Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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CHAPTER XIX. That the multiple is more ancient than the other species of inequality. To demonstrate this, let there be first a series of numbers in a natural order as far as to 10. In the second row let the duple order be arranged. In the third row, the triple order. In the fourth, the quadruple; and so on 2s far as to the decuple order. For thus we shall know how the species of the multiple precedes the superparticular, superpartient, and all the other species of inequality. We shall also at the same time perceive other things, which are exquisitely subtile, most useful in a scientific point of view, and which afford a most delightful exercise to the mind. This Table is called Pythagorean, from Pythagow, who is said to be the author of it. Length. 1 5 1 10 ( 15 ( 20 125- 30 135 140 ( 45 150 ( 1 6 1 12 1 18 124 130 136 Length. If therefore the two first sides which form an angle of the above table are considered, and each of which proceeds from 1 to 10, and if the inferior orders which begin from the angle
4, and terminate in 20, are compared to these, the double, i.e. the first species of multiplicity will be exhibited. Hence, the first will surpass the first by unity alone, as 2 surpasses 1. The second will exceed the second by 2, as 4 exceeds 2. The third will exceed the third by 3, as 6 exceeds 3 by 3. The fourth will surpass the fourth by 4, and in this way 8 surpasses 4. And after a similar manner in the rest. But if the third angle is considered, which begins from 9, and which extends both in length and breadth as far as to the number 30, and if this is compared with the first length and breadth, the triple species of multiplicity will present itself to the view, so that the comparison will take place through the black angle. And these numbers will surpass each other according to the natural progression of the even number. For the first number will surpass the first by 2, as 3 surpasses 1. The second surpasses the second by 4, as 6 surpasses 2. The third surpasses the third by 6, as 9 surpasses 3. And after the same mode of progression the rest are increased. If again, the boundary of the fourth angle is considered, which is distinguished by the quantity of the number 16, and which terminates its length and breadth in the number 40, here also a similar comparison being made with the preceding, will unfold the quadruple species of multiplicity. Hence, the first will surpass the first by 3, as 4 surpasses unity. The second will surpass the second by 6, as 8 surpasses 2. The third will exceed the third by 9, as 12 exceeds 3. And after a similar manner in all the following numbers. If the remaining angles likewise are considered, the same thing will take place through all the species of multiplicity, as far as to the decuple species. If however, in this description, the superparticular species are required, they may be found by the following method. For if we direct our attention to the second angle, the begirlning of which is 4, and above which is 2, and if the row in
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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
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 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 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
Page 113 and 114: BOOK Two 75 third, a pentagonal; th
Page 115 and 116: Two does not only not arrive at uni
Page 117 and 118: BOOK Two 79 to have four angles and
Page 119 and 120: 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
Page 125 and 126: 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
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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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