Page:The World and the Individual, First Series (1899).djvu/237

218 new ideal entities are called Derived Functions, or Differential Coefficients, and for a long time it was assumed as almost an axiom that every function continuous within given intervals must have, within those intervals, a derived function, or differential coefficient. This seemed as axiomatic as the assertion that every movement must take place at a given, even if constantly altering speed, or that a point moving on a curve must at every instant be moving in a given instantaneous direction. For the derived function, or differential coefficient, was an ideal entity corresponding to such facts as momentary velocity, or instantaneous direction of movement. This assumption, namely the existence of objects called the differential coefficients in question, persisted in the text-books until instances, first few, and then many, were produced, where beings of the type in question, namely continuous functions, were discovered, which had no differential coefficients whatever. How this was possible, I cannot pause to define, but I mention this now noted example of a pretty persistent mathematical error, because it exemplifies how, in the world of pure mathematical creations, you can have problems about existence which for a while seem as baffling as similar problems in physics and in natural history. Even mathematical science, then, has had, within the eternal shadowland of its creations, to deal, as it has grown, with sharp contrasts between myth and fact, between false report and real existence, — with contrasts, I once more insist, as striking as those known in the realm of astronomy or of history. The difference between the one science and the others lies in the fact