Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

More documents

Recommendations

Info

parts of these wholes have sometimes a natural, and sometimes an unnatural subsistence; for thus alone can the circle of generation unfold all the variety which it contains. The different periods in which these mutations happen, are called by Plato, with great propriety, periods of fertility and sterility. For in these periods, a fertility or sterility of men, animals, and plants, takes place; so that in fertile periods, mankind will be both more numerous, and upon the whole, superior in mental and bodily endowments, to the men of a barren period. And a similar reasoning must be extended to animals and plants. The so much celebrated heroic age, was the result of one of these fertile periods, in which men transcending the herd of mankind, both in practical and intellectual virtue, abounded on the earth. With respect to the epithet divinely generated, it is well observed by the Greek scholiast, "that Plato does not mean by this either the whole world, though the epithet is primarily applicable to it, nor the celestial regions only, nor the sublunary world, but every thing which is perpetually and circularly moved, whether in the heavens, or under the moon; so far as it is corporeal calling it generated, (for no body is self-subsistent), but so far as it is perpetually moved, divine. For it imt tates the most divine of things, which possess an ever-vigilant life. But with respect to the perfect number mentioned here by Plato, we must not only direct our attention to a perfect number in vulgar arithmetic,-for this is rather numbered than number, tends to perfection and is never perfect, as being always in generation,-but we must survey the cause of this number, which is indeed intellectual, but comprehends the definite boundary of every period of the world." In the third place, let us consider what Plato means by augmentations surpassing and surpassed; things assimilating and
ROOK Two dissimilating, increasing and decreasing, correspondent and eff able. Augmentations surpassing, are ratios of greater inequality, viz. when the greater is compared to the less, and are multiples, superparticulars, superpartients, m illtiple super-particulars, and multiple-superpartients. But multiple ratio is, as we have before shown, when a greater quantity contains a less many times; superparticular ratio is when the greater contains the less quantity once, and some part of it besides; and superpartient ratio is when the greater contains the less quantity once, and certain parts of it likewise. Again, multiplesuperparticular ratio is when the greater contains the less many times, and some part of it besides; and multiple-superpartient ratio, is when the greater contains the less many times, and also some of its parts. But augmentations surpassed, are ratios of less inequality, viz. when the less is compared with the greater quantity; as for instance, submultiples, subsuperparticulars, subsuperpartients, and those which are composed from these three. Those numbers are called by Plato assimilating and dissimilating, which are denominated by arithmeticians similar and dissimilar. And similar numbers are those whose sides are proportional, but dissimilar numbers those whose sides are not proportional. Plato also calls those numbers increasing and decreasing, which arithmeticians denominate superperfec~, and deficient, or more than perfect and imperfect. Things correspondent and effable, are boundaries which correspond in ratio with each other; and can be expressed in numbers either integral or fractional,-such as these four terms or boundaries 27, 18, 12, 8, which are in sesquialter and subsesquialter ratios; since these mutually correspond in ratio, and are effable. For effable quantities are those which can be expressed in whole numbers or fractions; and in like manner, ineffable quantities are such as cannot be expressed in either of
Page 1:
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 and 54:
agreeably to the form of the evenly
Page 55 and 56:
12, 20, and 28, the sum of the extr
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
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 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 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 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
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 270 and 271:
7X 1 and +1= 8 the 2nd 7 x 8 and +
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
show all

Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?