Johann Bernoulli's first lecture from the first integral calculus textbook ever written: an annotated translation

ABSTRACT

When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented translation of the first lecture and some general information about the whole treatise. In his introductory lecture, Johann Bernoulli develops many examples of how the integration of polynomials could lead to the integration of radicals and then provides some relevant changes of variables. When studying the integration of $\frac{a\mathrm{d}x}{\sqrt{2ax+x^2}}+\frac{x\mathrm{d}x}{\sqrt{2ax+x^2}}$ the integrable combinations appear for the first time; several developments will stem from them in the future.

KEYWORDS:

• Integral calculus
• history of mathematics
• Johann Bernoulli

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. As a matter of fact, the Instituzioni Analitiche ends, on page 1020, with the following words: ‘The Analyst must be careful to avoid in solving problems the first, and even more the second, fluxions (i.e. derivatives) by means of certain devices which are seen to be used by illustrious Mathematicians in problems of elastic curves, catenaries, veils, in isoperimetric and other problems, whose solutions can be read in Acta Eruditorum, as in other works, in order to acquire the necessary skill and dexterity’.

2. Such a mistake appears only in the first lecture, as we will see.

3. It is known that JB's lectures to de l'Hospital were held in Paris, in 1691–2, until JB had to return to Basel, were followed by correspondence between them, which was published in [29], but they never met again. After the discovery of JB's archives in 1922, it was concluded that de l'Hospital had not been a mere pupil, but had actually given some contributions of his own, and, instead, did not include some JB lectures, recently published in [30]. In any case in 1704, see [31], a few weeks after the Marquis's death, JB claimed that the famous rule for calculating limits with forms of indecision was his own, and explained how, when, and why the misunderstanding originated.

4. There are, however, other Birkhäuser collections devoted to Jakob Bernoulli (issues 2, 3, 4, and 5, plus the volume which was the subject of controversy with JB) and to Daniel Bernoulli.