Royal Society. 389 



of dividing the octave which can liave its notes arranged in a con- 

 tmuous series of equal fifths, and which returns to the same pitch 

 after a certain number of fifths, is called a regular cyclical system ; 

 and it is of the rth order, positive or negative according as twelve 

 of its approximate fifths exceed or fall short of seven octaves by r 

 units of the system. Thus the system of 53 is merely a particular 

 case of a regular cyclical system ; and it is of the first order positive, 

 because 12x31—7x53=1. 



In addition to the discussion of the positive systems, the mean- 

 tone or old unequal temperament is described, and some interesting 

 historical notices concerning it are given, in Chapters V. and VI., as 

 well as suggestions for adapting the generalized key-board (with a 

 smaller number of keys) to it. " The result of the writer's practical 

 experience is, distinctly, that there are many passages in ordinary 

 music which cannot be adapted with good effect to positive systems 

 — and that the rich and sweet masses of tone which characterize these 

 systems, with the delicate shades of intonation which they have at 

 command, offer to the composer a material hitherto unworked. 

 The character of music adapted for these systems is that of simple 

 harmony and slow movement : it is a waste of resource to attempt 

 rapid music ; for the excellence of the harmonies cannot be heard. 

 The mean-tone system is more suitable for such purposes " (p. 73). 



AVe cannot say that the arrangement of the book is satisfactory ; 

 a reader would probably be disappointed who goes to it hoping to 

 find an elementary discussion of musical intervals, such as a learner 

 would make out without excessive trouble, and such as the title- 

 page appears to promise. To mention a single point ; we cannot 

 help thinking that it would have been much better if the author 

 had assumed explicitly that his readers were able to use a common 

 table of logarithms of numbers, instead of setting them to reduce 

 a certain formula to numbers — a process which is in fact that of 

 calculating the logarithms for themselves. We do not see that the 

 least advantage is gained by not employing " the labours of those 

 who constructed the tables of common logarithms " (p. 88). How- 

 ever, it is not fair to press this point, as the author himself allows 

 that " the arrangement \i. e. of his book] is unsatisfactory from a 

 scientific point of view" (p. vii) ; but w^e must be allowed to add 

 that it seems to us unsatisfactory also from the point of view of ele- 

 mentary exposition. 



LIII. P7'oceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 310.] 



April 12, 1877.— Dr. J. DaltonHooker,C.B., President, in the Chair. 



^HE following paper was read : — 



-*- "On certain Molecular Changes which occur in Iron and Steel 

 during the separate acts of Heating and Cooling." By Professor 

 Norris, M.D., Queen's College, Birmingham. 



In the course of certain researches in Physiological Physics, in 

 which the author of this paper has been for some time past engaged, 



