Above is the title page of the 1686 volume of *Acta Eruditorum.*

This is the first page of the June 1686 issue (Number VI) of *Acta Eruditorum, *in which Leibniz published a second article describing the Calculus on pages 292-300.

In the June 1686 issue of *Acta Eruditorum, *Leibniz (G.G.L.) published *“De geometria recondita et analysi indivisibilium atque infinitorum*,” or "On a hidden geometry and analysis of indivisibles and infinites." In this article we find the first public occurrence of the integral sign \(\int\) and a proof of “The Fundamental Theorem of Calculus.” A partial translation from Latin to English of the article can be found in D. J. Struik's *A Source Book in Mathematics (1200-1800),* pp. 281-282. The remaining pages of the original article appear below.

On page 297 above, Leibniz pointed out that \(p\,dy=x\,dx\) implies \({\int{p}}\,dy={\int{x}}\,dx\), and therefore, in particular, \(d\left({\frac{1}{2}}xx\right)=x\,dx\) implies \({\frac{1}{2}}xx={\int{x}}\,dx.\) He then wrote, "... sums and differences or \({\int}\) and \(d,\) are reciprocals" ("summae & differentiae seu \({\int}\) & \(d,\) reciprocae sunt"), and concluded from his preceding equations that \({\int{p}}\,dy={\frac{1}{2}}xx.\)

*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.*

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