Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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mon measure of these except unity which is the generator and mother of all these. CHAPTER XIII. On the method of discovering the commensurability, or incommensurability of these n urn bers. THE art however of discovering whether such numbers are commensurable by some number besides unity, or whether each of them is measured by unity alone, is as follows: Two unequal numbers being given, it will be requisite to take the less from the greater, and if that which remains is still greater to take from it again the less, but if it is less to take it from the remaining greater number. And this must be done till unity at last prevents any farther subtraction, or till some odd number necessarily effects this, if both the proposed numbers are odd; and this number which remains will be the common measure of both. Thus for instance, let there be two numbers 9 and 29, of which it is proposed to investigate the common measure. From the greater let the less be taken, and there will remain 20. From this therefore let the less be again taken, and 11 will remain. From this again take 9, and 2 will remain. Let 2 then be taken from 9, and 7 will remain. From 7 let 2 be taken, and there will remain 5. Again, from 5 let 2 be taken, and 3 will remain. And farther still, if 2 be taken from 3, there will remain 1. And in the last place, if 1 be taken from 2, the subtraction will terminate in unity, which will alone be the measure of the two numbers 9 and 29. These numbers therefore, are called with reference to each other prime numbers. Let it, however, be proposed to investigate the common measure of two other numbers 21 and 9, that it may be ascer-
THE SIEVE OF ERATOSTHENES. Here 7 measures the first number according to 3, the second according to 5, the third according to itself, and so of the rest. Here the first number is measured by 5 according to 3, the second by 5 according to itself, the third by 5 according to 7, and so on. Here the first number is measured by 3 according to itself, the second by 3 according to 5, the third by 3 according to 7, and so of the rest. The sieve of Eratosthenes by which it is ascertained what numbers are primary, and what are composite. Eratosthenes appears, very properly, to have called the above invention a sieve; for in it the composite are separated from the incomposite numbers, just as in a sieve, the pure is separated from the impure, and that which is subtile from that which is dense and gross.
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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
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 and 42: since it is not possible for divisi
Page 43 and 44: dle. And these indeed are the commo
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: ever will give the mode of measurin
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
Page 93 and 94: 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
Page 99 and 100: BOOK Two 61 ratio is composed from
Page 101 and 102: BOOK Two CHAPTER IV. On the qztanti
Page 103 and 104: BOOK Two 65 distended by a triple d
Page 105 and 106: These triangular numbers are genera
Page 107 and 108: BOOK Two 69 But in these numbers al
Page 109 and 110: BOOK Two 71 triangle of a class imm
Page 111 and 112: 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
Page 133 and 134:
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
Page 161 and 162:
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
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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
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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
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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
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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
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ing this table from the Epanthematc
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for instance, 12 is a number longer
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