Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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The third proportion among the four, but the ninth in order is, when three terms being given, as is the middle to the least term, so is the difference of the extremes to the difference of the less terms; as in 4. 6. 7. For 6 to 4 is sesquialter, the difference of which is 2. But 7 differs from 4 by 3, which is sesquialter to 2. And the fourth proportion, but which is in order the tenth, is when in three terms, as is the middle to the least term, so is the difference of the extremes to the difference of the greater terms; as in 3. 5. 8. For 5 the middle term is to 3 in a superbipartient ratio. But the difference of the extremes is 5, which to the difference of the greater terms 8 and 5, i.e. 3, is also superbipartient." CHAPTER XXXII On the greatest and most perfect symphony, which is extended in three intervals;-and also on the less symphony. IT now remains for us to speak of the greatest and most perfect harmony, which subsisting in three intervals, obtains great power in the temperaments of musical modulation, and in the speculation of natural questions. For nothing can be found more perfect than a middle of this kind, which being extended in three intervals, is allotted the nature and essence of the most perfect body. For thus we have shown that a cube, which possesses three dimensions, is a full and perfect harmony. This, however, may be discovered, if two terms being given, which have themselves increased by three intervals, viz. + Jordanus Brunus has added an eleventh to these proportions; and this is when in three terms, as is the greatest to the middle term, so is the difference of the extremes to the difference of the grater terms; as in 6. 4. 3. For as 6 is to 4; so is the difference of 6 and 3, i.e. 3, to the difference of 6 and 4, i.c. 2.
BOOK Two 117 by length, breadth, and depth, two middle terms of this kind are found, which having three intervals, are either produced from equals, through equals equally, or from unequals to unequals equally, or from unequals to equals equally, or after some other manner. And thus, though they preserve an harmonic ratio, yet being compared in another way, from the arithmetical middle; and to these, the geometric proportionality which is between both, cannot be wanting. Let the following, therefore, be an instance of this arrangement, viz. 6. 8. 9. 12. That all these then are solid quantities, cannot be doubted. For 6 is produced from 1 X 2 X 3; but 12 from 2 X 2 X 3. Of the middle terms, however, between these, 8 is produced from 1 X2X4; but 9 from 1 X3X3. Hence all the terms are allied to each other, and are distinguished by the three dimensions of intervals. In these, therefore, the geometric proportionality is found, if 12 is compared to 8, and 9 to 6; for 12 : 8 : : 9 : 6. And the ratio in each is sesquialter. But the arithmetical proportionality will be obtained, if 12 is compared to 9, and 9 to 6; for in each of these the difference is 3, and the sum of the extremes is the double of the mean. Hence we find in these, geometrical and arithmetical proportion. But the harmonic proportionality may also be found in these, if 12 is compared to 8, and again 8 to 6. For as 12 is to 6, so is the difference between 12 and 8, i.e. 4, to the difference between 8 and 6, i.e. 2. We may likewise here find all the musical symphonies. For 8, when compared to 6, and 9 to 12, produce a sesquitertian ratio, and at the same time the symphony diatessaron. But 6 compared to 9, and 8 to 12, produce a sesquialter ratio, and the symphony diapente. If 12 also is compared to 6, the ratio will be found to be duple, but the symphony diapason. But 8 compared to 9, forms the epogdous, which in musical modulation is called a tone. And this is the common measure of all musical sounds; for it is of all sounds
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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
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: BOOK Two 115 For here the greatest
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 188 and 189: parts of these wholes have sometime
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
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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.
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$Fin of natural effects. Because li
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er is not alone even, nor alone odd
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the vegetative life. And the seed i
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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
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position; for I conjecture this to
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~ c ~ ~ ~ Or ~ Q rather, ~ v . whic
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The ratio of 12 to 9 is sesquiterti
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monad. Thus if from each of the num
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had these appellations in consequen
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and well-ordered digression (from t
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unity is added to the product, the
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And so of the rest, which abundantl
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Again, of any two numbers whatever,
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Again, if 4 multiplies any triangul
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With respect to the hexad, it is th
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and the double of it 12, the number
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incorporeal and corporeal essence;
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of the hebdomad 3 and 4 necessarily
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composition of which the hebomad is
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it is also surpassed by the last nu
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excite the glad husbandmen to the c
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mentioned particulars, but also to
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7X 1 and +1= 8 the 2nd 7 x 8 and +
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8X 1 and +1= 9 the 2ndl 8X 7and +1=
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10X 1 and +1= 11 the 2ndl i' 10x30
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tureX according to this number. In
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through a good allotment, or throug
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titude, and communion with itself,
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evenlyeven numbers, and for not dis
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ing this table from the Epanthematc
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for instance, 12 is a number longer
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