De l'Hôpital, Bernoulli, and the genesis of Analyse des infiniment petits

Abstract

The Marquis de l'Hôpital's Analyse des infiniment petits (1696) was the first ever calculus textbook. Its introductory chapters were based on Johann Bernoulli's Lectiones de calculo differentialium, a series of lessons written for the Marquis in 1691–92. Bernoulli also assisted with the later chapters, while the Marquis supported him with a stipend during 1694–96. We examine the text and its source material in this paper, which was presented at the research symposium ‘Editing historical mathematics: techniques and traditions since 1900’ at All Soul's College, Oxford, in December 2011.

With this in mind he published his Analyse des infiniment petits, a book that was both good and well-written … We may only find fault in that Mr. de l'Hôpital did not make well enough known the debt he owed to Mr. Bernoulli …

(Montucla 1798, 397)

Notes

¹ Montucla (p. 397) lists five: Leibniz himself, Jakob and Johann Bernoulli, Pierre Varignon and de l'Hôpital.

² De l'Hôpital spelled his name ‘Hospital’ and this was the spelling used in the posthumous second edition of 1716. By the time of the 1768 edition, this was replaced with the standard modern spelling of his name. English language authors in the early twenty-first century make use of both spellings.

³ Technically, this was the third edition, despite the words ‘Seconde Edition’ on the title page; see Figure 1. There was a ‘Seconde Edition’ of 1715, with many more typos, of which the 1716 edition is a corrected version (Bernoulli 1955, 499–500).

⁴ De l'Hôpital says ‘pension’.

⁵ Schafheitlin (1922) argued that the manuscript was a copy made in 1705 by Bernoulli's nephew Nicolaus.

⁶ We note that there is no mention of limits or any related notion in the discussion of this problem. Both Bernoulli and the Marquis clearly consider this fraction to have a determinate value when x = a.

⁷ ‘To obtain the value of the ordinate of the given curve in the given case, one must divide the differential of the numerator of the general fraction by the differential of the denominator; the quotient, after having set x equal to the supposed value of AB, will be the magnitude of BC.’ (Bernoulli 1955, 235)