Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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Since therefore number is the connective bond of all things, it is necessary that it should abide in its proper essence, with a perpetually invariable sameness of subsistence; and that it should be compounded, but not of things of a different nature. For what could conjoin the essence of number, since its paradigm joined all things. But it seems to be a composite from itself. Moreover, nothing appears to be composed from similars; nor yet from things which are conjoined by no analogy, and which are essentially separated from each other. Hence it is evident, since number is conjoined, that it is neither conjoined from similars, nor from things which mutually adhere by no analogy. Hence the primary natures of which number consists, are the even and the odd, which by a certain divine power, though they are dissimilar and contrary, yet flow from one source, and unite in one composition and modulation. CHAPTER 111. On the division of numbers, and the various definitions of the even and the odd. THE first division of number therefore is into the even and the odd. And the even number indeed is that which may be divided into two equal parts, without the intervention of unity in the middle. But the odd number is that which cannot be divided into equal parts, without unity intervening in the mid- too, that in the sensible universe, the first monad is the world itself, which comprehends in itself all the multitude of which it is the cause (in conjunction with the cause of all). The second monad is the inerratic sphere. In the third place, the spheres of the planets succeed, each of which is also a monad, comprehending an appropriate multitude. And in the fourth and last place are the spheres of the elements, which are in a similar manner monads. All these monads likewise, are denominated 0107q71~, wholmcsses, a d have perpetual subsistence.
dle. And these indeed are the common and known definitions of the even and the odd. But the definition of them according to the Pythagoric discipline is as follows: The even number is that which under the same division may be divided into the greatest and the least; the greatest in space, and the least in quantity, according to the contrary passions of these two genera. But the odd number is that to which this cannot hap pen, but the natural division of it is into two unequal parts. Thus for instance, if any given even number is divided, there is not any section greater than half, so far as pertains to the space of division, but so far as pertains to quantity, there is no division less than that which is into two parts. Thus, if the even number 8, is divided into 4 and 4, there will be no other division, which will produce greater parts, viz. in which both the parts will be greater. But also there will be no other division which will divide the whole number into a less quantity; for no division is less than a section into two parts. For when a whole is separated by a triple division, the sum of the space is diminished, but the number of the division is increased. As discrete quantity however, beginning from one term, receives an infinite increase of progression, but continued quantity may be diminished infinitely, the contrary to this takes place in the division of the even number; for here the division is greater in space, but least in quantity. In other words, the portions of continued quantity are greatest, but the discrete quantity is the least possible. According to a more ancient mode likewise, there is another definition of the even number, which is as follows: The even number is that which may be divided into two equal, and into two unequal parts; yet so that in neither division, either parity will be mingled with imparity, or imparity with parity; except the binary number alone, the principle of parity, which does not receive an unequal section, because it consists of two uni-
Page 1: THEORETIC ARITHMETIC IN THREE BOOKS
Page 4 and 5: IV INTRODUCTION the mathematical di
Page 6 and 7: INTRODUCTION works, like the remain
Page 8 and 9: VIII INTRODUCTION fore, that the so
Page 10 and 11: INTRODUCTION the other of such as a
Page 12 and 13: XII INTRODUCTION must her inherent
Page 14 and 15: XIV INTRODUCTION principles existin
Page 16 and 17: XVI INTRODUCTION indissoluble perma
Page 18 and 19: XVIII INTRODUCTION mathematics to p
Page 20 and 21: XX INTRODUCTION utility it administ
Page 22 and 23: XXII INTRODUCTION may not energize
Page 24 and 25: INTRODUCTION tradition, however, of
Page 26 and 27: XXVI INTRODUCTION Nicomachus, which
Page 28 and 29: INTRODUCTION perplexity, and that h
Page 30 and 31: INTRODUCTION ever, requisite that w
Page 32 and 33: XXXII INTRODUCTION that he is ignor
Page 34 and 35: XXXIV INTRODUCTION tirely the prais
Page 37 and 38: THEORETIC ARITHMETIC BOOK ONE CHAPT
Page 39 and 40: that all motion is after rest, and
Page 41: since it is not possible for divisi
Page 45 and 46: monad is either even or odd. It can
Page 47 and 48: is denominated the evenly-odd, and
Page 49 and 50: terms should accord with its proper
Page 51 and 52: ural order, 1.3.5.7.9.11.13.15.17.
Page 53 and 54: agreeably to the form of the evenly
Page 55 and 56: 12, 20, and 28, the sum of the extr
Page 57 and 58: numbers proceeding from them are re
Page 59 and 60: THE generation however and origin o
Page 61 and 62: ever will give the mode of measurin
Page 63: THE SIEVE OF ERATOSTHENES. Here 7 m
Page 66 and 67: pass the sum of the whole number, i
Page 68 and 69: CHAPTER XV. On the generation of th
Page 70 and 71: CHAPTER XVI. On relative quantity,
Page 72 and 73: consequent order, these even number
Page 74 and 75: For the sesquialter has for its lea
Page 76 and 77: CHAPTER XIX. That the multiple is m
Page 78 and 79: which 4 is, is compared with the ro
Page 80 and 81: number besides containing the whole
Page 82 and 83: superquadripartient, may likewise b
Page 84 and 85: to them, duple sesquiquartan ratios
Page 86 and 87: the impression of itself, defines a
Page 88 and 89: If, however, the multiples which ar
Page 90 and 91: tan, the superquadripartient ratio
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BOOK TWO CHAPTER I. How all inequal
Page 95 and 96:
BOOK Two 57 reduced to equality. Fo
Page 97 and 98:
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
Page 121 and 122:
BOOK Two 83 the principles of thing
Page 123 and 124:
BOOK Two 85 from the monad, are sai
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pared with the first number longer
Page 127 and 128:
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
Page 171:
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
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The same writer also observes, "tha
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position; for I conjecture this to
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~ c ~ ~ ~ Or ~ Q rather, ~ v . whic
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The ratio of 12 to 9 is sesquiterti
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monad. Thus if from each of the num
Page 242 and 243:
had these appellations in consequen
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and well-ordered digression (from t
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unity is added to the product, the
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And so of the rest, which abundantl
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Again, of any two numbers whatever,
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Again, if 4 multiplies any triangul
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With respect to the hexad, it is th
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and the double of it 12, the number
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incorporeal and corporeal essence;
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of the hebdomad 3 and 4 necessarily
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composition of which the hebomad is
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it is also surpassed by the last nu
Page 266 and 267:
excite the glad husbandmen to the c
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mentioned particulars, but also to
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7X 1 and +1= 8 the 2nd 7 x 8 and +
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8X 1 and +1= 9 the 2ndl 8X 7and +1=
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10X 1 and +1= 11 the 2ndl i' 10x30
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tureX according to this number. In
Page 278 and 279:
through a good allotment, or throug
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titude, and communion with itself,
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evenlyeven numbers, and for not dis
Page 284 and 285:
ing this table from the Epanthematc
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for instance, 12 is a number longer
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