Origins of L’Hôpital’s Rule

Guillaume-François-Antoine Marquis de l’Hôpital, Marquis de Sainte-Mesme, Comte d’Entremont et Seigneur d’Ouques-la-Chaise, known popularly as Guillaume L’Hôpital, was born in 1661 in Paris to a family with a powerful military legacy. However, against his family’s wishes and the widespread perception of nobility in France, he was passionate about math from a young age. During his military service, he pretended to rest in his tent and instead studied geometry. Bernard de Fontenelle wrote about him in his eulogy of L’Hôpital:

For it must be admitted that the French nation, although as well mannered as any other, is still in that sort of barbarism by which it wonders whether the sciences, taken to a certain point, are incompatible with nobility, and whether it is not more noble to know nothing. … I have personally seen some of those who served at the same time, greatly astonished that a man who lived like them was one of the leading mathematicians in Europe.

L’Hôpital left the French army due to a vision impairment, though it was rumored he simply wanted to pursue mathematics full-time. Now twenty-four, he attended the Congregation of the Oratory in Nicolas Malebranche’s circle (a group which gathers for discussion and fellowship,) which was populated by many of the leading mathematicians and scientists of Paris. There, he met Johann Bernoulli, the younger and more petulant brother of Jakob Bernoulli, who had taught Leibniz in his youth and was already considered a mathematical genius. L’Hôpital was Bernoulli’s most enthusiastic student and soon paid him to tutor him privately, instead.

L’Hôpital submitted a problem solution from the course Bernoulli had given him to Christiaan Huygens without saying it was not his own. Understandably, with no evidence to the contrary, Huygens assumed L’Hôpital had done it. Bernoulli was angry and broke off his frequent letter correspondence with L’Hôpital for six months–but broke his silence once L’Hôpital asked him for more “discoveries” on a three-hundred-pound (and increasing) retainer. He asked his tutor to also give him exclusive rights to his breakthroughs and lectures. Bernoulli quickly responded that he wouldn’t publish anything again in his life if L’Hôpital desired.

Drawing from Bernoulli’s discoveries and notes from his lectures, L’Hôpital published what would become the first calculus textbook: Analyse de infiniment petits pour l’intelligence des lignes courbes (Analysis of Infinitely Small Quantities for the Understanding of Curves.) In it, he outlines how to evaluate otherwise indeterminate limits:

1. Grant that two quantities, whose difference is an infinitely small quantity, may be taken (or used) indifferently for each other; or (which is the same thing) that a quantity which is increased or decreased only by an infinitely small quantity may be considered as remaining the same.

2. Grant that a curve may be considered as the assemblage of an infinite number of infinitely small straight lines; or (which is the same thing) as a polygon of an infinite number of sides, each infinitely small, which determine the curvature of the curve by the angles they make with each other.

Though not presented as formally as in contemporary calculus textbooks, as in Section 4.4 of Stewart’s Calculus: Early Transcendentals, which describes:

as the rule of L’Hôpital (cited in the book as L’Hospital,) his original statement and the modern iterations are conceptually identical. When L’Hôpital speaks about infinitely small differences, this is analogous to the representation of limits. The idea of “infinitely small straight lines” represents the geometric understanding of differentiation and is an ancestor of our current concept of derivative. Overall, as in Section 4.4., L’Hôpital’s original theorem says that indefinite forms can be solved by finding the functions’ rate of change.

Sympathizers of Johann Bernoulli claim he was coerced to submit to the will of nobility. Despite Bernoulli’s initial agreement out of financial desperation, the arrangement continued long into his successful professorship at Groningen. Bernoulli claimed L’Hôpital’s book was “essentially his” only after his former student’s death. At that point, Bernoulli’s reputation was murky after multiple rows with his older brother. At the time, it was standard for nobility to pay for services from high-powered professionals like politicians and lawyers, and many regarded L’Hôpital as a competent mathematician in his own right.

One early point of doubt in the integrity of L’Hôpital’s work was his solution to the brachistochrone problem (posed by Johann Bernoulli in 1696, a problem about the curve of quickest descent):

New Problem Which Mathematicians Are Invited to Solve: If two points A and B are given in a vertical plane, to assign to a mobile particle M the path AMB along which, descending under its own weight, it passes from the point A to the point B in the briefest time.

It was suggested that L’Hôpital’s answer to the question was not his own, probably that of his teacher Bernoulli himself.

Ultimately, L’Hôpital was skilled at synthesizing the teachings of Johann Bernoulli and published an essential opus in the quickly developing field of calculus, which made developments accessible to an enormous audience. However, his work would not hold up to current standards of academic integrity, and it could be said that he abused his financial position to become an academic celebrity in seventeenth-century France without the genuine innovation of his peers.

References

“Acta Eruditorum. 1696.” Internet Archive, Lipsiae : Apud J. Grossium et J.F. Gletitschium, 1 Jan. 1696, archive.org/details/s1id13206630.

Katz, Victor J. A History of Mathematics. 3rd ed., Pearson Education Limited, 2014.

L’Hospital, Guillaume François Antoine De, and M. Varignon. Analyse Des Infiniments Pettits, Pour l’intelligence Des Lignes Courbes. ALL-Éditions, 1988.

O’Connor, J J, and E F Robertson. “Guillaume François Antoine Marquis de L’Hôpital.” Maths History, University of St. Andrews School of Mathematics and Statistics, Dec. 2008, mathshistory.st-andrews.ac.uk/Biographies/De_LHopital/.

Stewart, James. Calculus: Early Transcendentals. Vol. 8.