*MATHEMATICAL PHYSICS.*

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 remarked 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 idealized 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 Kirchhoff's lectures on mechanics 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.^{[1]} 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 comprehension 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 service. 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 **ὰεὶ ὀ θεὸς γεωμετρεἲ,** that definite geometrical magnitudes and relations are at the back of phenomena. It is recognized indeed that all our measurements are necessarily to some degree uncertain, 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.