- 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 38 and 39: animal, you will not at the same ti
- Page 40 and 41: and again, that it is the judicial
- Page 42 and 43: Since therefore number is the conne
- Page 44 and 45: ties. Thus for instance, an even nu
- Page 46 and 47: CHAPTER IV. On the predominance of
- Page 48 and 49: even, one medium cannot be found. H
- Page 50 and 51: CHAPTER VI. On the evenly-odd numbe
- Page 52 and 53: CHAPTER VII. On the unevenly-even n
- Page 54 and 55: And after this manner, if all the n
- Page 56 and 57: ives something from both through it
- Page 58 and 59: Nothing however which can be dissol
- Page 60 and 61: seventh term in order is to be meas
- Page 62 and 63: mon measure of these except unity w
- Page 65 and 66: tained of what kind they are when c
- Page 67 and 68: submultiples deficient. Thus, for i
- Page 69 and 70: fect number is produced as follows:
- Page 71 and 72: as those of greater inequality, wit
- Page 73 and 74: duced, the name of the multiple bei
- Page 75 and 76: third; and let all the triples be a
- Page 77 and 78: 4, and terminate in 20, are compare
- Page 79 and 80: intermediate longilateral number. T
- Page 81 and 82: sides. And in the following numbers
- Page 83 and 84: particular comparison and habitude.
- Page 85 and 86: Multiple superpartient ratio howeve
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the numbers produced from these wil
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1-nf inittcrn, the order of superpa
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tripartient ratios 16. 28. 49. the
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lar, or superpartient ratios, or in
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similar to themselves, is as follow
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who understand what has been alread
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medium, it will be sesquialter to 6
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and third, nothing intervenes; for
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lar figure, whether it be square, p
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first and second of the one, will c
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tagonal numbers are such as the fol
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The heptagons formed from the addit
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CHAPTER X. On solid numbers.-On the
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The pyramids also which are from th
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and eight angles, each of which is
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speaking accurately. Farther still,
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duced from equal terms multiplied e
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such as those of multiples, or supe
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CHAPTER XIX. \ That from the nature
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tween them, is multiplied by 2, and
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great power of difference. For ever
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Unequal differences.( 1. 2. 2. 3. 3
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CHAPTER XXII. A demonstration that
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of which produces that which is pro
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that middle which is alone conversa
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as will be hereafter shown. But of
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etc. the greater term will always b
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tervals. For there is one interval
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CHAPTER XXVII On the harmonic middl
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And 6+2 is more than twice 3. The e
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the arrangement 3.4. 6., the middle
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equired to insert all these middles
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as in 3. 5. 6. For as 6 is to 3, so
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The third proportion among the four
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the least. Hence a tone is the diff
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$her I ( e~epo~ &YO) which is demon
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side less. And in the first monadic
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the number of terms is finite, the
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If the half of each term of this se
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friendship, and which consequently
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In the quintuple series, 5, 25, 125
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127, 511, 2047, 8191, &c. And the s
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BOOK Two
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BOOK Two 137 Hence it appears that
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BOOK Two CHAPTER XL On another spec
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BOOK Two 141 remainder is 383. But
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BOOK Two 143 In the fourth place th
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BOOK Two And these expressions when
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BOOK Two 147 Divisors, with the add
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BOOK Two 149 is generated by man, t
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ROOK Two dissimilating, increasing
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BOOK Two 153 Again, when he says, "
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BOOK Two 155 have a contrary nature
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BOOK Two If three gnomons of square
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BOOK THREE CHAPTER I On the manner
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mon the term first is adapted to al
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sav7ov. Syrianus adds, "But Philola
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from the art which he possesses, fa
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val things without interval are the
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intellect, male and female, God, an
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y Styx, viz. they continue through
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for they say that the monadic natur
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parent. Of figures, likewise, those
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the duad was called indefinite and
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ina, Triton, and the perfect of the
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ner (than the triad,) a manifold, o
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Pythagoreans every number, because
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analogy or proportion comprehends t
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The fifth is of figures. The sixth
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consequence of moving circularly* a
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Iie further informs us, that they d
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tion of parts, and is more properly
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With respect to the appellation tru
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"The heptad is called Minerva, beca
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ooddess, and this, as Proclus infor
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as far as to the monad which is nat
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this I suppose he alludes to the eq
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xav~a apt0~ov 89' eau709.) Proclus
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lastly, according to Chalcidius on
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8~ 1 and +1= 9 the second ) 8X 7 an
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"Hence it is the first number of wh
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tion of 2 to itself, as by the mult
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any triangular gnomon, and unity is
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have a masculine property.* The mal
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6X 1 and +1= 7 the 2ndj 6X 7 and +1
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ers is also most musical. For 6 has
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that which is generated, in order t
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whole body, which extends to four t
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anged in seven orders, exhibit an a
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ody; but from the reason of the imm
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est boundary of the duration in the
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19; for the sum of these is 64. And
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9X 1 and +1= 10 the 2nd 9 x 10 and
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Egyptians that the death of Osiris
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ADDITIONAL NOTES P. 3. The motion o
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geometry and masic, which are prior
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Such therefore is the doctrine of t
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For the purpose of facilitating the
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from the triangle above it, and thr