Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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And so of the rest, which abundantly proves that the monad is causally every number. CHAPTER XIV On the properties of the duad, and triad. THE duad, as it is beautifully observed by Proclus in his Commentaries on Euclid, is the medium between the monad and number. For unity, says he, by addition produces more than by multiplication; but number, on the contrary, is more increased by multiplication than by addition; and the duad, whether multiplied into, or added to itself, produces an equal quantity. Hence some of the ancients derive its appellation from 701~ Buyat, because number begins from hence to enter into multitude. Hence too, it was called Rhea, which signifies a certain flux, because from hence the streams, i.e. the progressions of multitude are derived. Again, the duad, as Martianus Capella observes, is the mother of the elements; for 4 is the offspring of 2. And it is also the first form of the even number. The hebdomad likewise 64 which is both a square and a cube, is produced by a continued multiplication by 2 from unity, as we shall show when we discuss the properties of the number 7. As Theo of Smyrna also observes,' matter, every sensible nature, generation, motion, increase, composition, association and relation, subsist according to the duad. And lastly, as is evident from the preceding chapter, if 2 is multiplied by any triangular gnomon, and unity is added to the product, the sum will always be the gnomon of a square. Again, from the junction of the duad with the monad the triad is generated, which says Theo of Smyrna, is the first number that has a beginning, middle and end. He adds, * Mathemat. p. 156.
"Hence it is the first number of which the word all is enunciated. For we do not enunciate the all of the numbers less than three, but we say one, and both. By this number also we make compacts, manifesting by this that we request every good. We likewise call those that are in every respect miserable, thricewretched, and those that are in every respect blessed, thriceblessed. The first origin also of a superficies is from this number. For the first subsistence of it is in a triangle; and on this account there are three genera of a triangle, viz. the equilateral, the isosceles, and the scalene. There are likewise three species of angles, the right, the acute, and the obtuse. And the right angle indeed is defined by the nature of unity, and consists of the equal and the similar. Hence all right angles are equal to each other, being media between the acute and obtuse, the surpassing and the surpassed. But the other angles are in£ i- nite and indefinite; for they consist from excess and defect. The triad also from the composition of the monad and duad with itself produces 6, which is the first perfect number, being equal to its parts. And this perfect number when added to the first square, which is 4, produces the decad." According to the Pythagoreans likewise, as we have before observed,* every transmission of divine and mortal concerns is accomplished through emission and reception, and in the third place is strengthened through restoration. Thus after a certain manner the etherial bodies disseminate; but terrestrial natures receive as it were (the etherial effluxions); and a restoration? is effected through the intermediate natures. See chap. 5. Bullialdus, from whom this information is derived, and who obtained it from the anonymous authar of Theologum. Arithmet. is very much mistaken in the meaning of the word av7aao8ogr~ rertoration, in this place. For he says, "per QIV~~~OBOQIV intelligere oportet effectus prductos a causis." Sa far however is this word from signifying in this passage effects produced by causes, that it signifies the restoration of parts to the whdes from which they were derived.
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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
Page 198 and 199: ples of these; referring the point
Page 200 and 201: If it be requisite, however, to spe
Page 202 and 203: the image of the never-failing and
Page 204 and 205: that they are separate, they are no
Page 206 and 207: 27, but they signify through these
Page 208 and 209: dissimilarly similar to divinity. I
Page 210 and 211: anonymous author, in consequence of
Page 212 and 213: It is also Justice, and Isis, Natur
Page 214 and 215: adytum of god-nourished silence, as
Page 216 and 217: called Cupid for the reason above a
Page 218 and 219: the diapente, and the diatessaron.
Page 220 and 221: $Fin of natural effects. Because li
Page 222 and 223: er is not alone even, nor alone odd
Page 224 and 225: the vegetative life. And the seed i
Page 226 and 227: and beauty. And in external things,
Page 228 and 229: which are productive of life after
Page 230 and 231: 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 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 282 and 283: evenlyeven numbers, and for not dis
Page 284 and 285: ing this table from the Epanthematc
Page 286: for instance, 12 is a number longer
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