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
* 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.
T1 MAIL'S OF PLATO.
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.
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.
[bouk m.
If, however, it was possible in the terms described by us, to divide the* sesepii- tertian ratios, inte> sesejuioctaves and leimmas, we* should leave no occasion to pree- eeed 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.
•took III.]
TINL-EU3 <>l PLATO.
If, therefore, we also wish to condense these terms with harmonic and arithme¬ tic midrib's, which I join;; inserted, make sesquialtcr and sesquiterlian intervals, the intermediate numbers will 1m* 384, and 788, the double of 381; 512, which produce* an harmonic, and 578, which makes an arithmetic middle. But if we
1 The number 61 has not a sesquitertian in whole numbers. Fur as 3 is lo 4 <o is 01 to SjJ.
* For as fl is to 8, so is 243 to 216.
* For as b is to 0 so is 576 to 618 ; auil also so is 64b to 7 -9*
[book III
wish to assume the above-mentioned middles of the triple interval, viz. of 304 and 1152, then 576 will preserve the harmonic ' middle, which filled for us the arithmetic * middle in the double 1 interval; and 760 will be the arithmetic middle, which was the greater extreme of the double interval. Again, if we wi-di to assume the same middles of the duple and quadruple, i. e. of the middles between the terms 760 and 1536, the former of which is the double of 301, and the latter the quadruple, the harmonic midtile will bo 1024, ami 1152 t ic arithuie- tic middle. If also we widi to condense the second triple, the terms of which are 1152, and 3456 [the former being the double of 576, and the latter the triple of 1152 ] then 1720 will give us the harmonic, and 2301 the arithmetic middle. And if we wish to condense the third double, w hich consists in the terms 1536 and 3072, then 2 ) 16 will be the harmonic, and 2301 the arithmetic middle. But if we wish to condense the third triple, with similar middles, but I mean the fifth and seventh part, the extremes will be for us 3456, and 10360; but the harmonic middle will be 5101, and the arithmetic 6012. If again, we should condense each of the sesquin rlians which present themselves from these mid¬ dles, and sesquialters, with sesquioctaves and a leimina, this will lie manifest to us after the whole exposition, when we exhibit the whole diagram with all the terms in a consequent order, which has indeed 24 s ( squioctaves hut 9 leimmas.
These things therefore, having been elucidated by us, we shall observe thus much concerning tlie leimmu, that as it is not possible to divide any superparti- tular into equal ratios, a semitone cannot be assumed in numb* rs ; hut taking the ratios which are contiguous to each other, vi/ the seventeenth and the sixteenth part, and demonstrating that the seventeenth part is greater than that which is called the leimina, and vvh eh is h ss than an accurate M ini- tone, it is inferred that the leimmu and also the seventeenth part are less than a semitone. But that it is h s S than a semitone, is demonstrated as follows; I>‘t there lie given the term 16 and the sesquioctave of it 16. Betw een these placin': 17 it will divide the scsquioctave into unequal ratios, vvhii i will he near to the semitonic interval, since 17 dillirs from the extremes by unity alone. And it is evident that it will make a greater ratio with the less t< rm; because in all arithmetical proportion, the ratio is greater w hieli is in the lcssteims, so that the seventeenth part is less than a semitone. .Moreover, the leimmu is less than the seventeenth part, as is evident from the terms exhibited by IMato. For since 256 has to 243 the ratio of the leimina, as we shall demonstrate in what
» For aottlfiiTTui)*' here. m the orijiinut it i» necessary lo read npfioy^y.
’ For mj» alto here, we mini re-d apiD^ntttv.
1 And for rp*»Xa<riy, il is requisite U> read }iw\avi{-
"'•] TJMyEUS OF PLATO. , ;5
follows where we shall show that the radical ratio of the leimma is in these nuinhers , anti since 256 exceeds 243 hv less than the seventeenth part of it;
for if exceeds it hy 13 unities, hut the seventeenth part of 2 13 is more than 13;_
tills bring the case, much umre is the ratio of the h iinma less than the seinitonie interval. Hence the ratio which remains to the completion of a tone, and which is called the ratio oi an apotome, is necossanlj greater than a semitone.
I arther still, this may also he deinonstrated after another manner, as follows: Lctthe iiuiiiIhts 2-i 6 and 243 Ire given, and let there lie assumed three numlMTS in a conseipient order, in a ratio of this kind ; from 256 indeed, 63336, but Irom 243 39049, and from both 6>Z08. These three numlien, therefore, are analogous in the ratio of the leimma, which, if it is a semitone, will be the tonic ratio ot the extremes. lint if it is greater than a semitone, that also wi ! he greater than a tone; hut if less, that also will he less. The sesrpiioctaTe how ever of 3309, is 664310 i. But this is greater than the greater term.
Afu r another and a third way the same thing may also Ire demonstrated, viz. that a tone cannot he divided into two erjual parts, hating the same ratio as that of 230 to 243. For if we take the eighth part of 213, w hicli is 30 \ and add this i irf to it, we shall make 73 w hich has a sesquioctave ratio to 243. You see therefore that 230 has to 2 43 a less ratio than 273’, to 230. For 250 has to 243 a superpartient ratio, exceeding it hy -Vj ; hut 273 } exceeds
2oO, hy ,\\\ __ But the ratio is greater which exceeds hy seven¬