The Commentaries of Proclus on the Timaeus of Plato — trans. Thomas Taylor (1820)

teen and more, than that which alone exceeds hy thirteen, according to the ratio of excess. A tone, therefore, cannot he divided into equal parts, hut this is ts < leimma , as I lato also calls it, and that which has the greater ratio is apotome, as musicians arc accustomed to denominate it. For let 273 have to 213 a srsqui- octave ratio, hut 230, to the same 213, the ratio of the leimma, which has a loss ratio than that of the seventeenth part, it is evident that 273’, which has- the ratio of a tone to 213, will bate to 250 the ratio of the apotome, which is the remainder of the leimma, lreing greater than the seventeenth part, which we have demonstrated to lx* less than the ratio of the leimma. If therefore we multiply these right times, we shall find the first numbers winch in perfect unities have the ratio of the apotome. For the octuple of 243 is 1914, of 25G is 2048, of 273j, 2187. Hence the ratio of the apotome in radical numlrers (tv Wj}it<Tiv ’) is that of 2107 to 2040. And we shall he in want of these three terms which arc in a consequent order, in the diagram. 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

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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 sesqui- t 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 Aristoxe- nus assume a semitone after two sesquioctaves, the ratio of a semitone is as- sii.iird, aswehav. said, according to their position, in order to discover what

* 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 super- pailit 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 ses- rjuioctave 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 leim- mas 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.

"T~ ; TjJSO mx Mi frlSSU.

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.

A-rain 570 is scsquioctavc to 512, fl-IH to 570, 72?) to 018, and 708 has the ratio of the 1« iuuna to 720. And from these the sesquialter is fdled, having three sesquioctaves, and ont leimma.' Hut the whole is duple, consisting of live ses- qiiioetavi-s, and two leimmas. A"ain, according to the above described terms 304, and 700, the term 512 produces an harmonic, but 570 an arithmetical medium. Farther still, 00 1 is placed as scsquioctavc to 700, but ‘*72 is sesquioctave to 001, and 1021 has the ratio of the leimma to 072. To 1021 also 1152-is scsquioctavc. And now after the duple the sesquialter ratio is produced, "hie i m «k* s a trip e ratio, viz. the ratio of 1152 to 301. Hut between tins triple interval, 570 is the harmonic middle to the extremes, but 700 the arithmetic middle. For a theorem of the f ullouiw' kind is universally demonstrated, that ij of the same let m, one number i t double, but another triple, ami a certain mean of the double is assumed according to arith¬ metical proportion, this m an u ill be to the triple number an harmonic middle. But the great!r term in the duple ratio,ill Fa me the arithmetical mean in the triple. Tims for instance, in the above terms, 708 is the double of 304, but lloi is the triple. Between also the duple terms 700 and 301, an arithmetical mean 570 is assumed ; and the same mean between the triple terms 3H1 and 1152 is seen to be an har¬ monic mean. And 700 which was duple, becomes between tin* triple terms an arithmetical mean. Afterwards, 12!>0 is sesquioctave to 1152, and of this 14513 is the sesquiuctave, to which 1530 lias the ratio of the leimma. And as far as to this, the second duple is tilled, bein^ composed ol the se>quialter and sesquifer- ti in ratios, tin* extremes of which are 700 and 153*1, and are divided into five sesquiuctaves and two leimmas. It likewise has for the harmonic mean 1021, am! for the arithmetical mean 1 152.

A^ain, 1720 is sesquioctave to 153*5, of this 1 J> 14 is sesquioctave, of tins 2107 Is sesquioctave, and to this 23o t has the ratio of the leimma. But the sesquioc¬ tave of 23**1, is 25!*2 ; of this 2010 is the sesqmoetave, and to this 3072 has the ratio of the leimma, which is octuple ol the t’lr-t [tart, filling the third double. And farther still, the sesquioctave of 3**72 i> 315*5. Ami as far as to this t ie second triple extends, having lor its extr« iiies 1 152 anil .115*5, and for its harmo¬ nic- mean 172*5, but for its arithmetical nn-aii 23**1. In addition to this also, the

* 1 ,ir 70 S i> tcitjuullcr In !>l t, and bctvveiu lluae Ihu tecno, ilnrc tire ihe above llin-c sevquioctavci. jud one leumna.

HOOK Ml.]

TIM /EL’S or PLATO.

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sesquioctave of 3450 is 3888, but of this tin? sesquioctave is 4374, to which 4008 has the ratio of the lcinuna. The sesquioctave also of 1008 is 3101, and of this again, the sesquioctave is 5832, to which 014 1 has the ratio of the leimuia, the sesquioctave of who li in (581*2. And this again is another duple' [ti/.. 3450 and 091*2] after the before-mentioned three dnples [and afterwards another sesquioc- tave : for 777fl is sesquioctave to flJ»12j :* lor in the third triple, there is also a certain duple. And again, the sesquioctave of 7770 is 07 10, to which 0210 has the ratio of the lcinuna, and of 0210 the sesquioctave is 10308. And as far as to this, the third triple is extended, lieiug comprehended in the terms 3150 and 10308, and having two means, the harmonic and the arithmetical, the former ol which is 3101, but the latter 0912.