# L’Hôpital

Guillaume François Antoine, Marquis de L’Hôpital, Analyse des infiniment petits: pour l’intelligence des lignes courbes (Paris, 1696), portrait.

Bos (2000) introduces his discussion of L’Hôpital’s work as follows:

Leibniz’s publications did not offer an easy access to the art of his new calculus, and neither did the early articles of the Bernoullis. Still, a good introduction appeared surprisingly quickly, at least to the differential calculus, namely L’Hôpital’s Analyse (1696).

As a good textbook should, Analyse starts with definitions of variables and their differentials, and with postulates about these differentials. The definition of differential is as follows: “The infinitely small part whereby a variable quantity is continually increased or decreased, is called the differential of that quantity.”

At the beginning of Analyse, we see this text (in French) together with an illustration indicating the lengths (or areas) together with the corresponding differentials (‘Différences’).

Guillaume François Antoine, Marquis de L’Hôpital, Analyse des infiniment petits: pour l’intelligence des lignes courbes (Paris, 1696), p. 2.

Thus the line segments, AP, PM and AM are identified, together with their respective differentials, Pp, Rm and Sm; likewise the arc, AM, with its differential, Mm; likewise the area between the line segment and arc, AM, with its differential, MAm; finally the area, APM, with its differential, MPpm (and, in this case, MA and mM are considered as arcs rather than line segments).

Guillaume François Antoine, Marquis de L’Hôpital, Analyse des infiniment petits: pour l’intelligence des lignes courbes (Paris, 1696), plate 1, figure 1.

The wide variety of books on calculus/analysis available to us nowadays had to start somewhere, and it started with this book. L’Hôpital (1661-1704) had met Johann Bernoulli (1667-1748) in 1691 in the circle of Nicolas Malebranche (1638-1715) in Paris. With the help of private lectures from Bernoulli, L’Hôpital wrote this textbook, including theorems and axioms that later mathematicians studied in depth and developed further. The best known legacy from L’Hôpital is his rule; this appears on page 5 of Analyse.

Guillaume François Antoine, Marquis de L’Hôpital, Analyse des infiniment petits: pour l’intelligence des lignes courbes (Paris, 1696), p. 5.

Analyse continued to appear for most of the eighteenth century in new editions. It is evident today that the books we use from our own library resemble the layout of the original. The preface of L’Hopital’s book talks about the aim of the book, which was the context of analysis. Ordinary analysis deals with finite quantities whereas the type of analysis described in Analyse focuses on the infinite ones. Analysis is capable of giving a true insight into the properties of curves. Curves are merely polygons with an infinite number of sides.

An overview of L’Hopital’s work would be incomplete without reference to Fontenelle’s ‘Eulogy’ which appeared for the first time in English translation in Bradley, Petrilli and Sandifer (2015), and indeed L’Hôpital is listed in the contents page of Worth’s copy of the Eloges of Fontenelle.

Sources

Bos, H. J. M., ‘Newton, Leibniz and the Leibnizian Tradition’, in Ivor Grattan-Guinness (ed.) From the Calculus to Set Theory 1630-1910: An Introductory History (Princeton University Press, 2000), 70-73.

Bradley, R., Petrilli, S. and Sandifer, C., L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli, (Basel: Birkhäuser, 2015).

O’Connor, J. J. and E. F. Robertson, Guillaume-François-Antoine Marquis de l’Hôpital, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 2008).

Stedhall, Jacqueline, Mathematics Emerging: A Sourcebook 1540-1900 (Oxford University Press, 2008).
Text: Maurice OReilly and Elaine Hickey.