Lopitals’ Rule or Lospital Rule or as I prefer to call it L’hospitals’ rule is used extensively in calculus to evaluate limits of the indeterminate forms 0/0 and 8/8. The rule was first published by the French mathematician Guillaum De’ Hopital (Giom de hospital) in 1696 in a book who title can be roughly translated to English as “Analysis of the infinitely small, for the understanding of curved line.” Now, this book is considered to be the very first book on Differential calculus. However, it is somewhat controversially believed that the rule which goes by the name of l’Hôpital was introduced to him in 1694 by his teacher Johann (Johenn) Bernoulli.
l’Hôpital met Bernoulli at the end of 1691, when Bernoulli was just 24 and was new to the mathematics world. By then l’Hôpital was already a member of important circle consisting of mathematicians and physicists. l’Hôpital identified the talent of Bernoulli and employed him to give him private lessons for a short period of time. There have been historical accounts which indicates that l’Hôpital may have paid Bernoulli a retainer in lieu of his discoveries and may have signed a pact that Bernoulli would hide his discoveries from others.
Initially, content with the agreement and the “princely sum” Bernoulli continued to honor their agreement. It was when the accolades bestowed on l’Hôpital’s work, he grew increasingly unhappy. Bernoulli complained that he had not received enough credit for his contributions, in spite of the preface of l’Hôpital’s book which says:
I recognize I owe much to the insights of the Messrs. Bernoulli, especially to those of the young (John), currently a professor in Groningen. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.
It was in 1704, when l’Hôpital died, Bernoulli publicly revealed of their agreement & claimed credit for almost everything of real interest in l’Hôpital’s book.
At that time barring Liebniz and few others, everyone in France regarded his claim as ridiculous primarily on two counts. One, l’Hôpital’s’ mathematical talent was well regarded and two, Bernoulli was involved in several other priority disputes.
In 1921, a manuscript of Bernoulli’s lectures on differential calculus from 1691-92 was discovered in the Basel University. The text showed remarkable similarities to l’Hôpital’s writing, substantiating Bernoulli’s account of the book’s origin.
Considering all that, somewhat arguably I can say that, if history had played itself differently, we would have been discussing the Bernoulli’s Rule today.