Above is the title page for the 1684 volume of Acta Eruditorum.

The first page of the October 1684 issue (number X) of Acta Eruditorum is shown above. Leibniz's first article describing the Calculus appeared on pages 467-473 of this issue.

A plate of diagrams for Leibniz's article on the Calculus was placed opposite page 467, the first page of the article. The reader is referred to it in the very first line of the article: note "TAB.XII," or Table XII, in the righthand margin of page 467, below.

Leibniz’s first article describing the Calculus was published in October, 1684. The article, “Nova methodus pro maximis et minimis, itemque tangentibus,…, & singulare pro illis calculi genus” [“New method for the maximum and minimum, and also tangents, …, and a singular type of calculus for them”], was a definitive exposition of his differential calculus. The adjective "singulare" is often translated as "remarkable": "a remarkable type of calculus for them." The initials "G.G.L." are for Gottfried Wilhelm Leibniz with Wilhelm latinized to Guilielmus.

Do you recognize the product and quotient rules for differentials in the second paragraph above? It helps to know that "aequ." is used for "equals" or "=". The equals sign, =, is used in Leibniz's articles from 1686 and 1693 on the next two pages.

The remainder of the article is presented below in its entirety. A translation of the article from Latin to English can be found in D. J. Struik's A Source Book in Mathematics (1200-1800), pp. 271-280.

On the page above (page 469), Leibniz discussed differentials of powers (potentiae) and of radicals (radices). He also named the methods discussed in his article "differential calculus": see "calculi hujus, quem voco differentialem," or "this calculus, which I call differential" at the start of the last paragraph.

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

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.

Reference

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

Index to Mathematical Treasures

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