270 Mr. E. H. M. Bosanquet on the History of the 



ledged by Mr. Venn (' Symbolic Logic,' p. 271), and practi- 

 cally in the construction of bis logical-diagram machine 

 ('Symbolic Logic/ p. 122) ; 



I have constructed a logical-diagram machine on the basis 

 of these diagrams. Blocks, corresponding to the squares or 

 rectangles on the diagrams and resting on two sets of slides, 

 the one set below and at right angles to the other, may be 

 made to drop when the combinations which they indicate are 

 inconsistent with the premises. 32 such slides are required 

 to operate ihe 256 blocks of an 8-term machine. 



But this machine, though simple enough, is practically 

 inferior to the diagrams which I have had printed at a trifling 

 expense for problems of 7, 8, 9, and 10 terms. 



XXXIV. On the History of the Theory of the Beats of Mistimed 

 Consonances. By B. H. M. Bosanquet, Fellow of St. John's 

 College, Oxford. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



IN a paper recently published in the Philosophical Maga- 

 zine, I have given an account of my recent experiments 

 on an important branch of this subject. I wish to contribute 

 further to the appreciation of the bearing of my results, by 

 means of a critical notice of a few of the most important 

 points connected with the history of the subject; to which I 

 propose to add some remarks on Konig's recent paper 

 \Annalen der Physik und Chemie, 1880, p. 857). 



Passing over the early history of the subject, we come to 

 Smith's ' Harmonics ' (1759), a treatise always regarded as 

 important, but difficult. Smith was mainly concerned with 

 the reduction of the phenomena to rule, for the purpose of his 

 studies of musical temperament. His fundamental principles 

 do not really reach beyond the numerical or geometrical ap- 

 pearances or patterns which arise from the superposition of 

 certain sequences of forms at regular intervals. Causal expla- 

 nation, reference to laws of sensation, analysis of the functions 

 of the ear, were not attempted by Smith. But his work is of 

 great importance. Is is still much appreciated as the first 

 and, in one sense, almost the most powerful exposition of prin- 

 ciples still largely held, which are entirely incompatible with 

 the point of view originated by Helmholtz, or its developments. 



Freed from the singular phraseology in which Smith's pro- 

 positions are enveloped, they are not difficult in themselves. 

 I will shortly state in an example the effect of his reasoning, 

 so far as it purports to deal with the causes of beats. 



