Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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And after this manner, if all the numbers in the duple series arc multiplied by those in the upper series, the products wil! be found to be unevenly-even numbers. This also is admirable in this species of numbers, that if the disposition and description of them according to breadth is regarded, the property of the evenly-odd numbers will present itself to the view, but the property of the evenly-even, if the disposition of them is regarded according to length. For according to breadth, the two extremes are equal to the two media, or if there is but one medium, the double of it is equal to the extremes. But according to length, the property of the evenly-even number will be discovered; for here the product of the extremes is equal to that of the two media, or if there is but one medium the square of it is equal to the product of the extremes. The description of them however, according to length and breadth, is as follows; The products arising from the multiplication of evenly-even numbers in an orderly series by 3, are to be placed in the first row. Again, the products arising from the multiplication of the same evenly-even numbers by 5, are to be placed in the second row. Those arising from the multiplication by 7, are to be placed in the third row, and so of the rest. 3 5 7 9 etc. Odd numbers. 4 8 16 32 etc. Evenly-even numbers. Length. 12 24 48 % 20 40 80 160 Breadth, 28 56 112 224 36 72 144 288 Length. Breadth, Here in the breadth, if three terms are taken, as for instance
12, 20, and 28, the sum of the extremes is double the middle term; for 12+28=40. Thus also 20+36=56 equal twice 25. But where there are two media, the sum of the extremes will be equal to the sum of the means. Thus 12+36=48=20+28. Thus also 24+72=%=40+56; and so of the other parts of the breadth. This however takes place according to the form of the evenly-odd number, in which, as we have before observed, this property is found. Again, i£ we direct our attention to the length, where two terms have one medium, the product of the extremes is equal to the square of the medium or middle term ; for 12 )< 48=576=24 )< 24. Again, 24 X 96= 2304=48)
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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 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: agreeably to the form of the evenly
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
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
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BOOK Two 73 22 is formed from the s
Page 113 and 114:
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
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
Page 127 and 128:
BOOK Two 89 and twice 16, has for i
Page 129 and 130:
atios. Let them, therefore, be alte
Page 131 and 132:
BOOK Two 93 ity. For 1 is the whole
Page 133 and 134:
BOOK Two 95 ticipate of an immutabl
Page 135 and 136:
BOOK Two 97 are opposite to those a
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BOOK Two is equal to the sum of the
Page 139 and 140:
BOOK Two 101 will be the same multi
Page 141 and 142:
BOOK Two 103 terms there is a great
Page 143 and 144:
BOOK Two 105 from 8, and one side f
Page 145 and 146:
BOOK Two 107 by the greater, by the
Page 147 and 148:
BOOK Two 109 tertian ratio, or that
Page 149 and 150:
BOOK Two 111 ratio, arises from a c
Page 151 and 152:
BOOK Two 113 to 10 is a quadruple r
Page 153 and 154:
BOOK Two 115 For here the greatest
Page 155 and 156:
BOOK Two 117 by length, breadth, an
Page 157 and 158:
BOOK Two The greatest harmonies. Pr
Page 159 and 160:
BOOK Two 121 which is 18416, we mus
Page 161 and 162:
BOOK Two former diameter, exceeds i
Page 163 and 164:
BOOK Two In the geometrical fractio
Page 165 and 166:
BOOK Two The parts of 51=17)
Page 167 and 168:
BOOK Two 129 In the second place, i
Page 169 and 170:
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
Page 186 and 187:
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
Page 232 and 233:
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
Page 238 and 239:
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
Page 246 and 247:
unity is added to the product, the
Page 248 and 249:
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
Page 278 and 279:
through a good allotment, or throug
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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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