50 SECTION PHYSICS. 



artistic and the more so as the discrepancy and difficulty lie 

 not in the unfortunate mathematician whom he so sarcastically alludes 

 to, but in a law of nature. As well may the surveyor say, " when 

 mathematicians have made the diameter of the circle commensurate 

 with the circumference, when they have established a simple formula 

 for the circular measurement of an angle, then we will proceed to 

 mensuration and go into the art of surveying." In both instances 

 we might reply that there is no time but now ; and we must struggle 

 in spite of the difficulties which the accidental or possibly the 

 intended conformation of nature, at present hidden from us, puts in 

 our way, to produce as great an amount of perfection and as great 

 a union between science and practice, as we can possibly obtain. 

 This incommensurability of nature it would take too long to explain at 

 length, but it is so admirably given in a few words by Mr. Alexander 

 Ellis, in his translation of the great work of Helmholtz, that I may 

 venture to quote the single sentence which contains its exposition. " It 

 is impossible," says Mr. Ellis, in his appendix, " to form octaves by 

 just fifths or just thirds, or both combined, or to form just thirds by 

 just fifths, because it is impossible by multiplying any one of the 

 numbers %, or $, or -|, each by itself, or one by the other any number 

 of times, to produce the same result as by multiplying any other one of 

 these numbers by itself any number of times." Thus, having this initial 

 difficulty, how has it been met, and how may it be met ? The number 

 of plans brought forward for meeting it are numerous almost number- 

 less. I will only venture to mention, on this occasion, one, two, or 

 three which stand out beyond the rest. The first is the old unequal 

 temperament. In former times players and composers of music were 

 content to restrict themselves to a certain number of keys. Modulation 

 was not so rapid nor so extensive as it has become of late years. The 

 early effects of music had to be developed before the more compli- 

 cated. Indeed, Handel's and Bach's music keep in one key. Even 

 Mozart in his early works keeps very much in one or two keys, but 

 with Mozart and Beethoven there came a time when equal tempera- 

 ment became known, and immediately these giants began to throw 

 their gigantic arms about and travel into all the irrelevant parts 

 of the scale. Thus the difficulty soon became manifest. But 

 the older composers, as I was saying, being content to keep to a 



