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

«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.

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 sesqiiioetavi-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 sesquiferti 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.]

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.

fly 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 sesquioctave : 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.

The double and triple intervals therefore are filled with middles, and with wwjuioctaves and leimmas. The whole likewise of this diagram has nine lcimmas, and twenty-four sesqnioctaves. Tor the intervals are le^s in number than the terms by one. It also proceeds as far as to a quadruple diapason, and a diapente ami tone. Adrastus how ever, who was a lover of the arts, makes the figure, as we have said, in the form of the letter 7 ,; and places the terms in certain triangles. And in the interior triangle, indeed, he places the ratios that are in monadic imml>ors [i. e. that consist in the numl»ers within t* nj ; but in the trian¬ gle next to this, the sextuple of these numbers, w hich have two middles according to each duple or triple interval. And in the outermost triangle he places the terms winch make the whole of the lieforc-mentioned diagram. What we have said, however, will become manifest from the delineation. But between the double and triple intervals, lie inscribes all the above-menlioiii d numbers, which we have not thought tit to add. being unwilling to introduce a [needless] multitude of terms. Tor such a disposition of terms, and the insertion of the same numbers twice, is unmethodical. Tor many of the same media are found between the duple and triple intervals; since the triple intervals themselves consist of duple terms and sesquiallers. What is s,*ul by Plato, therefore, has been elucidated by ns. Tor two media have been discov* red between all tb*‘ duple and triple inter¬ vals. And from these media scsquialter anil sesquiterlian ratios having lieen produced, these are divided by the sesquioctave; a portion being left in both, which has the ratio of the leimma. From these likewise, assumed in an orderly

* Fur tiaTatrvr here, it 11 necessary lo read iir\nirtoy.

And from these media scsquialter anil sesquiterlian ratios having lieen produced, these are divided by the sesquioctave; a portion being left in both, which has the ratio of the leimma. From these likewise, assumed in an orderly

* Fur tiaTatrvr here, it 11 necessary lo read iir\nirtoy.

* The word* within the brackets arc supplied from the version of Thoma-us, where however it ■■ necessary to read scsquioctavuin instead of sesqutallerum.

p. 70

PROCLUS ON THE

[book inmanner, the terms which comprehend the whole diagram will be found to be thirty-four only.*

Siuce, however, the Pythagoric Timieus says that the terms of the diagram are thirty-six, and vet assumes the same extremes as Plato, viz. 301 and 10308, in order that these philosophers may not appear to Ik* in any respect discordant with each other, let us show how the other two ti rm.s are inserted. These tin n there¬ fore li. e. the Pythagoreans] were willing that there should not only he the ratio of the leimma in the diagram, hut also that of the apotome, whieh they twire dis¬ covered, both m radical iuuiiIrts, and in those alone which are the triple of these. Adding likewise one term to each, they introduced this into the diagram. But Plato makes no mention of the apotome-; whence aKo we being satisfied with the leimma, have alone employed the above-enumerated terms. Pur how, since lie assumes the diatonic genus, could he make use of the apotome, the sesquioetave not being divided in this genus; the apotome being produced when the sesquioctave is divided ? For the part of the sesquioetave which remains idler the leimma, is the apotome. Hence, since Plato does not mention the apotome, and it is not possible for it to occur in the dialotiii genus, it would lx ridiculous in us to end« avuur to insert other terms, in ordt r that we may h ive the apotome, the thirty-four terms Ih mg sufhi i« lit to the completion of the sesquioct; ves and lcimmas. It seems also, that the number 3 1 is adapted to the diatonic genus, in

* Thu will he evident from the following diagram, which also will he found to contain a quadruple diapason, together with the diapente and tone.

Sesq. ^esq Lemma, ‘'esq. Sesq Sesq l eimma.
JS4, 432. 4Sb. 512. 576. 648. 779. 768.
Tlie first duple interval.

Sesq. Sesq. Leimma. Sesq. Vsq. Sesq. Leimma.
86 4. 972. 102 4. 1152. 129<i. 1 45S. 1536.
The second duple interval.

Sesq Sesq Lviinma. Sesq. Sesq. Leiuima.
1728. 194 4. 2187.* 2304 2592. 2‘Jl6. 3l>72.

The third du|ile interval.

Sesq. St-sq Leimma. Sesq. Sesq. la-imma.
3456. 3SS8. 437 1. 4008. 5184. 5832. Mil.
The fourth duple interval.

* 2187 •» the octuple of 273

Sesq. Sesq. Ijcimma. Sesq.

6912. 7770. 8718. 9316. 10308.

I lie third triple interval.