Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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These numbers are as follows: In the first place 2 X 5 and +1=11, 2 )( 23 and + 1=47, 2X 191 and +1=383, 2X 1535 and +1=3071, 2X 12287 and + 1=24575, and 2 X 98303 and +1=196607. And thus the double of the first number in each rank added to unity is equal to the second number in the same rank. In the next place, the first number of the first rank multiplied by 4 and added to 3, will be equal to the first number of the second rank. Thus also the second number 11 X 4 and added to 3=47 the second number of the second rank. But 23 >< 8 and +7=191, 47x8 and +7=383, 191x8 and +7=1535, 383x8 and +7=3071, 1535x8 and +7=12287, 3071x8 and +7=24575, 12287x8 and +7=98303, and 24575~ 8 and +7= 196607. Again, 47x4 and +3=191, 383x4 and +3=1535, 3071 X 4 and +3=12287, and 24575 >< 4 and +3=98303. 1151 73727- Again, T=16, with a remainder 15. -64, with a 4718591- remainder 63. 3 -64, and the remainder is 63. And 301989887- 7 5 -64, with the same remainder 63. And so of the rest ad infin. the quotient and remainder being always 64 and 63. Hence infinite series of all these may easily be obtained, viz. of the two first terms in each rank, the first rank excepted, and of the third term in each rank the two first ranks excepted. For 23+7+7+7, kc. ad infin.
BOOK Two And these expressions when reduced will be 23-16 47-40 1-9+8,1-9+8, But and Another remarkable property of these numbers is this, that the product of the two first in each rank subtracted from the third number in the same rank, leaves a remainder equal to the aggregate of those two first. Thus 5X 11--55, and 71-55 =16. But 16=5+11. Thus too, 23)
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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
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tan, the superquadripartient ratio
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BOOK TWO CHAPTER I. How all inequal
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BOOK Two 57 reduced to equality. Fo
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BOOK Two 59 alters as much distant
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BOOK Two 61 ratio is composed from
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BOOK Two CHAPTER IV. On the qztanti
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BOOK Two 65 distended by a triple d
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These triangular numbers are genera
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BOOK Two 69 But in these numbers al
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BOOK Two 71 triangle of a class imm
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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
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
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 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
Page 190 and 191: these, and are called by modern mat
Page 192 and 193: The numbers, therefore, that form 1
Page 194 and 195: with each other, it will be found,
Page 196 and 197: PYTHAGORAS
Page 198 and 199: ples of these; referring the point
Page 200 and 201: If it be requisite, however, to spe
Page 202 and 203: the image of the never-failing and
Page 204 and 205: that they are separate, they are no
Page 206 and 207: 27, but they signify through these
Page 208 and 209: dissimilarly similar to divinity. I
Page 210 and 211: anonymous author, in consequence of
Page 212 and 213: It is also Justice, and Isis, Natur
Page 214 and 215: adytum of god-nourished silence, as
Page 216 and 217: called Cupid for the reason above a
Page 218 and 219: the diapente, and the diatessaron.
Page 220 and 221: $Fin of natural effects. Because li
Page 222 and 223: er is not alone even, nor alone odd
Page 224 and 225: the vegetative life. And the seed i
Page 226 and 227: and beauty. And in external things,
Page 228 and 229: which are productive of life after
Page 230 and 231: 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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