Page:A History of Mathematics (1893).djvu/241

 Since the symbol of summation $$\scriptstyle{\int}$$ raises the dimensions, he concluded that the opposite calculus, or that of differences d, would lower them. Thus, if $\scriptstyle{\int l=ya}$, then $\scriptstyle{l=\frac{ya}{d}}$.|undefined The symbol d was at first placed by Leibniz in the denominator, because the lowering of the power of a term was brought about in ordinary calculation by division. The manuscript giving the above is dated October 29th, 1675.[39] This, then, was the memorable day on which the notation of the new calculus came to be,—a notation which contributed enormously to the rapid growth and perfect development of the calculus.

Leibniz proceeded to apply his new calculus to the solution of certain problems then grouped together under the name of the Inverse Problems of Tangents. He found the cubical parabola to be the solution to the following: To find the curve in which the sub-normal is reciprocally proportional to the ordinate. The correctness of his solution was tested by him by applying to the result Sluze's method of tangents and reasoning backwards to the original supposition. In the solution of the third problem he changes his notation from $$\scriptstyle{\frac{x}{d}}$$ to the now usual notation dx. It is worthy of remark that in these investigations, Leibniz nowhere explains the significance of dx and dy, except at one place in a marginal note: "Idem est dx et $\scriptstyle{\frac{x}{d}}$,|undefined id est, differentia inter duas x proximas." Nor does he use the term differential, but always difference. Not till ten years later, in the Acta Eruditorum, did he give further explanations of these symbols. What he aimed at principally was to determine the change an expression undergoes when the symbol $$\scriptstyle{\int}$$ or d is placed before it. It may be a consolation to students wrestling with the elements of the differential calculus to know that it required Leibniz considerable thought and