The Commentaries on the Timaeus of Plato — trans. Thomas Taylor

But tlie harmonic medium is that in which the

* \ocv is omillttl in the original.

BOOK III.]

TIM /HUS OF PLATO.

middle term is exceeded by the greater, I»y tin* same part of the greater, liy which it exceeds the loss tmn, as in tin* mimlirrs 3, 1 ami 3. For here 1 is exceeded l>v ti 1»v *2, \\ is tin* tliird part of 3, ami it exceeds 3 the less term l»y 1, which is the third part of 3.’ Ami the geomi trical medium is that, in which there is the same ratio of the greater to the middle term, as there is of the middle to the less term.

The methods however of discovering these, must in the next place he unfolded hy us. Id two terms, then fore, he given, hetwivn which it is proposed to find an harmonic, and also an aritlum tie medium ; and let the terms have a duple ratio, as for instance 1*2 and 0. I take, therefore, the excess of the greater mini her above the less, which is evidently 3, and dividing it into two equal parts, 1 arid the half to the less nu miter, and make this the middle term. Hence is the arith¬ metical medium In-tween 12 and 3. For the excess is three, hotli of the greater ahovc the middle, and of the middle above the least term. Again, taking the difference of the extremes, w hich is 0, I multiply this by the less term, and the product is 30, and dividing this l»y the sum ol the extremes, i. e. hy 13, the quo¬ tient 2 is produced, which is the breadth ol the comparison * lo tins also, I add 0, ami I have the harmonic middle 3. y For hy that part of the greater term 1*2 hy which 3 is exceeded hy it, hy this part of the less term 0, 0 exceeds the less. For it is exceeded h\ the third pari of 1*2, and hy a third part ol 0 it exceeds 0. Again, let there lie a triple interval, as for instance lit and <»,’ adding these together I make 24, of which taking the half, 1 have the arithmetical middle 12. Again, taking the i \c< ss of 13 above 0, i e. 12, I multiply it hy the less term 0, and the product is 7*2. This 1 divide hy 24 the sum of the extremes, and 3 the breadth of the comparison is produced. Afterwards, 1 add this to 0, and 1 have ft for the harmonic medium, which exceeds ami is exceeded hy the same part of the extremes. Thus also, if 1 and *2 were the extremes, hy adding them together, and taking the half of both, 1 shall have 1 and the half of I, for the middle term ol the aritlm.etieal middle. Hut taking the excess of the greater term above unity, and multiplying it hy the h*ss term, viz. unity hy unity, I have t from both. Afterwards dividing* this hy 3, the sum of the extremes, I shall

' Il.immmr proportion mav also lir d, liool In lie that, in which the difference between the greatest anil niiddic li-rin, ;■> l<> the dilVcrrnic IkIwicii Ihe iniilillc and least term, as (lie greatest term is to the l< ait 4 Inis in i!ie iionihi rv (>, l, 3, as ti— t t—3 :: (i: 3 ; via. as 2 is In I, so is C to 3.

as 2 is In I, so is C to 3.

1 Fv the brtndth cf the con/iari-rn, Prochis means the ratio of the terms first proposed to etch other, winch m this iii'l nne i'duple.

* er is mmltcil ill the orii,uidl.

* For TapaXafjtir here, it i> obviously necessary to read firfuiur.

PHOBIA'S ON THE

[book 111.

<K> have the breadth of the third part of 1 ; in order that 3 being compared to one may make the third part of I. Addin" therefore, this * to !, I shall have \ whicliis the harmonic medium Ix-twcen 1 and «, in the same manner as before. Hence, by employing these methods, we shall in a liecomimr manner fill all the double and triple intervals, with arithmetical and harmouiacal middles; which Tiimeus has comprehended in the geometric middle, and which he increase's by the insertion of the other middles.

In short, sinca* Plato makes mention of the three middles, which are compre¬ hended in the "cometric middle, let tin* following theorem In* added [as a corol¬ lary] to what has been said. If the analogy consists in four terms, and one of the intermediate numbers produce s an arithme tical middle, tin* other will produce an harmonic middle, and vice versa. For let there be four terms, it, A, c, d, so that the first a,' is to b, as c is to d, and let b he an arithmetical middle, [so that a, b, d, are in arithmetical proportion,] 1 say that c is an harmonic middle*. For because the product of a by d is equal to the product of b by c, but b is an arithmetic middle and tin* product of c by a added to tin* product of c by d is the double of the product of A by c, as m the arithmetic middle; this luring tin? case, it follows that the product of c by a add* d to the product of c by </, is tin* double of the pro¬ duct of a by d . 1 I tut this was the property of the harmonic middle, viz. that the product of the middle by the e\tn men, is the double of tbe product of the ex¬ tremes. Again, let c In* an harmonic middle, I say that A is an arithmetic middle. For sinee the product of r by a added to the product of c by d, is tin* double of the product of A by c, the sum of a added to d is the double of A. 1 13ut tins is an arithmetical middle, when the sum of the t xtrenies is the double of the middle term. Airain of these four terms, let A be an arithmetic, hut c an harmonic mean, I say that as a is to A, so is c to d. For because tin* product of t by a, added to tbe product of e by d, is the double of the product of a by d, on account of the harmonic middle, hut tin* sum of u added to d, is tin double of A on at count of tin* arithmetic middle, hence tin* product of a by d will Ih* espial to tin* product of A bv c. As a therefore is to A, so is c to d.* But this was the peculiarity of the geometric middle. Hence those* two middles are contained in the geometric

* a is omitted in tins place in llie original.

* a u to b!!c: d !iy hvpothesis, ami therefore ad—be. Hut ca -fed— Cbc ; and because be—ad, therefore Cbc—Cad

Hence those* two middles are contained in the geometric

* a is omitted in tins place in llie original.

* a u to b!!c: d !iy hvpothesis, ami therefore ad—be. Hut ca -fed— Cbc ; and because be—ad, therefore Cbc—Cad

* Since ta + cd=Cbc, it follows since c multiplies all tbe three terms ca, cd, Cbf, that a-*-d—Cb.

* a+dxC—Cad. But Cbxc=2ad, and therefore bxc=axd. Hence a: b^c : d. Tbe trulbof lL>s nu\ be seen in numbers, by |>uUini; (j. 12. 1*. lt>. for a. b. c d.

BOOK III.]

T1 MAIL'S OF PLATO.

p. 01

middle, 1 and reciprocate with each other, Since however we have premised thus much, let us proceed to tiie text of Plato.

“ In the first place, he took one part from the whole. After this, he separated a second part double of the first: and again, a third part, sesipiialtcr of the second, but triple of the first.”

The mathematical theory is neither to he entirely despised [in the present discussion] nor to he alone embraced itself by itself. For the latter will not exhibit to us the things which Plato intended to represent to us in images, and the former will cause the whole exposition to Is* unproductive of advantage. For it is uecessarv to consider the essence of the things which are the subject ot discussion, as on a scour ■ foundation. As we observed therefore Indore, we shall proceed in a middle way, first mathematically, in a manner adapted to the sub¬ jects, and after this we. shall unfold the division presented to our view in the text. The Pvthagi.re.ins then conceive magnificently, respecting the division or -a etion of the rule ill this place, viz. that Plain unfolds in it the essential causes, and the reasons which are generative of mathematical theorems. I<et us, therelore, as I have said, lir-t mathematically exercise the reasoning power of the reader, by contractedlv explaining w hat is asserted by many, at tin: same time abstaining from controversy, and investigating the truth by itself. Our discourse, however, will he in short, concerning these five particulars; viz. concerning multiple ratios; the media that subsist between these; the scscpiitertian and sesrpiialtrr ratios, which present themselves in the middles; the scsipiioctaves which fill these inter¬ vals; and the Icimma. For it is necessary that the diagram should In* comprehen¬ sive of all these, and he condensed with all these ratios.