# Mathematical Treasure: Leibniz's Papers on Calculus - Fundamental Theorem

Author(s):
Frank J. Swetz (The Pennsylvania State University)

Shown above is the title page of the 1693 volume of Acta Eruditorum.

The September 1693 issue (No. IX) of Acta Eruditorum, began with an article by Leibniz (G.G.L.). In “Supplementum Geometriae Dimensoriae…” ("Supplement on geometric measurement"), he showed that the general problem of quadrature can be reduced to finding a curve that has “a given law of tangency.” A modernization of this accomplishment would be showing that the general problem of definite integration can be reduced to finding a function that has a given derivative – that is, finding an antiderivative function – which is essentially the Fundamental Theorem of Calculus. A partial translation of the article from Latin to English can be found in D. J. Struik's A Source Book in Mathematics (1200-1800), pp. 282-284.

For diagrams accompanying the article, see the second page below, "Tab[ula] V," inserted between pages 386 and 387.

For modern calculus notation, see page 390.

Figures 1-3 in Table V, above, are referenced in the margins of the pages that follow.

On page 390, above, at the start of the first full paragraph, Leibniz seemed to get to the mathematical point of his article, writing, "I shall now show the general problem of quadratures [integration] to be reduced to the invention [finding] of a line [curve] having a given law of declivity [tangency]." Also on page 390, be sure to find the integral sign $\int$ near the bottom of the page, in the sentence, "Ergo $a\,dx=z\,dy,$ adeoque $ax={\int{z}\,dy}={\rm{AFHA,}}$" or "Therefore, $a\,dx=z\,dy,$ so that $ax={\int{z}\,dy}={\rm{AFHA.}"}$

The images above are used through the courtesy of the Lilly Library, Indiana University, Bloomington, Indiana. You may use them in your classroom; for all other purposes, please seek permission from the Lilly Library.

##### Reference

D. J. Struik (editor), A Source Book in Mathematics (1200-1800), Harvard University Press, Cambridge, Mass., 1969.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Leibniz's Papers on Calculus - Fundamental Theorem," Convergence (June 2015)