Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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etc. the greater term will always be found to differ from the less by the quantity of the ratio less by unity. In continued geometrical proportionality also, the rectangle under the extremes is equal to the square of the mean. Thus in the three terms 2.4. 8. which are in geometrical proportion, 2 X 8=4X4=16. The same thing will also take place though there should be a greater number of terms than three, if the number of terms is odd. Thus in the five terms 16. 8. 4. 2. 1, the rectangle under the extremes, i.e. 16x1 is equal to the square of the middle term 4. Thus too in the seven terms 64. 32. 16. 8. 4. 2. I., 64X1=8)(8. But where the terms are not continued, the rectangle under the extremes is equal to the rectangle under the means; as is evident in the terms 2. 4. 8. 16; for 2X 16=4X8=32. And if the terms are more than four, if only they are even, the same thing will take place. Thus in the six terms 2. 4. 8. 16. 32. 64, 2 X64=128=4X32=8 X 16. Another property of this middle is, that there is always an equal ratio both in the greater and the less terms. Thus in all the terms 2. 4. 8. 16. 32. 64, there is a duple ratio. This is likewise the case in the terms, 3. 9. 27. 81. 243. 729. And in a similar manner in others. And in the last place, it is peculiar to this middle, squares and numbers longer in the other part, being alternately arranged, to proceed from the first multiple into all the habitudes of superparticular ratios, as will be evident from the following scheme. , , , , Squares and numbers longer in the other part, aIternately arranged. 1 2 4 6 9 12 16 20 25 30 37 Puple, Duple slqui- . Sesqui- Sesqui- Suqui- Sesqui- Sesqui- Stsqui- Suquiquintan alter I alter ttrtian I tertian quartan ( quatan qintan 1 Hence the arithmetical middle is compared to an oligarchy, or the republic which is governed by a few, who pursue their own good, and not that of the community; because in its less
BOOK Two 103 terms there is a greater ratio. But the harmonic middle is said to correspond to an aristocracy, because there is a greater ratio in its greater terms. And the geometric middle is analogous to a popular government, in which the poor as well as the rich have an equal share in the administration. For in this there is an equality of xatio both in the greater and the less termsox CHAPTER XXVI That plane numbers are conjoined by one medium only, but solid numbers by two media. IT is now time, however, that we should discuss certain things very useful to a knowledge of the fabrication of the world, as delivered in the Timaeus of Plato, but not obvious to every one. For all plane figures are united by one geometrical medium only; and hence in these there are only two intervals, viz. from the first to the middle, and from the middle to the third term. But cubes have two media according to a geornetrical ratio; whence also solid figures are said to have three in- + With respect to these three middles, the arithmetic, the geometric and the harmonic, Proclus in Tim. p. 238. observes "that they pertain to the thre daughters of Themis, viz. Eunomia, Dice, and Irene. And the arithmetic middle indeed pertains to Irene or peace, which surpasses and is surpassed by an equal quantity, which middle we employ in our contracts during the time of peace, and through which likewise the elements are at rest. But the geometric middle pertains to Eunomia or equitable legislation, which also Plato denominates the judgment of Jupiter, and through which the world is adorned by geometrical analogies. And the harmonic middle pertains to Dice or Juctice, through which greater terms have a greater; but less, a less ratio." The geometric middle also comprehends the other two. For let there be any three terms in arithmetical proportion, for instance 1. 2. 3, and let a fowth term be added which shall cause all the four to be in geometrical proportion. This fourth term will be 6. For 1:2::3:6. This analogy therefore, wiIl comprehend both arithmetical and harmonica1 proportion. For 2. 3. and 6 arc in harmonic, and 1. 2 and 3 in arithmetic proportion.
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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
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CHAPTER XIX. That the multiple is m
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which 4 is, is compared with the ro
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number besides containing the whole
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superquadripartient, may likewise b
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to them, duple sesquiquartan ratios
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the impression of itself, defines a
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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
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
Page 137 and 138: BOOK Two is equal to the sum of the
Page 139: BOOK Two 101 will be the same multi
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
Page 176 and 177: Farther still, if 2 be subtracted f
Page 178 and 179: 24575, and 301989887. And the two i
Page 180 and 181: In like manner if 9437056 be divide
Page 182 and 183: These numbers are as follows: In th
Page 184 and 185: That the reader may see the truth o
Page 186 and 187: all perfectly as well as all imperf
Page 188 and 189: 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
show all

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