Dr WHEWELL, ON PLATO'S SURVEY OF THE SCIENCES. 589 



an interval, and that this is the smallest possible interval, by which others are to be measured; 

 while others say that the two notes are identical : both parties alike judging by the ear, not by 

 the intellect. 



" You mean, says Socrates, those fine musicians who torture their notes, and screw their 

 pegs, and pinch their strings, and speak of the resulting sounds in grand terms of art. We 

 will leave them, and address our inquiries to our other teachers, the Pythagoreans." 



The expressions about the small interval in Glaucon's speech appear to me to refer to a 

 curious question, which we know was discussed among the Greek mathematicians. If we take 

 a keyed instrument, and ascend from a key note by two octaves and a third, (say from A L to 

 C 3 ) we arrive at the same nominal note, as if we ascend four times by a fifth (A } to E x , E x 

 to fi 2 , B 2 to F-i, F 2 to C 3 ). Hence one party might call this the same note. But if the 

 Octaves, Fifths, and Third be perfectly true intervals, the notes arrived at in the two ways will 

 not be really the same. (In the one case, the note is ^ x -1 x 4- ; in the other % x % x % x |- ; 

 which are |- and 1A, or in the ratio of 81 to 80). This small interval by which the two notes 

 really differ, the Greeks called a Comma, and it was the smallest musical interval which they 

 recognized. Plato disdains to see anything important in this controversy ; though the con- 

 troversy itsef is really a curious proof of his doctrine, that there is a mathematical truth in 

 Harmony, higher than instrumental exactness can reach. He goes on to say : 



" The musical teachers are defective in the same way as the astronomical. They do indeed 

 seek numbers in the harmonic notes, which the ear perceives : but they do not ascend from 

 them to the Problem, What are harmonic numbers and what are not, and what is the reason 

 of each"?* " That, says Glaucon, would be a sublime inquiry." 



Have we in Harmonics, as in Astronomy, anything in the succeeding history of the Science 

 which illustrates the tendency of Plato's thoughts, and the value of such a tendency ? 



It is plain that the tendency was of the same nature as that which induced Kepler to call 

 his work on Astronomy Harmonice Mundi ; and which led to many of the speculations of 

 that work, in which harmonical are mixed with geometrical doctrines. And if we are disposed 

 to judge severely of such speculations, as too fanciful for sound philosophy, we may recollect 

 that Newton himself seems to have been willing to find an analogy between harmonic numbers 

 and the different coloured spaces in the spectrum. 



But I will say frankly, that I do not believe there really exists any harmonical relation in 

 either of these cases. Nor can the problem proposed by Plato be considered as having been 

 solved since his time, any further than that the recurrence of vibrations, when their ratios are 

 so simple, may be easily conceived as affecting the ear in a peculiar manner. The imperfec- 

 tion of musical scales, which the comma indicates, has not been removed; but we may say that) 

 in the case of this problem, as in the other ultimate Platonic problems, the duplication of the 

 cube and the quadrature of the circle, the impossibility of a solution has been already esta- 

 blished. The problem of a perfect musical scale is impossible, because no power of 2 can be 

 equal to a power of 3 ; and if we further take the multiplier 5, of course it also cannot bring 

 about an exact equality. This impossibility of a perfect scale being recognized, the practical 

 problem is what is the system of temperament which will make the scale best suited for musical 

 purposes; and this problem has been very fully discussed by modern writers. 



TIMS %v/i<t>uwoi dpidfiol, &C. 



