in 1666, and employed it in the composition of his Principia, although in the work as printed any special algebraic notation is avoided. But he did not print a direct statement of his method till 1693. Leibniz published his first statement in 1684. He was accused by Newton's friends of having got it from a MS. by Newton, which he had been shown privately. Leibniz also accused Newton of having plagiarized from him. There is now not very much doubt but that both should have the credit of being independent discoverers. The subject had arrived at a stage in which it was ripe for discovery, and there is nothing surprising in the fact that two such able men should have independently hit upon it.
These joint discoveries are quite common in science. Discoveries are not in general made before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea. If we merely keep to discoveries in which Englishmen are
concerned, the simultaneous enunciation of the law of natural selection by Darwin and
Wallace, and the simultaneous discovery of
Neptune by Adams and the French astronomer,
Leverrier, at once occur to the mind.
The disputes, as to whom the credit ought to be given, are often influenced by an unworthy spirit of nationalism. The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races. Indians, Egyptians, Assyrians, Greeks, Arabs, Italians, Frenchmen, Germans, Englishmen, and Russians, have all made essential contributions to the progress of the science. Assuredly the jealous exaltation of the contribution of one particular nation is not to show the larger spirit.
The importance of the differential calculus
arises from the very nature of the subject, which is the systematic consideration of the rates of increase of functions. This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity. Thus the fundamental idea of change, which is at the basis of our whole perception of phenomena, immediately suggests the enquiry as to the rate of change. The familiar terms of "quickly" and "slowly" gain their meaning from a tacit
reference to rates of change. Thus the differential calculus is concerned with the very key of the position from which mathematics can be successfully applied to the explanation of the course of nature.
This idea of the rate of change was certainly in Newton's mind, and was embodied in the 32 language in which he explained the subject. It may be doubted, however, whether this point of view, derived from natural phenomena, was ever much in the minds of the preceding mathematicians who prepared the subject for its birth. They were concerned with the more abstract problems of drawing tangents
to curves, of finding the lengths of curves, and of finding the areas enclosed by curves. The
last two problems, of the rectification of curves and the quadrature of curves as they are named, belong to the Integral Calculus, which
is however involved in the same general subject as the Differential Calculus.