Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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CHAPTER VI. On the evenly-odd number, and its propertie$. The evenly-odd number however is that which is itself indeed allotted the nature and essence of parity, but is opposed in a contrary division to the nature of the evenlycven number. For it will be shown that it is divided in a very dissimilar way. For because it is even, it receives a section into equal parts, but the parts of it immediately become indivisible. And such are the numbers, 6.10.14.18.22 and others similar to these. For in dividing these numbers into equal parts, we are immediately stopt by the odd number which cannot be divided equally. But it happens to these numbers that all their parts have denominations contrary to the quantities of the parts that are denominated; nor can it ever be possible that any part of an evenly-odd number should receive a denomination and quantity of the same kind. For always, if the denomination is even, the quantity of the part will be odd. Thus for instance, in the number 18 the half of it, which term half is the appellation of parity, is 9, which is odd in quantity. But its third part, which is an uneven denomination, is 6, which is even in quantity. Again, by conversion, the sixth part which is an even denomination is 3, but the ternary is odd. And the ninth part, which is an uneven appellation, is 2 which is an even number. And the same thing will be found to take place in all other evenlyodd numbers. Nor can it ever be possible that the name and number of any part are of the same kind. These numbers are produced by disposing from unity all the numbers that differ from each other by 2, viz. all the odd numbers in a natural series. For if each of these is multiplied by 2, all the evenly-odd numbers will be generated. Let there be given then the following odd numbers from unity in a nat-
ural order, 1.3.5.7.9.11.13.15.17. 19. These, if each of them is multiplied by 2, will form the series 2.6.10.14.18.22.26.30. 34.38. And each of these if divided will only receive one section, a second division being excluded by the intervention of imparity. These numbers also differ from each other by 4: and this arises from the mode of their generation. For the odd numbers which are the foundation of them, exceed each other by 2; and because each of these is multiplied by 2, the progression receives a four-fold increase. But these species of numbers, the evenly-even, and the evenly-odd, are said to be contrary, because in the evenly-odd number, the greater extremity alone receives division,* and the less term is in this alone liberated from section; and because in the form of the evenly-even number, the product of the extremes is equal to the product of the means, as far as to the two media, if the number of terms is even; but if the number of terms is odd, the square of the medium is equal to the pre duct of the extremes. In the evenly-odd number however, if the number of terms is odd and there is therefore only one middle term, this term will be the half of the sum of the terms placed about it. It will also be the half of the sum of the terms placed above these; and this will be the case as far as to the extremes of all the terms. Thus in the series of evenly-odd numbers 2.6. lo., the middle term 6 is the half of 10+2. And if there are two media, the sum of these will be equal to the sum of the terms placed above them. Thus in the series 2.6. 10.14, the sum of the media 6f10 is equal to 2f14 the sum of the extremes. Viz. In the evenly-even number, it is the least part alone, i.e. unity which receives no division; but on the contrary in the evenly-odd numbers, the greater extremity alone receives division, i.e. the whole number. Thus 10 can be divided into two equal parts, but in thwe parts the division stops; so that in this number the section is stopped in the greatest pan, while on the contrary, in the evenly-even numbcr, it is stopped in the least pan, unity.
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: terms should accord with its proper
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
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These triangular numbers are genera
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BOOK Two 69 But in these numbers al
Page 109 and 110:
BOOK Two 71 triangle of a class imm
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BOOK Two 73 22 is formed from the s
Page 113 and 114:
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
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
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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
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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
Page 157 and 158:
BOOK Two The greatest harmonies. Pr
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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
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BOOK Two them; (see Chap. 15, Book
Page 171:
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
Page 180 and 181:
In like manner if 9437056 be divide
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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
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parts of these wholes have sometime
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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
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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
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dissimilarly similar to divinity. I
Page 210 and 211:
anonymous author, in consequence of
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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.
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$Fin of natural effects. Because li
Page 222 and 223:
er is not alone even, nor alone odd
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the vegetative life. And the seed i
Page 226 and 227:
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
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

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