JS'otices respecting New Books. 387 



Irate various methods of solution. There can be little doubt that 

 most students only obtain a firm hold of the principles of Mechanics 

 gradually, by applying them to particular cases. 



The printing, form, and binding of the volume are such as to 

 suggest that it is designed as a companion to Mr. Williamson's 

 Treatises on the Differential and Integral Calculus, which have 

 been recently noticed in our pages. Wq cannot perhaps better 

 express our sense of its merits than by saying that it is well fitted 

 for the position it appears to be designed to fill. 



An Elementary Treatise o?i Musical Intervals and Teynjperament, with 

 an account of an Enharmonic Harmonium exhibited in the Loan 

 Collection of Scientific Instruments, South Kensington, 1876 ; also 

 of an Enharmonic Organ exhihited to the Musical Association of 

 London, May 1875. By E. H. M. Bosajstqtjet, Pelloiv of St. John's 

 College, Oxford. London: Macmillan and Co. 1876. (8vo. 

 Pp. XX and 94.) 



Dr. Staiuer, in the course of a work on "Harmony founded 

 on the Tempered Scale," used the following words : — " When 

 musical mathematicians shall have agreed amongst themselves upon, 

 the exact number of divisions necessary in the octave ; when me- 

 chanists shall have invented instruments upon which the new scale 

 can be played ; when practical musicians shall have framed a system 

 of notation which shall point out to the performer the ratio of the 

 note he is to sound to its generator ; when genius shall have used all 

 this new material to the glory of art — then it will be time enough to 

 found a Theory of Harmony on a mathematical basis." Mr. Eosan- 

 quet, who quotes this passage (p. xi), seems to have regarded it as 

 a challenge, to which the book before us is in some sort an answer. 

 The main points on which the answer turns he states as follows : — 

 " The theory of the division of the octave has now been completely 

 studied ; a generalized key-board has been invented and constructed, 

 upon which all the new systems can be played ; a notation has been 

 framed by which, in systems of perfect and approximately perfect 

 fifths and thirds, the exact notes can be indicated ; and it has been 



shown that other systems require no new notation The new 



material may be therefore said to be ready " (p. xij. In explanation 

 of these points we hope the following brief statement will be suffi- 

 cient, though far from complete. The division of the octave, on 

 which the principal stress is laid, and to which our statement must 

 be limited, is into 53 equal intervals. This is called the system of 

 53 ; and in it 31 intervals give a very nearly perfect fifth. In 

 fact, if the interval of an octave be taken as unity, the perfect fifth 

 is 0-58496, the approximate fifth in the system of 53 is 0-58491, 

 while in equal temperament it is 0-58333. The (major) third in 

 the system of 53 is also a close approximation to the perfect third, 

 at all events much closer than the equal-temperament third — the 

 numerical values of the intervals being respectively 0-32193, 

 0-32075, and 0-33333. Another important point is the following : 

 by tuning up seven fifths and down four octaves, and by tuning 

 down five fifths and up three octaves, two semitones are obtained, 



2C 2 



