Page:The World and the Individual, First Series (1899).djvu/236

Rh ingenuity of a Gauss was taxed to furnish some of the known solutions. The problem has proved fundamental for algebraic theory. And so my Western friend was wrong.

Of course this is but a single instance. Very many other mathematical cases can be found where problems as to real Being, of the type here at issue, have been the topic not only of inquiry, but of serious and sometimes pretty persistent error on the part of even rioted mathematicians themselves. Such was the fortune of the older Theory of Functions with regard to the existence of the differential coefficients of continuous functions. This case cannot be fully explained in non-mathematical language. It is enough here to say that the mathematical world contains countless ideal entities of the type called Functions, and these are beings which have values corresponding to the values of certain quantities called “independent variables.” The values of the “functions” therefore, in general, vary when the “independent variables” vary. If the functions vary continuously, whenever the variables vary continuously, the variation of the functions may correspond to such a physical process as a movement, or to such a process as the description of a curve, on a surface, by a continuous motion. Now such an ideally definable process generally has properties corresponding to the rate of the physical motion, or to the instantaneous direction of movement of a point on the curve. And these properties of the functions in question may be investigated by constructing certain other ideal entities, related to the original functions, and derived by a well-known process from them. The