Oct 30, 1870] 



NATURE 



62^ 



production of beats in the ear between pure tones depends 

 on the interval : — 



(Ellis's "Helmholtz," p. 260.) "On the other hand we 

 have seen that distinctness of Ijeating and the roughness 

 of the combined sounds do not depend solely on the 

 number of beats. For if we could disregard their magni- 

 tudes all the following intervals, which by calculation 

 should have 33 beats, would be equally rough : — 



" The semitone ^' /' 



„ whole tones c'al'andi/'^ 



,, minor third eg 



,, major third ce 



„ fourth G c 



„ fifth CG 



(to which we may add the octave Cj C). 



" and yet we find that the deeper intervals are more and 

 more free from roughness." 



Helmholtz then proceeds to give an approximate deter- 

 mination of this important law, for which we must refer 

 to his work. Our own impression is that this law is 

 almost solely concerned in the variation of the roughness 

 of different combinations. We ourselves hear the roughness 

 of beats up to very high numbers, and consider that up 

 to high numbers beats of sensible intensity do not fail to 

 be heard by reason of their number only. If this is the 

 case the rapidity of beats must be of less importance in 

 the theory of consonance than the law of dependance on 

 intervals exhibited in the above quotation from Helm- 

 holtz. 



To show the practical importance of this : — 



(Pole, p. 213.) " Here we find the two fundamental notes 

 themselves {c" - /) beating at the rate of 64 per second. 

 . . . This is, therefore, a less perfect combination than 

 the fifth ; but still the beats are quick, and the effect is 

 not disagreeable." 



This seems to us incorrect. If the 64 fundamental 

 beats per second were present with any intensity to speak 

 of, the combination would certainly be most dissonant. 

 It is because the ear receives the two notes on different 

 parts of the sensorium, and so gets them out of each 

 other's way, that the beats do not exist in sensible inten- 

 sity, and do not produce dissonance. 



In the appendix on Beats, and an essay there referred 

 to, Dr. Pole has developed doctrines which arise to some 

 extent from the point of view above indicated. The 

 statement made is substantially that the beats described 

 by Robert Smith (" Harmonics," 1749), have a real exist- 

 ence, besides the various types of beats described by 

 Helmholtz. 



Smith's cycles are best seen if the sum of two har- 

 monic curves be described by Donkins's harmonograph, 

 or some such machine. Smith's doctrine consists of the 

 statement that the cycles which appear in the resulting 

 curves are the cause of the beats. (Of course Smith did 

 not use pendulum-vibrations, but the use of these adapts 

 the doctrine to our modem knowledge.) 



Now in order that these cycles maybe seen, it is neces- 

 sary that one and the same scribing point should describe 

 the sum of the two motions simultaneously. If the 

 motion be analysed and its two components be described 

 separately on the paper, the cycles fail to appear. 



This is what must happen in the ear if the doctrines 

 of Helmholtz are even approximately true. The two 

 sounds (if beyond the minor third apart) fall more or less 

 completely on different parts of the sensorium, and the 



conditions requisite in the first instance for the formation 

 of Smith's cycles are not fulfilled. Whether, if the cycles 

 existed, the beats could arise out of them in the way in 

 which we hear them, is quite a different question, on which 

 we will not now enter. 



The great importance of this question has induced us 

 to prolong our remarks on it. On these points every 

 student should consult Helmholtz's work. But on the 

 more purely musical questions Dr. Pole's book has its own 

 value. 



OUR BOOK SHELF 



A Treatise on Chemistry. By H. E. Roscoe, F.R.S., and 

 C. Schorlemmer, F.R.S., Professors of Chemistry in 

 Owens College, Manchester. Volume II. Metals. Part 

 II. (London; Macmillan and Co., 1879.) 



This portion of Professors Roscoe and Schorlemmer's 

 work treats of the metals of the iron, chromium, tin, 

 antimony, and gold groups, also of spectrum analysis, 

 the natural arrangement of the elementary bodies, and the 

 condensation of the gases formerly called permanent. 

 The treatment of these subjects is characterised by the 

 same accuracy of description and clearness of explanation 

 and arrangement that were so conspicuously displayed in 

 the former parts, and the illustrations of metallurgical 

 operations, &c., are well chosen and admirably executed, 

 such, indeed, as are not to be found in any other English 

 manual of chemistry. Amongst them may be especially 

 noticed the figures of the plant for Weldon's method of 

 regenerating manganese dioxide from chlorine residues, 

 of the various forms of blast-furnace, of the Bessemer 

 and Siemens-Martin processes for making steel, and of 

 hydrauhc gold-mining as practised in California. The 

 best methods of detecting and estimating the several 

 metals are carefully described, and interesting details are 

 given relating to their history, some of which will, we 

 think, be new to many readers. 



Spectrum analysis, in which Prof. Roscoe is known to 

 be a high authority, is well treated and illustrated, and 

 attention is drawn to recent speculations, founded on 

 spectroscopic observation, respecting the possible resolu- 

 tion of the bodies now regarded as elementary, into still 

 simpler forms of matter. In the chapter on the Natural 

 Arrangement of the Elements, a clear view is given of 

 the remarkable relations between the properties of the 

 elements and their atomic weights, first pointed out by 

 Mr. Newlands, and further developed by Lothar-Meyer, 

 and Mendelejeff; and the volume concludes with an 

 account of the condensation of the gases formerly re- 

 garded as permanent, in which the ingenious forms of 

 apparatus employed for the purpose by MM. CaiUetet 

 and Pictet are fully described and illustrated. 



Altogether the two volumes of the work now published 

 form a treatise on Inorganic Chemistry of which English 

 science may well be proud ; and the student who masters 

 their contents will not fail to acquire a sound elementary 

 knowledge of the subject. H. Watts 



Elementary Mechanics, including Hydrostatics ami 

 Pneumatics. By O. J. Lodge, D.Sc. Chambers's 

 Elementary Science Manuals. (Edinburgh, 1879.) 

 This is one of the comparatively sound text-books which, 

 since the publication of Thomson and Tait's work, have 

 been every year more effectually thrusting aside the 

 cumbrously artificial and often erroneous introductions to 

 Physical Science which reigned almost unchallenged till 

 about sixteen years ago. Dr. Lodge knows his subject 

 well, and lias evidently bestowed very careful thought 

 upon it. Still we cannot unreservedly commend his 

 book ; and this for several reasons. First, he evidently 

 proceeds under the idea that the subject can be made 



