# L'Hôpital's Analyse des Infiniments Petits

###### Robert E. Bradley, Salvatore J. Petrilli, and C. Edward Sandifer
Publisher:
Birkhäuser
Publication Date:
2015
Number of Pages:
311
Format:
Hardcover
Series:
Science Networks Historical Studies 50
Price:
169.00
ISBN:
9783319171142
Category:
Monograph
BLL Rating:

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on
09/21/2015
]

Even though he does not seem to have proved any important theorems nor made significant mathematical discoveries, the Marquis de L’Hospital had a significant role to play in the early days of the calculus. In 1684, Leibniz had announced his “new method” in a short article which described a “calculus of differentials” without giving any explanation of how or why it worked. Jakob and Johann Bernoulli seem to have been the first to really understand the differential calculus. The latter, finding himself in financial need, agreed to tutor L’Hospital in the new theory. The result was this book, published in 1696.

The Analyse des Infiniment Petits (Analysis of the Infinitely Small) treated the differential calculus as a tool “for the understanding of curved lines.” The book, accordingly, has two parts: in the first, L’Hospital gives an account of the basics of differential calculus; in the second, he treats topics in the differential geometry of curves. There are two final chapters which function almost as appendices: one on “other applications” and one comparing the new calculus with older ideas of Descartes and Hudde.

Originally published in French, the book is given here in English translation with (brief) notes. The authors have provided an introduction which helps the reader understand L’Hospital’s calculus, in some cases providing “translations” into modern mathematics. This introduction is simply marvelous: in less than fifty pages the authors give a surprisingly thorough account of what is in the book, of L’Hospital’s biography, and of his sources.

There has been much talk about how L’Hospital’s book relates to Bernoulli’s original notes. In his preface, L’Hospital says that most of the ideas came from Leibniz and the Bernoulli brothers, but in such a way as to avoid revealing just how dependent he was on the younger Bernoulli. On the other hand, late in life Johann Bernoulli seemed to argue that all that L’Hospital had really done was to translate from Latin to French. Since the Latin notes were rediscovered in 1921, we can now know exactly what each author contributed. The conclusion of Bradley, Petrilli, and Sandifer is that the book should be viewed as a collaborative work, a prime example of mathematical originality (Bernoulli) married to expository talent (L’Hospital).

The authors go the extra mile, however: they provide English translations of both Bernoulli’s notes and the relevant parts of Bernoulli’s correspondence with the Marquis. This allows readers to directly compare the published book with the raw material provided by Bernoulli.

The result is a truly valuable book: here is the original calculus textbook, with the sources on which it was based, presented in English with an excellent introduction that makes it much easier to follow what is going on. This one is not to be missed!

Notes:

1) Note that the authors of the book prefer L’Hôpital while I prefer L’Hospital. As the footnote on page v points out, the first is the modern spelling of the name, while the second is the one the Marquis himself used. There is no consensus on how best to spell it today, and clearly the authors disagree with me.

2) The one mathematical result traditionally connected to L’Hospital is the famous “rule.” As the authors note, the $0/0$ case of the rule (more specifically, the situation when $f(x)/g(x)$ is of the form $0/0$ but $f'(x)/g'(x)$ is not) is indisputably the work of Bernoulli, who provided it to L’Hospital with a proof. In fact, it is clear that even the question is due to Bernoulli.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.