Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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XVIII INTRODUCTION mathematics to philosophy, and other sciences and arts, will, from all that has been said, be apparent. Some, however, endeavour to subvert the dignity of the mathematical science, by depriving it of beauty and good, because it does not make these the subjects of discussion, and others by endeavouring to evince that sensible experiments are more useful than the universal objects of its speculation, as for instance, geodesia than geometry, vulgar arithmetic than that which consists in theorems, and nautical astronomy than that which demonstrates universally. For, say they, we are not made rich by our knowledge of riches, but by the use of them; nor do we become happy by a knowledge of felicity, but by living happily. Hence we must confess that not speculative but practical mathematics contribute to human life and actions. For those who are ignorant of the reasons of things, but are experienced in particulars, excel in every respect, in what is useful to human life, those who are engaged in theory alone. Against objections then, of this kind, we shall reply by showing the beauty of the mathematical disciplines from those arguments by which Aristotle endeavours to persuade us. For these three things are in a remarkable degree effective of beauty, in bodies and in souls, viz. order, symmetry, and the definite. Since corporeal deformity indeed, arises from material irregularity, privation of form, and the dominion of the indefinite in the composite body. But the baseness of the soul originates from its irrational part being moved in a confused and disorderly manner, and from its being discordant with reason, and not receiving from thence its proper limitation. Hence beauty has its essence in the contraries to these, viz. in order, symmetry, and that which is definite. These, however, we may survey in the most eminent degree in the mathematical science; order indeed, in the perpetual exhibition of things posterior and more various, from such as are first and more
INTRODUCTION XIX simple. For things subsequent are always suspended from those that precede them; the former having the relation of a principle, but the latter, of things consequent to the first hypotheses. But we may perceive symmetry, in the concord of the things demonstrated with each other, and the reference of all of them to intellect. For intellect is the measure of all science, from which it receives its principles, and to which it converts the learners. And the definite is seen in the perpetually stable and immutable objects of its theory. For the subjects of its knowledge, do not subsist differently at different times, like the objects of opinion and sense, but they present themselves to the view invariably the same, and are bounded by intellectual forms. If therefore these particulars are in an eminent degree effective of beauty, but the mathematical sciences are characterized by these, it is manifest that in these the beautiful subsists. Indeed, how is it possible this should not be the case with a science which is supernally illu,minated by intellect, to which it tends, and to which it hastens to transfer us from the obscure information of sense? But we ought to judge of its utility, not looking to the conveniences and necessities of human life. For thus also we must acknowledge that contemplative virtue itself is useless; since this separates itself from human concerns, to which it does not tend, nor is in short desirous of making these the objects of its knowledge. For Socrates in the Theaetetus, speaking of the coryphxan philosophers, or those that philosophize in the most eminent degree, says, that through intellectual energy they are separated from all habitude to human life, and from an attention to its necessities and wants, and that they extend the reasoning power of the soul without impediment to the contemplation of real beings. The mathematical science, therefore, must be considered as desirable for its own sake, and for the contemplation it affords, and not on account of the
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 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
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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
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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
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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
show all

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