Newton to Euler

Since the symbol of summation $\int$ raises the dimensions, he concluded that the opposite calculus, or that of differences $d$, would lower them. Thus, if $\int l=ya$, then $l={\displaystyle \frac{ya}{d}}$. 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 ${\displaystyle \frac{x}{d}}$ 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 ${\displaystyle \frac{x}{d}}$, 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 $\int$ 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 attention[]39

to determine whether $dxdy$ is the same as $d(xy)$, and ${\displaystyle \frac{dx}{dy}}$ the same as $d{\displaystyle \frac{x}{y}}$. After considering these questions at the close of one of his manuscripts, he concluded that the expressions were not the same, though he could not give the true value for each. Ten days later, in a manuscript dated November 21, 1675, he found the equation $yd\bar{x}=d\overline{xy}-xd\bar{y}$, giving an expression for $d(xy)$, which he observed to be true for all curves. He succeeded also in eliminating $dx$ from a differential equation, so that it contained only $dy$, and thereby led to the solution of the problem under consideration. "Behold, a most elegant way by which the problems of the inverse methods of tangents are solved, or at least

are reduced to quadratures!" Thus he saw clearly that the inverse problems of tangents could be solved by quadratures, or, in other words, by the integral calculus. In course of a

half-year he discovered that the direct problem of tangents, too, yielded to the power of his new calculus, and that thereby a more general solution than that of Descartes could be

obtained. He succeeded in solving all the special problems of this kind, which had been left unsolved by Descartes. Of these we mention only the celebrated problem proposed to Descartes by De Beaune, viz. to find the curve whose ordinate is to its sub-tangent as a given line is to that part of the ordinate which lies between the curve and a line drawn from the vertex of the curve at a given inclination to the axis.