Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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7X 1 and +1= 8 the 2nd 7 x 8 and + i 1= 57 the 9th enneagonal gnomon. 7x21 and +1=148 the 22nd &c. 7x1 and +1= 8 the 2nd 1 7x4 and +1=29 the 5th enneagonal gnomon. 7x9 and +1=64 the 10th J &c. 7X 1 and +1= 8 the 2nd i 7X 6 and +1= 43 the 7th enneagonal gnomon. 7X 18 and + 1=127 the 19th &c. 7 X 1 and 1= 8 the 2nd 7X 3 and + 1=22 the 4th enneagonal gnomon. 7x5 and +1=36 the 6th &c . 1 7X 1 and +1= 8 the 2nd 7X 7 and +1=50 the 8th enneagonal gnomon. 7X 13 and + 1=92 the 14th &c. CHAPTER XVII On the ogdoad, ennead, and decad. THE ogdoad, says Martianus Capella, is perfect, because it is covered by the hexad;* for every cube has six superficies. Likewise it derives its completion from odd numbers in a following order. For the first odd number is 3 and the second 5, and 3+5=8. Thus also the cube which is formed from the triad, viz. 27, is composed of three odd numbers in a following order, viz. of 7,9, and 11. Thus too, the third cube which is formed from the tetrad, viz. 64, derives its completion from four odd numbers in a following order, viz. from 13, 15, 17, * In my edition which is Lugd. 1619, the original is, "Perfechls item quod a septc~rio tcgitur;" but instead of septenatio, it is evident it should be rcnario.
19; for the sum of these is 64. And thus all cubes will be found to consist of as many odd numbers as there are unities in its root. Moreover, this octonary cube is the first of all cubes, in the same manner as the monad is the first of all numbers." Thus far Capella. Again, in the chapter On the properties of the monad, we have shown from Plutarch that if 8 multiplies any triangular number, and unity is added to the product, the sum will be a square number. And we have also shown that if the triple of 8, i.e. 24, multiplies any pentagonal number, and unity is added to the product, the sum will also be a square. Farther still, conformably to what we have shown of the former numbers within the decad, if 8 multiplies any triangular or enneangular gnomon, or any gnomon of a square, or any square, or first hexangular pyramid, or any enneagon, and unity is added to the product, the sum will be a decagonal gnomon. For, 8x1 and +1= 9 the 2ndj 8x2 and +1=17 the 3rd decagonal gnomon. 8x3 and +1=25 the 4th J &c. 8X 1 and +1= 9 the 2nd] 8~ 8 and +1= 65 the 9th ) decagonal gnomon. 8 X 15 and + 1=121 the 16th J &c. 8x1 and +1= 9 the 2nd 1 8 ~3 and +1=25 the 4th decagonal gnomon. 8x5 and +1=41 the 6th J &c. 8x1 and +1= 9 the 2nd 8x4 and +1=33 the 5th decagonal gnomon. 8x9 and + 1=73 the 10th I &c.
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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
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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.
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
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Page 228 and 229: which are productive of life after
Page 230 and 231: one side, and 1 on the other, each
Page 232 and 233: The same writer also observes, "tha
Page 234 and 235: position; for I conjecture this to
Page 236 and 237: ~ c ~ ~ ~ Or ~ Q rather, ~ v . whic
Page 238 and 239: The ratio of 12 to 9 is sesquiterti
Page 240 and 241: monad. Thus if from each of the num
Page 242 and 243: had these appellations in consequen
Page 244 and 245: 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
Page 250 and 251: Again, of any two numbers whatever,
Page 252 and 253: Again, if 4 multiplies any triangul
Page 254 and 255: With respect to the hexad, it is th
Page 256 and 257: and the double of it 12, the number
Page 258 and 259: incorporeal and corporeal essence;
Page 260 and 261: of the hebdomad 3 and 4 necessarily
Page 262 and 263: composition of which the hebomad is
Page 264 and 265: it is also surpassed by the last nu
Page 266 and 267: excite the glad husbandmen to the c
Page 268 and 269: mentioned particulars, but also to
Page 272 and 273: 8X 1 and +1= 9 the 2ndl 8X 7and +1=
Page 274 and 275: 10X 1 and +1= 11 the 2ndl i' 10x30
Page 276 and 277: tureX according to this number. In
Page 278 and 279: through a good allotment, or throug
Page 280 and 281: 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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