MATHEMATICAL PHYSICS. 5*5 



having established their equations somehow, they can proceed to build 

 securely on these. This has led some people to the view that the laws 

 of nature are merely a system of differential equations; it may be re- 

 marked in passing that this is very much the position in which we 

 actually stand in some of the more recent theories of electricity. As 

 regards dynamics, when once the critical movement had set in, it was 

 easy to show that one presentation after another was logically defective 

 and confused; and no satisfactory standpoint was reached until it was 

 recognized that in the classical dynamics we do not deal immediately 

 with real bodies at all, but with certain conventional and highly ideal- 

 ized representations of them, which we combine according to arbitrary 

 rules, in the hope that if these rules be judiciously framed the varying 

 combinations will image to us what is of most interest in some of the 

 simpler and more important phenomena. The changed point of view 

 is often associated with the publication of KirchhofPs lectures on me- 

 chanics in 1876, where it is laid down in the opening sentence that the 

 problem of mechanics is to describe the motions which occur in nature 

 completely and in the simplest manner. This statement must not be 

 taken too literally ; at all events, a fuller, and I think a clearer, account 

 of the province and method of abstract dynamics is given in a review 

 of the second edition of Thomson and Tait, which was one of the last 

 things penned by Maxwell in 1879.* A ' complete ' description of even 

 the simplest natural phenomenon is an obvious impossibility ; and, were 

 it possible, it would be uninteresting as well as useless, for it would 

 take an incalculable time to peruse. Some process of selection and 

 idealization is inevitable if we are to gain any intelligent comprehen- 

 sion of events. Thus, in astronomy we replace a planet by a so-called 

 material particle — i. e., a mathematical point associated with a suitable 

 numerical coefficient. All the properties of the body are here ignored 

 except those of position and mass, in which alone we are at the moment 

 interested. The whole course of physical sciences and the language 

 in which its results are expressed have been largely determined by the 

 fact that the ideal images of geometry were already at hand at its serv- 

 ice. The ideal representations have the advantage that, unlike the 

 real objects, definite and accurate statements can be made about them. 

 Thus two lines in a geometrical figure can be pronounced to be equal 

 or unequal, and the statement is in either case absolute. It is no 

 doubt hard to divest oneself entirely of the notion conveyed in the 

 Greek phrase azi b 6eb^ j-ecouerpsF^ that definite geometrical magni- 

 tudes and relations are at the back of phenomena. It is recognized 

 indeed that all our measurements are necessarily to some degree uncer- 

 tain, but this is usually attributed to our own limitations and those 

 of our instruments rather than to the ultimate vagueness of the entity 



* Nature, Vol. XX., p. 213; Scientific Papers, Vol. II., p. 776. 



