Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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mentioned particulars, but also to the best of the sciences, to grammar and music. For the lyre indeed consisting of seven chords, analogous to the choir of the seven planets, produces the most noble harmonies, and is nearly the leader of all instrumental music. Among the elements also of the grammatical art, the vowels truly so called are seven, since they are seen to produce a sound from themselves, and when they are conjoined with other letters, they form articulate sounds. For they supply indeed, what is wanting in the semivowels, by furnishing entire sounds; but they turn and change the natures of the mutes, by inspiring them with proper power, in order that things which were before ineffable may become effable. Hence those who first gave names to letters, as being wise men, appear to me to have denominated this number ET~CX, from the veneration and inherent sanctity pertaining to it, (ax0 ~ 0 T~Q! 3 aoso.+ c~Ba~ou, 7rj~ Q O U O ~ O W+VOVW.) ~ ~ S But the Romans adding the letter 0, which is wanting in the Greek, more clearly exhibit its origin, by calling it septem, from, as I have said, veneration and sanctity." Again, according to Herophilus, as we are informed by Theo of Smyrna,* the human intestine is 28 cubits long, i.e. four times seven; and 28 is a perfect number. The moon also is said to impart from the etherial to the sublunary regions celestial powers while she uses the tetrad and the hebdomad. The increase likewise and decrease of things, and especially of such as are humid, follow the phases of the moon. Hence, a Greek poet cited by Baptista Carnotius, in his Comment on the Metaphysics of Theophrastus, says of the moon, "that when she increases she augments, and when she decreases, injures all things." This number too is the shortest boundary of the birth of children. And according to the Theban Hephzstion, the long- *: Mathemat. p. 162.
est boundary of the duration in the womb of those children that are born in the tenth month is 288 days and 8 hours. The mean duration is 273 days and 8 hours. And the shortest duration is 258 days and 8 hours. But of those that are born in the seventh month, the longest duration in the womb is 206 days and 8 hours; the least duration is 176 days and 8 hours; and the mean duration is 191 days and 8 hours. The ancient physicians likewise, as we learn from the anonymous writer, observe, that seven hours prior to the birth the navel of the foetus is spontaneously separated from the mother; within which space of time the foetus is able to support life, without receiving any nutriment from the mother." Again, Plato, says Theo Smyrnaeus, following nature constitutes the soul from seven numbers. These numbers are 1, 2, 3,4, 8,9, 27, concerning the remarkable properties of which, T refer the reader to Plutarch's treatise On the Generation of the Soul according to Plato in the Timaeus. Farther still, conformably to what I have shown in the preceding numbers, if 7 multiplies a triangular gnomon, and unity is added to the product, the sum will be an enneagonal gnomon. For, 7x1 and +1= 8 the 2nd 7X 2 and +1=15 the 3rd enneagonal gnomon. 7X 3 and +1=22 the 4th 7x4 and +1=29 the 5th &c . i By a similar process likewise with octagons, with squares, with pentagonal column. triang. pyramids, with square gnomons, and with the gnomons of octangles, enneagonal gnomons will be produced. Thus, + Vid. Notas Bulliald. in Theon. Mathemat. p. 284.
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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
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 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 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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