Page:Popular Science Monthly Volume 75.djvu/124

120 may be set for him by the physicist or the engineer. If he had done this in the past he would not have created the instruments necessary to solve such problems, and hence it is unreasonable to make such restrictions as to the future.

If the physicists of the eighteenth century had abandoned the study of electricity because it seemed to serve no useful end, we should not have had the many useful applications of electricity during the nineteenth century. Similarly, if the mathematician had abandoned the study of negative and imaginary numbers because they seemed to point only to impossibilities, we should not have had the many powerful instruments of thought which enable us to cope more successfully with many problems of nature. Just as the physicist is largely guided in his work by those facts which seem to point to general laws, so the mathematician is guided in his work by the desire to discover extensive relations and laws having a wide range of application. Millions of isolated facts present themselves to the investigator, some of which are of striking interest to the initiated, but they are of practically no value in the development of mathematics except that they may sometimes serve as an exercise in secondary instruction.

At a first thought the statement that "Mathematics is the art of giving the same name to different things" may appear to be entirely contrary to fact, but from a certain standpoint this statement conveys a very fundamental truth. It should be borne in mind that these different things must have in common the property to which this common name refers, and that it is the duty of the mathematician to discover and exhibit this common property. By way of illustration we may recall the use of x for various unknowns in algebra and the (1,1) correspondence between the two series of operators. When the language has been properly chosen it is often surprising to find that the demonstrations, as regards a known object, apply immediately to a large number of new objects without even a change of name.

Just as the boundaries between the elementary subjects of mathematics—arithmetic, algebra and geometry—vanish when the knowledge of these subjects is sufficiently extended, so the boundaries between subjects in pure and applied mathematics are disappearing, and it is exactly in these bordered lands, or in this common territory of two or more subjects, where the greatest recent progress has been made and where the greatest future activity may be expected. The work in this common territory is made possible by observing similarity of form where there is dissimilarity of matter, or by observing some other common properties which admit mathematical treatment.

In Poincaré's address some of these general observations were illustrated by numerous examples chosen from various fields of higher mathematics. On the contrary, we shall confine our illustrative