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

Let then these terms be, 243, 2«>6, 273 m . But on account of [the fraction] let the octuple of these be the

II .-fyiijr is a primary ratio, being as it were a bottom or root, from which other ratios arise.

Tim. Flat. Vou II. j

I'KOCLUS ON THE

p. 111

Gf>

numbers 1944, 2048, 2187, in order that the terms may be in perfect unities, and not in the parts of unity. Because however, it is necessary that the rat io of the leimma should !*e that of 250 to 2 18, we may demonstrate it to he so as follows: If from the sesquitertian interval, two •sesquioctaves are taken away, the terms which comprehend the remaining interval, will have to each other the ratio of 2od to 243. For let a b Is* sesquitertiau off, and let c he takt n away, which is suhses-. quioctave of the sesquioctave a b. And in a similar matin* r let */ he t.iki n from c. 1 say that d will have to < the proposed ratio. For from n b let c he taken which is «qual to ; b, and #/which is eipial c b. Since therefore, as ab is to c, so is c to d ; for they are sesquioct nos ; it will al-n he as a z is to b z, so is b z to be. Hence the remainder a z will he to the remainder ; r, as whole to whole, i. e. as*/ b In b z. But a b is sesquioctave ol b z. I leiice b ; is sesquioctave of c I. l>t - /, | H . placed equal to ze. Hence 2 //is octnph o (ha. But * e is equal to 2 //. Hence f h is eighteen times h a. Again, since z b is sesquioctave of be, tore is ses¬ quioctave of*/, hence b e is octuple oft =. Of such numbers then fore as z c is 8, of such t b is hi, and zb 72, For 72 is sesquioctave oft»4. But tin* whole a b is 81 ; for this is tin soqiiioctave of 72. The numhers, therefore, are quadruple.

1 fence of mr h numhers as <* b is 32 t, of such e b, i e. </, is 2o(>. F or 32 1 is qua¬ druple of 81, and 258 of 04. But numhers which are equally multiplied, have the panic ratio as their parts. Since tin r* loie </ b is sesquitertiau ol e, of such num¬ bers as */ b is 32 I, of such t w .11 he 2 13. For 32 I contains 2 13, and a third part of it, viz. Hi. But it has appeared, that of Midi numhers as ab is 321, of such d is 250. Hi-nee of sin li numbers as </is 250, of such * is 2 13.

It is manifest, however, that this ratio of the leimma is m the least terms. l or they are first terms with relation to each oilier. And this is evident from sub¬ traction. For they end m unity, the less bring always taken from the greater. But if they are first terms, it is evident they are the least of those that have the sain, i i(io with ihrin. If, tin leforc, two srsquinrtaves are taken From the sesquit rtian inti ival, the remaining terms will have the ratio of 2*>0 to 213.

But if they are first terms, it is evident they are the least of those that have the sain, i i(io with ihrin. If, tin leforc, two srsquinrtaves are taken From the sesquit rtian inti ival, the remaining terms will have the ratio of 2*>0 to 213.

'This therefore being demonstrated, let then- betaken in a consequent or«h r */ b fm the tonic ratio, be for the ratio of the leimma, ad for the ratio of that which is called a semitone, and </ to c for the ratio of the comma. I’or the ratio of the excess of the apotome, above that winch is truly a semitone, and which cannot !>e obtained in numhers, is thus called. I his then is demonstrated. Io wl.a, k,.s Imvii said however, it must be added, that we leave called the ratio of db a semi¬ tone, not that a sesquioctave is divided into two equal ratios ; for no superparti- * ulai ratio is capable of bt mg so divided; hut because the followers of Aristoxenus assume a semitone after two sesquioctaves, the ratio of a semitone is assii.iird, aswehav. said, according to their position, in order to discover what

HOOK 111.]

TIMJEUS OF PLATO.

* ho r:itio is of the comma and aj>otom<* to the ratio of the leimma. This therefore is asserlt-d through the cause w hicli has been mentioned by us. For that every superpailit ular ratio is incapable of being divided into two espial ratios, is one among the things that are demonstrated. Thus much, however, must be added, for the sake of elegant erudition, that as the Pythagoreans neither admit that there is a semi¬ tone from which together with two sesijuioctaves a sesquitertian ratio is produced, nor the symphony diapason and diatessaron, as the followers of Aristoxenus admit;—this being the case, the musicians posterior to him, the disciples of Ptolemy, grant with the Pythagoreans, that what is called a semitone, is not truly so, but reject the opinion, that the diapason and diatessaron are not sym¬ phonies, We, however, necessarily demonstrate the former, on account of the opinion of Plato; but not Im mg compelled to demonstrate the latter, liecause Plato says nothing about it, we shall at present omit it.

.Since then we have shown in what numbers the ratio of the leimma, and the ratio of the apotome are first found, we must likew ise show, in what numliers the ratio of the comma, by which the apotome exceeds the leimma, is first disco¬ vered. This ratio therefore is in perfect [i. e. in undivided] unities, as the ancients say, that of 53144 1 to 524288.' But if to divide unity makes no difference, let the ratio of tin* leimma be taken in that of the numbers 25G to 213. But the sesrjuioctave of 213 is 273 and of 258 208. 1 Another leimma is that of 209 to 213 For this is the ratio of the leimma. For 20.9 contains 258 and thirteen units, and 250 also contains 243 and thirteen units. Because tin refore 250 con¬ sists of 213, and besides this of thirteen units, which arc the numerator of 243 ; lienee the 13 hv w hich 250 exceeds 243, contains in itself A’ parts of 243. Fach likew ise of the thirteen units by w hich 250 exceeds 243 contains in itself F of 243. Hence 200 + A\ w ill have tli^* same ratio to 250, as 250 to 213,’ being in a Miperpartient ratio to it, and having A’» parts of it, and 243 units. lit nee that which remains, vi/:. 273 has the ratio of the comma to 200 and A\. So that it is shown in what numliers of the monad when divided, and in what two leimmas taken from the sesijuioctave, the ratio of the comma is first found. It is evi¬ dent therelore, from what has been said, that we have effected what we promised to do. The terms likewise, and all the intervals, are condensed with harmonic and arithmetic middles, and the divisions of the sesqui alter and sesquitertian ratios, into sesquioctaves and Icimmas, have been effected. For as there is a

* Lcomcu* Thoinaeu* has in his version -SCV’98.

* 288 H oiui.icri in the Greek, and also in the version ofTbomirus.

2S« 1-VK52I* 1T90*

"T~ ; TjJSO mx Mi frlSSU.

«8

BKOCLUS ON THE

[book lit.

«!up!e interval lietwoen 301 and 700, the term 432 which i« sesquioctave to 301, and 48 'J which is sesquioctave to 432, fill between them, and also 512 which makes a leiintna with 400. And thus far the sesquitertian ratio consists of two tones and the leimma.