Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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evenlyeven numbers, and for not distinguishing one of these from evenlyeven numbers. And certainly, as Bullialdus observes, Euclid ought to have distinguished odd numbers from the unevenly-even, and these again from the evenly-even, which are confounded by him. P. 68. Chap6 A square number, &c.-In this and the two following chapters, the generation of squares, pentagons, hexagons, kc. is delivered. And in the first place it is necessary to observe, that another generation of squares is delivered by ancient authors, and is called by them %rao~o~, or, a circuitous course from the same to the same. For in this method, says Jamblichus on Nicomachus, "the monad becomes both the barrier and the goal in each composition. For from it the progression begins in the generation of every square, from the barriers as it were, as far as to the goal, which is the side of the square." Thus for instance in order to produce a square whose side is 4, let numbers be disposed from unity, as from the barriers of a Circus, as far as to 4, and from 4 again in a retrograde order as far as to unity, as below: For the sum of all these is 16, a square number, the side of which is 4. Thus too, in order to produce a square whose side is 5, numbers must be disposed from unity in a similar manner as follows: For the sum of these is 25, and consequently the side is 5. And so in all other instances. But numbers will be thus disposed, if a number is distributed into units, and in this distributed form multiplied into itself, as may be seen below:
For the purpose of facilitating the knowledge of the generation of polygonous numbers, the following Table is added: A Table of Polygons The Gnomons of Triangles . . . 1 2 3 4 5 6 7 8 9 10 11 12 Triangles collected . . . . . . . 1 3 6 10 15 21 28 36 45 55 66 78 The Gnomons of squares . . . . 1 3 5 7 9 11 13 15 17 19 21 23 Squares collected . . . . . . . . . 1 4 9 16 25 36 49 64 81 108 121 144 The Gnomons of Pentagons . . . 1 4 7 10 13 16 19 22 25 28 31 34 Pentagons collected . . . . . . . 1 5 12 22 35 51 70 92 117 145 176 210 The Gnomons of Hexagons . . . 1 5 9 13 17 21 25 29 33 37 41 45 Hexagons collected . . . . . . . . 1 6 15 28 45 66 91 120 153 190 231 270 The Gnomons of Heptagons . . . 1 6 11 16 21 26 31 36 41 46 51 56 Heptagons collected . . . . . . . 1 7 18 34 55 81 112 148 189 235 286 342 The Gnomons of Octagons . . . . 1 7 13 19 25 31 37 43 49 55 61 67 Octagons collected . . . . . . . . 1 8 21 40 65 96 133 176 225 280 341 408 The Gnomons of Enneagons . . . 1 8 15 22 29 36 43 50 57 64 71 78 Enneagons collected . . . . . . . 1 9 24 46 75 111 154 204 261 325 396 474 The Gnomons of Decagons . . . 1 9 17 25 33 41 49 57 65 73 81 89 &cagons collected . . . . . . . . 1 10 27 52 85 126 175 232 297 370 451 540 Jamblichus has extracted the following particulars respect-
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THEORETIC ARITHMETIC IN THREE BOOKS
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IV INTRODUCTION the mathematical di
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INTRODUCTION works, like the remain
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VIII INTRODUCTION fore, that the so
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INTRODUCTION the other of such as a
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XII INTRODUCTION must her inherent
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XIV INTRODUCTION principles existin
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XVI INTRODUCTION indissoluble perma
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XVIII INTRODUCTION mathematics to p
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XX INTRODUCTION utility it administ
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XXII INTRODUCTION may not energize
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INTRODUCTION tradition, however, of
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XXVI INTRODUCTION Nicomachus, which
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INTRODUCTION perplexity, and that h
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INTRODUCTION ever, requisite that w
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XXXII INTRODUCTION that he is ignor
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XXXIV INTRODUCTION tirely the prais
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THEORETIC ARITHMETIC BOOK ONE CHAPT
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that all motion is after rest, and
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since it is not possible for divisi
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dle. And these indeed are the commo
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monad is either even or odd. It can
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is denominated the evenly-odd, and
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terms should accord with its proper
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ural order, 1.3.5.7.9.11.13.15.17.
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agreeably to the form of the evenly
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12, 20, and 28, the sum of the extr
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numbers proceeding from them are re
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THE generation however and origin o
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ever will give the mode of measurin
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THE SIEVE OF ERATOSTHENES. Here 7 m
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pass the sum of the whole number, i
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CHAPTER XV. On the generation of th
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CHAPTER XVI. On relative quantity,
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consequent order, these even number
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For the sesquialter has for its lea
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CHAPTER XIX. That the multiple is m
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which 4 is, is compared with the ro
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number besides containing the whole
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superquadripartient, may likewise b
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to them, duple sesquiquartan ratios
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the impression of itself, defines a
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If, however, the multiples which ar
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tan, the superquadripartient ratio
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BOOK TWO CHAPTER I. How all inequal
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BOOK Two 57 reduced to equality. Fo
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BOOK Two 59 alters as much distant
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BOOK Two 61 ratio is composed from
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BOOK Two CHAPTER IV. On the qztanti
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BOOK Two 65 distended by a triple d
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These triangular numbers are genera
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BOOK Two 69 But in these numbers al
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BOOK Two 71 triangle of a class imm
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BOOK Two 73 22 is formed from the s
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BOOK Two 75 third, a pentagonal; th
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Two does not only not arrive at uni
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BOOK Two 79 to have four angles and
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BOOK Two 81 parity. Hence, in conse
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BOOK Two 83 the principles of thing
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BOOK Two 85 from the monad, are sai
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pared with the first number longer
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BOOK Two 89 and twice 16, has for i
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atios. Let them, therefore, be alte
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BOOK Two 93 ity. For 1 is the whole
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BOOK Two 95 ticipate of an immutabl
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BOOK Two 97 are opposite to those a
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BOOK Two is equal to the sum of the
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BOOK Two 101 will be the same multi
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BOOK Two 103 terms there is a great
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BOOK Two 105 from 8, and one side f
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BOOK Two 107 by the greater, by the
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BOOK Two 109 tertian ratio, or that
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BOOK Two 111 ratio, arises from a c
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BOOK Two 113 to 10 is a quadruple r
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BOOK Two 115 For here the greatest
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BOOK Two 117 by length, breadth, an
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BOOK Two The greatest harmonies. Pr
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BOOK Two 121 which is 18416, we mus
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BOOK Two former diameter, exceeds i
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BOOK Two In the geometrical fractio
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BOOK Two The parts of 51=17)
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BOOK Two 129 In the second place, i
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BOOK Two them; (see Chap. 15, Book
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BOOK Two 133 of this series into th
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Here it is evident in the first ins
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Farther still, if 2 be subtracted f
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24575, and 301989887. And the two i
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In like manner if 9437056 be divide
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These numbers are as follows: In th
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That the reader may see the truth o
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all perfectly as well as all imperf
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parts of these wholes have sometime
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these, and are called by modern mat
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The numbers, therefore, that form 1
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with each other, it will be found,
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PYTHAGORAS
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ples of these; referring the point
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If it be requisite, however, to spe
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the image of the never-failing and
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that they are separate, they are no
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27, but they signify through these
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dissimilarly similar to divinity. I
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anonymous author, in consequence of
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It is also Justice, and Isis, Natur
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adytum of god-nourished silence, as
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called Cupid for the reason above a
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the diapente, and the diatessaron.
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$Fin of natural effects. Because li
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er is not alone even, nor alone odd
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the vegetative life. And the seed i
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and beauty. And in external things,
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which are productive of life after
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one side, and 1 on the other, each
Page 232 and 233: The same writer also observes, "tha
Page 234 and 235: position; for I conjecture this to
Page 236 and 237: ~ c ~ ~ ~ Or ~ Q rather, ~ v . whic
Page 238 and 239: The ratio of 12 to 9 is sesquiterti
Page 240 and 241: monad. Thus if from each of the num
Page 242 and 243: had these appellations in consequen
Page 244 and 245: and well-ordered digression (from t
Page 246 and 247: unity is added to the product, the
Page 248 and 249: And so of the rest, which abundantl
Page 250 and 251: Again, of any two numbers whatever,
Page 252 and 253: Again, if 4 multiplies any triangul
Page 254 and 255: With respect to the hexad, it is th
Page 256 and 257: and the double of it 12, the number
Page 258 and 259: incorporeal and corporeal essence;
Page 260 and 261: of the hebdomad 3 and 4 necessarily
Page 262 and 263: composition of which the hebomad is
Page 264 and 265: it is also surpassed by the last nu
Page 266 and 267: excite the glad husbandmen to the c
Page 268 and 269: mentioned particulars, but also to
Page 270 and 271: 7X 1 and +1= 8 the 2nd 7 x 8 and +
Page 272 and 273: 8X 1 and +1= 9 the 2ndl 8X 7and +1=
Page 274 and 275: 10X 1 and +1= 11 the 2ndl i' 10x30
Page 276 and 277: tureX according to this number. In
Page 278 and 279: through a good allotment, or throug
Page 280 and 281: titude, and communion with itself,
Page 284 and 285: ing this table from the Epanthematc
Page 286: for instance, 12 is a number longer
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