- NEWTON AND LEIBNIZ 471
yÆXFigure 5. Leibniz' proof of the fundamental theorem of calculus.the horizontal axis.^2 The second curve has an ordinate proportional to the area
under the first curve. That is, for a positive constant a, having the dimension of
length, aF(x) is the area under the curve y = f(x) from the origin up to the point
with abscissa x. As we would write the relation now,In this form the relation is dimensionally consistent. What Leibniz proved was that
the curve æ = F(x), which he called the quadratrix (squarer), could be constructed
from its infinitesimal elements. In Fig. 5 the parentheses around letters denote
points at an infinitesimal distance from the points denoted by the same letters
without parentheses. In the infinitesimal triangle CE(C) the line E(C) represents
dF, while the infinitesimal quadrilateral HF(F)(H) represents dA, the element of
area under the curve. The lines F(F) and CE both represent dx. Leibniz argued
that by construction, adF = f(x)dx, and so dF : dx = /(÷) : a. That meant that
the quadratrix could be constructed by antidifferentiation.
Leibniz eventually abbreviated the sum of all the increments in the area (that
is, the total area) using an elongated S, so that A — I dA = I ydx. Nearly all
the basic rules of calculus for finding the derivatives of the elementary functions
and the derivatives of products, quotients, and so on, were contained in Leibniz'
1684 paper on his method of finding tangents. However, he had certainly obtained
these results much earlier. His collected works contain a paper written in Latin
with the title Compendium quadraturae arithmeticae, to which the editor assigns
a date of 1678 or 1679. This paper shows Leibniz' approach through infinitesimal
differences and their sums and suggests that it was primarily the problem of squar-
ing the circle and other conic sections that inspired this work. The work consists
of 49 propositions and two problems. Most of the propositions are stated without
aF(x) = / /(f) dt.(^2) The vertical axis is to be assumed positive in both directions from the origin. We are preserving
in Fig. 5 only the lines needed to explain Leibniz' argument. He himself merely labeled points on
the two curves with letters and referred to those letters.