90, 119, 141, 154 : on EuclidI. 47, 145, 147: attempt to proveparallel-postulate 358, ii. 534 : onloci 219 : on porisms 433-4 : onEuclid's music 444: comm. onRepublic 92-3, ii. 536-7 : Hypotyposisof astronomical hypothesesii. 535-6.Prodicus, on secondary education20-1.Prolate, of numbers 108, 204.P/>/370, ii.533.Proportion : theory discovered byPythagoras 84-5, but his theorynumerical and applicable to commensurablesonly 153, 155, 167:def. of numerical proportion 190:the perfect ' proportion 86 :'Euclid's universally applicabletheory clue to Eudoxus 153, 155,216,325-7."Proposition, geometrical : formaldivisions of, 370-1.Protagoras 202 : on mathematics23, 179.Prou, V. ii. 309.Psammites or Sand-reckonerofArchimedes40, ii. 3, ii. 81-5.Psellus, Michael, 223-4 w., ii. 453,ii. 545-6.Pseudaria of Euclid 430-1.Pseudo-Boetius 47.Pseudo-Eratosthenes: letter on duplicationof cube 244-5.Ptolemies :coinst)f,with alphabeticnumerals 34-5: Ptolemy I, storyof, 354.Ptolemy, Claudius, 181, ii. 198, ii.216, ii. 218, ii. 273-97: sexagesimalfractions 44-5, approximationto TT 233 :attempt to proveparallel-postulate 358, ii. 295-7 :Syntazisii. 273-86, commentariesand editions ii. 274-5, contentsof, ii. 275-6, trigonometry in, ii.276-86, 290-1, Table of Chordsii. 259, ii. 283-4, on obliquity ofeclipticii. 107-8 : Analemmaii. 286-92: Planispheriumii.292-3, Opticsii. 293-4, other works ii.293 :ircpi pon&v ii. 295 : rrepl 8iaoTttcr*a>$ ib.Pyramids : origin of name 126 :measurements of, in Bhind Papyrus126-8: pyramids of Dakshur,

ENGLISH INDEX 583Gizeh, and Medum 128: measurementof height by Thales 129-30 167, 216: construction of155,; regular pentagon 160-2 astronomicalsystem (non-geocentric):volume of pyramid 176, 180, 217,ii. 21, &c., volume of frustum ii. 163-5 : definitions 166 : on order334.of planetsii. 242.Pythagoras 65-6, 121, 131, 133, 138:travels 4-5, story of bribed pupil Qay en heru, height (of pyramid)24-5: motto 25, 141: Heraclitus, 127.Empedocles and Herodotus on, Quadratic equation solved by Pythagoreanapplication of areas:65 : Proclus on discoveries of, 84-5,90,119,141,154: made mathematicsa part of liberal education merical solutions ii. 344, ii. 448,150-2, 167, 394-6, 422-3: nu-141, called geometry * inquiry 1ii. 463-5.166, used definitions 166 : arithmetic(theory of numbers) 66-80, 225-30, ii. 379-82.QuadraMx 2, 23, 171, 182, 218, 219,figured numbers 76-9 :gnomons Quadrivium of Pythagoreans 11.77, 79: * friendly 1 numbers 75: Quinary system of numerals 26.formula for right-angled trianglesin rational numbers 79- Qusta b. Luqa, translator of EuclidQuintilian ii. 207.80 : founded theory of proportion 362, ii. 453.84-5, introduced 'perfect* proportion86 discovered dependenceof musical intervals on Ratdolt, Erhard, first edition ofRangabe, A. R. 49-50.:numerical ratios 69, 75-6, 85, Euclid 364-5.165: astronomy 162-3, earth Reductio ad absurdum 372 :spherical ?&., independent movementof planets 67, 163: Theorem Reduction (of a problem) 372.of Pythagoras 142, 144-9, how Reflection: equality of angles ofalreadyused by Pythagoreans 168.discovered? 147-9, general proof, incidence and reflection 442, ii.how developed i&., Pappus's -extensionii. 369-71.Refraction 6-7, 444 : first attempt294, ii. 353-4.Pythagoreans 2, 11, 220: quadrivium11 : a Pythagorean first Regiomontanus 369, ii. 27, ii. 453-4.at a law (Ptolemy)ii. 294.taught for money 22 : first to Regula Nicomachi 111.'advance mathematics 66 : allthings are numbers Rhabdas, Nicolas, 40, ii. 324 n., ii.* '67-9 : number' '550-3.of an object 69, number in Rhind Papyrus: mensuration in,the heaven 168: figured numbers 122-8: alffebra in, ii. 440-1.69 : definition of unit 69 : 1 isRight-angled triangle inscribed:odd-even 71 : classification of by Thales in circle 131 : theorem'numbers 72-4 :friendly numbers75: 10 the 'perfect number75: Pythagoras 142, 144-5, supposed'of Eucl. I. 47, attributed to1oblong numbers 82-3, 108, Indian origin of, 145-6.114 : side- and diameter- numbersgiving approximations to V2, Right-angled triangles in rational91 numbers :Pythagoras's formula3 : first cases of indeterminate 80, Plato's 81, Euclid's 81-2,analysis 80, 91, 96-7: sum ofangles of = 405: triangle (3, 4, 5) known totriangle 27?, 135, Egyptians 122: Indian examples143 :geometrical theorems attributedto, 143-54 : invented appli-146 :Diophantus's problems on,ii. 507-14.cation of areas and geometrical Robertson, Abram,ii. 27.algebra 150-4: discovered the incommensurable65, 90-1, 154, Rodolphus Pius ii. 26.Rodet, L. 234.with reference to \/2 155, 168: Roomen, A. van, ii. 182.theory of proportion only applicableto comniensurables 153, Rudio, F. 173, 184, 187-91, ii.539.

684 ENGLISH INDEXRudolph of Brugesii. 292.Ruelle, Ch. tm. ii. 538.Rttstow, F. W. ii. 309.Ruler - and - compasses175-6.restrictionSachs, Eva, 209 n.Salanrinian table 48, 50-1.Salmon ii. 23, ii. 103.Sampi = ("^ 900) derived fromSsade q.v.Sar (Babylonian for 60 2 ) 28, ii. 215.Satapcttha Brdhmana, 146.Savile, Sir H., on Euclid 360, 369.Scalene: of triangles 142: of certainsolid numbers 107: of an oddnumber (Plato) 292: of an obliquecone ii. 134.Schiaparelli, G. 317, 330, ii. 539.Schmidt, W. ii. 308, 309, 310.Schflne, H. ii. 308.SchCne, R. ii. 308, 317.Scholiast to Charmides 14, 53.Schooten, P. van, 75 n., ii. 185.Schulz, 0. ii. 455.Scopinasii. 1.Secondary numbers 72.Sectio canonis 17, 215, 444.Seelhoff, P. 75 n.Seleucus ii. 3.Semicircle :angle in, is right(Thales) 131, 133-7.Senkereh, Tables 28, 29.Senti, base' (of pyramid) 127.Se-qet, * that which makes th e nature '(of pyramid= cot an ) gent of angleof slope 127- 8, 130, 131.Serenus ii. 519-26: On section ofcylinder ii. 519-22, On section ofcone ii. 522-6.Sesostris (Ramses II) 121.Sexagesimal system of numeralsand fractions 28-9: sexagesimalfractions in Greek 44-5, 59, 61-3,233, ii. 277-83.Sextius 220.Sicily 8.4Side- 1 and 'diameter-numbers 1 91-3, 112, 153, 308, 380, ii. 536.Simon, M. 200.Simplicius extract from Eudemus:on Hippocrates's quadrature oflunes 171, 182-99 : on Antiphon221-2 : on Eudoxus's theory ofconcentric spheres 329 commentaryon Euclid 358, ii. 539-40 : :onmechanical works of Archimedesii. 24: ii. 538-40.Simson, R., edition of Euclid'sElements 365, 369, and of Euclid'sData 421 : on Euclid's Porisms435-6: restoration of Plane Lociof ii.Apollonjus 185, ii. 360.Simus of Posidonia 86.Sines, Tables of, ii. 253, ii. 259-60.Sinus rectus, sinus versus 367.Sluse, R. F. de, 96.Smith, D. E. 49, 133w.'Solid' loci and problems 218, ii.117-18: Solid Loci of Aristaeus438, ii. 118-19.1Solid ' numbers, classified 106-8.Solids, Five regular:discovery attributedto Pythagoras or Pythagoreans84, 141, 158-60, 168,alternatively (as regards octahedronand icosahedron) to Theaetetus162: all five investigatedby Theaetetus 159, 162, 212, 217:Plato on, 158-60: Euclid's constructionsfor, 415-19 :Pappus'sconstructions ii. 368-9 : contentof, ii. 335, ii. 395-6.Solon 4, 48.Sophists : taught mathematics 23.Sosigenes 316, 329.Soss = sussu = 60 (Babylonian) 28,ii. 215.Speusippus 72, 73, 75, ii. 515: onPythagorean numbers 76, 318 :ôn the five regular solids 318 on:theorems ib.Sphaerlc 11-12: treatises on, by Autolycusand Euclid 348-52, 440-1 : earlier text-book presupposedin Autolycus 349-50 Sphaerica:of Theodosius ii. 245, 246-52, ofMenelaus ii. 252-3, 260, 261-73.Sphere-making 18: Archimedes on,ii. 17-18.Spiric sections ii. 203-6.Sporus 226 criticisms on :quadrn-Mx 229-30 :Krjpin 234 : duplicationof cube 266-8.Square root, extraction of, 60-3:ex. in sexagesimal fractions(Theon) 61-2, (scholiast to Euclid)63 : method of approxima-^ting to surds ii. 51-2, ii. 323-6,ii. 547-9, ii. 553-4.Square numbers 69 : formation byadding successive gnomons (odd

ENGLISH INDEX 585numbers) 77 :any square is sumof two triangular numbers 83-4 :8 times a triangular number-f 1 square, 84, ji.516.Ssade, Phoenician sibilant (signsT A m T) became "^ (900) 32.'Stadium/ l/60th of 30,ii. 215.Stadium of Zeno 276-7, 281-3.Star-pentagon, or pentagram, ofPythagoreans 161-2.Stereographic projection (Ptolemy)ii. 292.S. ii. 455.Steyin,*Stigma,' name for numeraloiiginally C (digamma) 32.5",Strabo 121, ii. 107, ii. 220.Strato ii. 1.Subcontrary (= harmonic) mean,defined 85.Subtraction in Greek notation 52.Surds: Theodorus on, 22-3, 155-6,203-9, 304: Theaetetus's generalization203-4, 205, 209, 304 : seealso * Approximations '.Surface-Loci 219, ii. 380-5 Euclid's:439-40, ii. 119, ii. 425-6.Silrya-Siddhanta ii. 253.tiit88u soss (Babylonian for 60) 28,ii. 215.Synesius of Cyreneii. 293.Synthesis 37 1-2: defined by Pappusii. 400.Syracuse 8.Table of Chords 45, ii. 259-60, ii.283.Taittiriya Samhitd 146.Tannery, P. 15, 44, 87, 89, 119, 132,180, 182, 184, 188, 196 w., 232,279, 326, 440, ii. 51, ii. 105, ii.204-5, ii. 215, ii. 218, ii. 253,ii. 317, ii. 453, ii. 483, ii. 519,ii. 538, ii. 545, 546, ii. 550, ii. 556,ii. 561.Teles on secondary education 21.Teos inscription 32, 34.Tetrads of Apollonius 40.Tetrahedron : construction 416, ii.368 : volume of, ii. 335.Thabit b. Qurra : translator of Euclid362, 363: of Archimedes'sLiber assumptorumii. 22 : ofApollonius's Conies V-VII ii, 127:of Menelaus's Elements of Geometryii. 260 : of Ptolemy ii.274-5.Thales 2, 4, 67 : one of Seven WiseMen 128, 142: introduced geometryinto Greece 128: geometricaltheorems attributed to, 130-7 : measurement of height ofpyramid 129-30, and of distanceof ship from shore 131-3 : definitionof number 69 :astronomy137-9, ii. 244: predicted solareclipse 137-8.Theaetetus 2, 119, 170: on surds22-3,155, 203-4, 205, 209, 304:investigated regular solids 159,162, 212, 217 : on irrationals 209-12, 216-17.Themistius 221, 223, 224.Theodorus of Cyrene taught mathematics22-3 : on surds 22-3, 155-:6, 203-9, 304.Theodosius ii. 245-6: SphaericaSW-50, ii. 246-52: other works ii.246 : no trigonometry in, ii. 250.Theologumena arithmetices 96, 97,318.Theon of Alexandria :examples ofmultiplication and division 58,59-60 : extraction of square root61-3: edition of Euclid's Elements360-1, ii. 527-8 of : Optics 441,ii. 528 :Catoptrica ib. : commentaryon Syntaxis 58, 60, ii. 274,ii. 526-7.Theon of Smyrna 12, 72, 73, 74, 75,76, 79, 83, 87, ii. 515 : treatiseof, ii. 238-44 on ' side- ' and:4diameter-numbers 1 91-3, 112:forms of numbers which cannotbe squares 112-13.Theophrastus 158, 163: on Plato'sview of the earth 315.Theudius 320-1.Theuth, Egyptian god, reputed in*ventor of mathematics 121.Thevenot, M. ii. 308.Thrasyllus 97, 176, 177, ii. 241, ii.243.Thucydidesii. 207.:Thymaridas definition of unit 69:1rectilinear = ' prime numbers72 :cTrdvQrujLn, a system of simpleequations solved 94.Tiinaeus of Locri 86,Tittel ii. 300, 301, 304.Tore (or anchor-ring) use :by Archytas219, 247-9: sections of(Perseus),ii. 203-6: volume of

586 ENGLISH INDEX(Dionysodorus and Heron),ii.218-19, ii. 334-5.Torelli, J. ii. 27.Transversal: Menelaus's theoremfor spherical and plane trianglesii. 266-70: lemmas relating toquadrilateral and transversal(Pappus)ii. 419-20.'Treasury of Analysis' 421, 422,439, ii. 399-427.Triangle: theorem about sum ofangles Pythagorean 135, 143,Geminus and Aristotle on, 135-6.Triangle, spherical called :rptTrXetpov(Menelaus) ii. 262 :tions proposi-analogous to Euclid's onplane trianglesii. 262-5 : sum ofangles greater than two rightii.angles 264.Triangular numbers 15, 69 : formation76-7 : 8 times triangularnumber -f 1 = a square 84, ii.516.Trigonometry ii. 5, ii. 198, ii. 257-9,ii. 265-73, ii. 276-86, ii. 290-1.Trisection of any angle solutions:235-44 :Pappus on, ii. 385-6.Tschirnhausen, E. W. v., 200.Tycho Brahe 317, ii. 2, ii. 196.Tzifra ii.(= 0) 547.Ukha-fhebt (side of base in pyramid)126,127.Unit : definitions (Pythagoreans,Euclid, Thymaridas, Chrysippus)69.Usener, H. 184, 188.Valla, G. : translator of extractsfrom Euclid 365, and from Archimedesii. 26.Venatorius, Thomas Gechauff: ed.princeps of Archimedes ii. 27.Venturi, G. ii. 308.Vieta 200, 223, ii. 182, ii.456, ii.480,ii. 557.Vigesimal system (of numerals) 26.Vincent, A. J. H. 50, 436, ii. 308,ii. 545, ii. 546.Vitruvius 18, 147, 174, 213, ii. 1,ii, 245: Vitruvius and Heron,ii. 302-3.Viviani, V. ii. 261.Vogt, H., 156 M., 203 n.Wescher, C. ii. 309.Wilamowitz - Moellendorff, U. v.,158 w., 245, ii. 128,Xenocrates 24, 319 : works onNumbers 319 :upheld 'indivisiblelines '181.Xenophon, on arithmetic in education19.Xylander (W. Holzmann) ii. 454-5,ii. 545.Yahya b. Khalid b. Barmak ii. 274.Zamberti, B., translator of Euclid365, 441.ZenoofElea 271-3: arguments onmotion 273-83.Zeno of Sidon on Eucl. I. 1. 859, ii.221-2.Zenodorus ii. 207-13.Zero in Babylonian notation 29 :in Ptolemy 39, 45.Zeuthen, H. G. 190, 206-9, 210-11,398, 437, ii. 52, ii. 105, ii. 203,ii. 290-1, ii. 405, ii. 444.Zodiac circle :obliquity discoveredby Oenopidcs 138, 174, ii. 244.

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