TKANSACTIONS OF THE SECTIONS. 9 



Both these theories are found to explain not only the phenomena by the aid of 

 which they were originally constructed, hut other phenomena, which were not 

 thought of or perhaps not known at the time ; and both have independently arrived 

 at the same numerical result, which gives the absolute velocity of light in terms of 

 electrical quantities. 



That theories apparently so fundamentally opposed should have so large a field of 

 truth common to both is a fact the philosophical importance of wliich we cannot 

 fully appreciate till we have reached a scientific altitude from wliich the true rela- 

 tion between hypotheses so difterent can be seen. 



I shall only make one more remark on the relation between Mathematics and 

 Physics. In themselves, one is an operation of the mind, the other is a dance of 

 molecules. The molecules have laws of their own, some of which we select as most 

 intelligible to us and most amenable to our calculation. AVe form a theory from 

 these partial data, and we ascribe any deviation of the actual phenomena from this 

 theory to disturbing causes. At the same time we confess that wliat we call dis- 

 turbing causes are simply those parts of the true circumstances which we do not 

 know or have neglected, and we endeavour in future to take account of them. We 

 thus acknowledge that the so-called disturbance is a mere figment of the mind, not 

 a fact of nature, and that in natural action there is no disturbance. 



But this is not the only way in which the harmony of the material with the 

 mental operation may be disturbed. The mind of the mathematician is subject to 

 many disturbing causes, such as fatigue, loss of memory, and hasty conclusions ; and 

 it is found that, from these and other causes, mathematicians make mistakes. 



1 am not prepared to deny that, to some mind of a higher order than ours, each 

 of these errors might be traced to the regular operation of the laws of actual think- 

 ing ; in fact we ourselves often do detect, not only errors of calculation, but the 

 causes of these errors. This, however, by no means alters our conviction that they 

 are errors, and that one process of thought is right and another process wrong. 



One of the most profound mathematicians and thinkers of o\u time, the late 

 George Boole, when reflecting on the precise and almost mathematical character of 

 the laws of right thinking as compared with the exceedingly perplexing though 

 perhaps equally determinate laws of actual and fallible thinking, was led to another 

 of those points of view from which Science seems to look out into a region beyond 

 her own domain. 



" We must admit," he says, " that there exist laws " (of thought) "which even 

 the rigour of their mathematical forms does not preserve from violation. We must 

 ascribe to tliem an authority, the essence of which does not consist in power, a 

 supremacy which the analogy of the inviolable order of the natural world in no 

 way assists us to comprehend." 



Mathematics. 



Chi the Problem of the in-cmd-clrciimscrihcd TriawjJe. 

 By Professor A. Catlet, LL.D., F.E.S. 



I have recently accomplished the solution of this problem, which I spoke of at 

 the Meeting in 1864. The problem is as follows : required the number of the tri- 

 angles the angles of which are situate in a given cuito or curves, and the sides of 

 which touch a given curve or curves. There are in aU 52 cases of the problem, 

 according as the curves which contain the angles and are touched by the sides are 

 distinct curves, or are any or all of them the some curve. The first and easiest 

 case is when the curves are all of them distinct ; the number of triangles is here 

 = 2aceBJ)F, where a, c, e are the orderx of the cui-ves containing the angles (or, say, of 

 the angle-cunes) respectively; andB, D, Fare the classes oit]xe cmTes touched by 

 the sides (or, say, of tne side-curves) respecti'Nely. An interesting case is when the 

 angle-cur\'es are one and the same cuiTe; or, say, a=c=e (where the sign = is 

 used to denote the identity of the curves); the number of triangles is here 



