Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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tween them, is multiplied by 2, and added to the sum of the two squares, the aggregate will be a square number. Thus 1 +4=5, and 5 added to twice 2 is equal to 9. Thus too 4+9 =13, and 13 added to twice 6, is equal to 25; and so of the rest. But if the process is converted, and between the two first terms longer in the other part, the second square is placed, which is in order the second, but in energy the first; and if this square is doubled, and is added to the sum of the two other terms, the aggregate will again be a square. Thus 2 added to 6, added to twice 4, is equal to the square 16. Thus too, if the second and third terms longer in the other part are added together, and to their sum the double of the third square is added, viz. if to 12+6=18 the double of 9 is added, the aggregate will be the square 36. And thus also, 12+20=32, added to twice 16, is equal to the square 64; and so of the rest. This too, is no less admirable, that when a number longer in the other part is placed as a medium between two squares, and a square by the above process is produced from the three terms, the square so produced has always an odd number for its side. Thus from 1+4 and twice 2 the square 9 is produced, the side of which is the odd number 3. And the square 25 which is produced from 4+9, and twice 6, has for its side the odd number 5. Thus likewise, the square 49, arising from the addition of 9, 16, and twice 12, has for its side the odd number 7. And the like will be found to take place in the rest. But when one square subsists between two numbers longer in the other part, all the squares produced from them will have even numbers for their sides. Thus the square 16, produced from the addition of 2,6, and twice 4, has for its side the even number 4. Thus too, the square 36, arising from the addition of 6, 12, and twice 9, has for its side the even number 6. And thus also the square 64, produced by the addition of 12, 20,
BOOK Two 89 and twice 16, has for its side the even number 8; and so of the rest. CHAPTER XX. That from squares and numbers LONGER IN THE OTHER PART, all numerical figures consift.-How nurnt5er.s LONGER IN THE OTHER PART, are produced from squares and uice versa, the Iatter from the former, etc. NOR does it less deserve to be considered, that from these two, all figures are produced. For triangles which are the elements of all the other arithmetical forms, as we have before shown, arise from the aggregates of these. Thus from 1 which is the first square in power or capacity, and 2 which is the first number longer in the other part, the triangle 3 is formed. Thus also from 2, and 4 the second square, the triangle 6 is generated. From 4 likewise and 6, the triangle 10 arises; and so of the rest. For let squares and numbers longer in the other part, be arranged alternately, and let the triangles produced by the addition of them, be placed under them as follows. Squares and heteromekeis alternately arranged. 1. 2. 4. 6. 9. 12. 16. 20. 25. 30. 36. 42. Triangles. But every square, if its side is either added to, or taken from it, becomes a number longer in the other part. Thus, if to the square 4 its side 2 is added, the sum is 6, and if 2 be taken from 4, the remainder is 2, and both 6 and 2 are numbers longer in the other part. Thus too, by adding 3 to 9, and by taking 3 from it, 12 and 6 are produced; and these are numbers longer in the other part. This, however, arises from the
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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
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
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: pared with the first number longer
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
Page 137 and 138: 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
Page 174 and 175: 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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