David Nightingale: L'Hopital's Rule (l'Hopital, 1661-1704)

May 27, 2013

There's been talk over the years about whether or not a famous rule, called l'Hopital's Rule, after the French nobleman Guillaume-Francois-Antoine Marquis de l'Hopital, Comte d'Entremont, Seigneur d'Ouques-La-Chaise – which is not even his full name – possibly 'bought' the rule from the famous mathematician Johann Bernoulli, who was born in 1667.

In this kind of thing one is reminded of controversies such as Johanson vs the Leakeys on whose fossils were oldest, or, who invented the internet – not Vice President Al Gore of course, but rather Tim Berners-Lee – or who invented calculus, Leibniz or Newton? Such situations arise in music, history, art and so on all the time.

That 17th century Bernoulli family, of Basle, Switzerland, sprouted not a few brilliant people. There was a Jacob Bernoulli, his brother Johann Bernoulli, both highly regarded in mathematics, and Johann's son Daniel Bernoulli – famous today for the principle that explains why planes and birds stay up, and why the tails of super-boomer automobiles stay down.

The Marquis de l'Hopital, born in 1661, was 6 years older than Johann Bernoulli, and the nobleman was himself well-regarded as a mathematician. For example, at the age of 15, he had solved, by himself, a problem concerning cycloids that the eminent Pascal had proposed.

The marquis had started life as a cavalry officer, but was known for working math problems in his tent. Because of short-sightedness he had resigned, retiring to devote himself to mathematics. He was thus able to attend lectures in Paris, including some by Leibniz, as well as to spend time at his country estate.

Hearing about the brilliance of Johann Bernoulli, he offered considerable monies to the relatively poor Johann Bernoulli to tutor him.

At the age of 35 the Marquis de l'Hopital published the first book ever written on Differential Calculus [ref.2], and that famous rule – today universally called l'Hopital's rule, appears in Ch.9.  But there are many who believe this isn't right, and that it should be Bernoulli's rule.

However, in the acknowledgments for his text the Marquis wrote '… I own myself very much obliged to … Messieurs Bernoulli, and Leibniz ... and … whatever they please to claim their own ...' [Refs.1&2].  There is also in existence a letter from Johann Bernoulli that said the nobleman could use the lessons as he wished.

So whose rule should it really be?  Well, since Johann Bernoulli developed it, and l'Hopital published it, and we already have Daniel Bernoulli's name in hydrodynamics, perhaps the fairest thing would be to call it the “Bernoulli-l'Hopital” rule. A compromise, as with Leibnitz and Newton.

As for Al Gore and the internet, well, that could be the subject of a different essay.


1. www-history.mcs.st-andrews.ac.uk

2. “Analyses des infiniment pour l'intelligence des lignes courbes.” (“Infinitesimal calculus with applications to curved lines.”) This book was translated into various European languages.

Dr. David Nightingale is Professor Emeritus of Physics at the State University of New York at New Paltz, and is the co-author of the text,  A Short Course in General Relativity.

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