Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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incorporeal and corporeal essence; the species of the incorporeal indeed, according to the superficies which is formed by squares; but of the corporeal according to the other dimension (depth) which is formed by cubes. But the credibility of what is said is most manifest in the above mentioned numbers. For the hebdomad 64 which is immediately increased from unity in a duple ratio, is a square produced by the multiplication of 8 by 8; and it is also a cube, the side or root of which is 4. And again, the hebdomad which is increased in a triple ratio from the monad, viz. 729, is a square indeed, formed by the multiplication of 27 by itself, and is also a cube, the side of which is 9.+ By always making too a hebdomad the principle instead of the monad, and increasing according to the same analogy as far as to the hebdomad, you will always find that the increased numbers is both a square and a cube. The hebdomad therefore compounded in a duple ratio from 64, will be 4096,t which is both a square and a cube; a square indeed, having for its side 64; but a cube, the side of which is 16. Let us now pass to the other species of the hebdomad which is comprehended in the decad, and which exhibits an admirable nature no less than the former hebdomad. This therefore is composed of one, two and four, which possess two most harmonic ratios, the duple and the quadruple; the former of which forms the symphony diapason, and the latter the symphony disdiapason. This hebdomad also comprehends other divisions, consisting after a manner of certain conjugations. For it is in the first place indeed, divided into the monad and hexad, afterwards into the duad and pentad, and lastly into the triad and tetrad. But this analogy or proportion of num- + Thus lXZX2~2~2~2X2=64; and 1~3~3X3X3X3~3=729. f For 64 x 2 X 2 X 2 X 2 X 2 X 2 =4096. And thus also the hebdomad compounded in a triple ratio from 64 will be 46656, which is both a square and a cube; for the square root of it is 216, and the cube root is 36.
ers is also most musical. For 6 has to 1 a sextuple ratio; and the sextuple ratio produces the greatest interval in tones, by which the most sharp is distant from the flattest sound, as we shall demonstrate when we make a transition from numbers to harmonies. Again, the ratio of 5 to 2 exhibits the greatest power in harmony, nearly possessing an equal power with the diapason, as is most clearly exhibited in the harmonic canon. But the ratio of 4 to 3 forms the first harmony the sesquitertian, which is diatessaron. Another beauty likewise of this hebdomad presents itself to the view, and which is to be considered as most sacred. For since it consists of the triad and the tetrad, it exhibits that which is undiverging and naturally in a direct line in things. And it must be shown after what manner this is effected. The rectangular triangle which is the principle of qualities, consists of the numbers 3, 4 and 5.* But 3 and 4 which are the essence of this hebdomad, form the right angle. For the obtuse and the acute exhibit the anomalous, the irregular and the unequal; since they admit of the more and the less. But the right angle does not admit of comparison, nor is one right angle more right than another, but it remains in the similar, and never changes its proper nature. If however the right angled triangle is the principle of figures and qualities; but the essence + Viz. The first rectangular triangle whose sides are commensurable consists uf the numbers 3, 4, and 5. For the area of such a triangle is 6, being equal to half 3x4 the product of the two sides 3 and 4, i.e. to -. But the sides of any rec- 2 tangular triangle, whose area is less than 6, will be incommensurable. Thus, if 2x5 1x10 5 is the area of a rectanguhr triangle, it will be equal to - or to -. Hence 2 2 the two least sides will be either 2 and 5, or 1 and 10; and the hypothenuse will either be 2d29, or 24101, each of which is incommensurable. This also will be the case if the area is 4, or 3, or 2. And as the commensurable is naturally prior to the incommensurable, the rectangular triangle, whose sides are 3, 4, and 5, will be the principle of the rest. Hence too, it is evident why 3 and 4 form the right angle.
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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
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 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
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