Modern calculus**, **which can be defined as **“**the mathematical study of continuous change,” was developed independently by two of the great thinkers of the 17th and 18th centuries, namely, Isaac Newton and Gottfried Wilhelm Leibniz.

In this article, my focus will be on the work of Leibniz, and I will show, based mainly on the analysis in Dunham and Mena, how he derived the well-known formula for the integration by parts.

Figure 1: A portrait and a statue (without the wig) of Leibniz.

For an article about Isaac Newton’s early mathematical achievements, see the link below.

A Beautiful Proof of a Well-Known Mathematical Result

Gottfried Wilhelm Leibniz was the quintessential polymath. He made fundamental contributions to a broad range of topics, including philosophy, mathematics, linguistics, theology, engineering, jurisprudence, law, computer science, and geology. Leibniz once remarked that he often needed to use a substantial part of his week to document the thoughts of just a single morning (for an outstanding biography of Leibniz, see Antognazza).

Below, on the left, is a picture of his working space in his last residence in Hannover. On the right, the folding chair that he took with him when he traveled.

Figure 2: On the left, a picture of Leibniz’s workroom at his last residence in Hanover (source). On the right, the folding chair that he took with him when he traveled (source).

The titles of the three most famous articles on the Calculus that Leibniz published in the first scientific journal of German-speaking Europe, the *Acta* *Eruditorum*were:

*Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus*(1684)*De geometria recondite et analysi indivisibilium atque infinitorum*(1686)*Supplementum geometriae dimensoriae, seu generalissima omnium Tetragonismorum effectio per motum: similiterque multiplex constructio lineae ex data tangentium conditions*(1693)

In English, the titles are, respectively:

- “A new method for maxima and minima, and for tangents, that is not hindered by fractional or irrational quantities, and a singular kind of calculus for the above mentioned”
- “On a hidden geometry and analysis of indivisibles and infinite”
- “A Supplement to the Geometry of Measurements, or the Most General of all Quadratures to be Effected by a Motion: and likewise the various constructions of a curve from a given condition of the tangent.”

The title pages are shown in Fig. 3 below.

Figure 3: The three most famous articles on the Calculus that Leibniz published in the scientific journal Acta Eruditorum (source)

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