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

For it is necessary that the diagram should In* comprehen¬ sive of all these, and he condensed with all these ratios.

That we may proceed therefore in order, we shall assume the ratios which are first mentioned by Plato, in the numbers from unity. Let unity then be posited, and the double of this g; afterwards .3, which is sescpiialter indeed of 2, but triple of 1; then i which is the double* of 2 ; afterwards ft, the triple of 3; afterwards 0 , the oc tuple of 1 ; and after all, the seventh term, whii h is twenty-seven times 1. Some, therefore, as we have: said, arrange these munis rs in the form of the letter X, making the monad the summit, and arranging the* double umiuIkts here, but the triple there. Ihit others more conformably to Plato, arrange them m one order only. For he does not say, that lla* triple were apart from the duple numbers,

' After the wont »J>io y in the* original, it in m-cessArv to mipply from the vcr»*n of Iconic - uf Thoma u» I he woriU, rrptryi rra« ap« n« tvri prooTrjTti <r rp yfu/ifTfiirrj

1 Tor rpirXaeia heir, il if obviously necessary to reicl fiwKniua.

1MIOCLUS ON THE

[bouk m.

<32

1MIOCLUS ON THE

[bouk m.

<32 hut he alternately mixes them, as proceeding in a right line. If, however, Plato had stopped here, there would have been nothing further for us to dismiss. But since he Ionise If exhorts us to hind the douhh and triple intervals with harmonie and arithmetic middles, and it is not possible to discover these middles between 1 and 2, some first number must he assumed, which leeing the least, may have a half and a third part. For eu ry number may have a double, and this must there¬ fore 1 h- investigated. 1/ t then b he assumed, ami the double of it 12, the former having the same ratio to the latter as 1 to 2. Between these then fore, \i/. 1 and 2 multiplied bv b, placing as media 8 ami M, we shall have the above-mentioned middles. For 0 exceeds and is exet edi d by the same part of the extremes ; hut 9 exceeds and is exceeded aecending to an t-ipial number. lienee by multiply¬ ing: 1 and 2 six times, we shall find numbers receiving the befure-meiitiimed middles. In a similar maimer by multiplying by b the remaining double and tuple numbers m the hefore-inelitioned order, we shall liml the terms which we may Ik- able to condense with arithmetic and harmonic middles. For sextuple numbers will Ik* produced from all the before-mentioned orders, by arranging other iiuiiiImts, only observing that 48 eiught to Ik- placed before 5-1; ill this respect departing from the. arrangement of Plato, who plac es !» before 8, m order that he might alternately change the duple and triple* ratios. \\ c, however, make this alteration, as consentaneous to the* multitude ol the monads, and the nature ot increasing number. Hence 8 and it come between band 12; lent between 12 and tlu* double of it 21, the harmonic mean is lb, and the arithme tic 18. And between the* third double 1 21 and -lit, the harmonic mean is 82, but the arithmetic 38. But in triple* nimilie.s, between b and 18 which are thu lirst triple*, the har¬ monic middle is !>, but the* arithmetic 12. Betwe en the second triple 18, and 51, the harmonic middle is 27, but the* arithmetic 3b. And In-tween the* third triple 54 and lb2, the harmonic middle is 81, but the arithmetic 108. The double and triple intervals therefore, are divided by these* two middles. So that these terms will he successive to e ach other, viz. (i. 8. 0. 12. lb. 18. 24. 27. 32. 3b. 48. 51. 81 108. 182.

If, however, it was possible in the terms described by us, to divide the* sesepiitertian ratios, inte> sesejuioctaves and leimmas, we* should leave no occasion to preeeeed ativ further. But now, as this is not possible, we are in want of another method. Since, there fore, it was proposed at first, tee condense? the duple ratio, with the he*fori*-tncntione*d middles, and with scsepiioctaves, it is necessary that tlu subduple term, should have the sesejuitertiaii together with the two sesepiioctaves.

’ For rj>ir\(t(ucu lien , it is obviously necessary to reael?i*Xa<uoL.

the duple ratio, with the he*fori*-tncntione*d middles, and with scsepiioctaves, it is necessary that tlu subduple term, should have the sesejuitertiaii together with the two sesepiioctaves.

’ For rj>ir\(t(ucu lien , it is obviously necessary to reael?i*Xa<uoL.

•took III.]

TINL-EU3 <>l PLATO.

p. 83

Ix-t there he taken then in the first place, the third number from unity, according to an octuple ratio, viz. 84. From this it is possible to form two sesquioctaves. For every multiple number is the leader of as many multiple ratios denominated ('mi i itsi If, as it i» it"< If distant from unity. But it has not a sesipiiti itian. 1 Itv tripling therefore 84, we shall have 192, the seMpiitertian of which 1 - 258, but the sesiplioclave 218. and of this tin* sesqnioctave is ‘213.* But the ratio of the leimma is that which remains after the ablation of the two sesquioctaves '213 and 2 !8. For from every sesquitt rtian two sesquioctaves licinjr taken, the ratio of the leimma is left. But of 2-'#G the sesqnioctave is 233, which preserves an arithme¬ tical mean Iwtween 192 and 331, which has a duple ratio to 192, and a srsquitertiau to 233. If, therefore, it were possible to form two sesquioetaves from 283, we ini^ht also condense this sesquiterlian with sesquioctaves and a leimma. iSovv, however, this is not possible. For the sesqnioctave of it, 324, has not an eighth part. Hence if we wish to preserve unity always undivided it is impos¬ sible there should be a sesqnioctave ratio to it. For the eighth part of it is 40 and i. By doubling this, therefore, in order that we may make the half a whole, we shall be able to assume the eighth part of it. On this account, however, we shall be compelh d to double all the numliers prior to it, and also those posterior to it. Hence instead of 192, we shall have 331 ; instead of 210, 432; instead of 213, 18f>; instead of 258, 312; ami instead of 283, .'#73. And of tln> the s< Mpiioctave, is (#18, and of this 72.9.' Afterwards 788, which is the double of 381, has the ratio of a leimma to 729. After this manner, therefore, the double interval is Idled with sesquialtcr, s) squiterlian, and sesqnioctave ratios, in the numbers, 381. 132. 488, .#12. .#78.848.729. 7G8, Hence if we wish lo till the whide diagram, and to describe all the munhers in a consequent order, instead of the first part we must assume 381; instead of tin* double of the first, 7G8; instead of tin* triple of the first, hut tin* sesquialtcr of the second, 1152; instead of the quadruple of the lir<t, 1538; instead of the fifth part which is triple of the third, 3158; instead of the sixth part which is octuple of the first, 3072; and instead of the seventh part, which is the twenty-seventh pait of the fust, 103G8.