The seventeenth-century mathematicians that conceived of L’Hopital’s Rule are unknown to most calculus students. They are concerned with calculating limits that are initially indefinite, such as infinite over infinite or zero over 0. Guillaume Francois Antoine Marquis de L’Hopital grew in a military environment. His mother Elisabeth Gobelin is the daughter to the Intendant for the King’s armies. L’Hopital’s initial aptitude for math was demonstrated by his ability to solve Pascal’s cycloid problem in just a few short days. L’Hopital had both the desire and ability to be a proficient mathematician, but he enrolled initially in the French army, where he worked his way to the position of captain, leading a regiment of cavalry. But even at night, L’Hopital would study geometry alone in his tent. L’Hopital, growing weary of the military and wishing to be closer to his true calling, left the military. He claimed that his nearsightedness was preventing him from performing his duties effectively. However, reports differ on this condition. L’Hopital, despite his nearsightedness, turned to calculus as a new field.
L’Hopital is an aristocrat from Paris who joined many intellectual circles and organizations in Paris. These groups were committed to learning all they could through science, mathematical pursuits, and similar inquisitive activities. L’Hopital was introduced to Johann Bernoulli by one of his intellectual circles. Johann Bernoulli is one of four leading world figures who dominated the differentiable calculus field in the late seventeenth century. Isaac Newton and Gottfried Wilhelm leibniz were at the time considered to be four of the most influential and important thinkers on the subject of differentiable calculation. Jacob and Johann Bernoulli were also among the group. L’Hopital, recognizing that he could benefit from Bernoulli’s interaction, encouraged him both to speak in public and to lecture at intellectual forums, including the French Academy of Sciences of which L’Hopital twice served as vice president, but also to do so in the ancestral house of the de L’Hopital’s. Bernoulli accepted to tutor the French mathematician who was still a novice. He received about three hundred francs each year from the deal, but lost his exclusive rights over the mathematical discoveries and research. As a result of this growing exchange, Johann Bernoulli and other pioneers in calculus gained prominence by the late 1600’s.
L’Hopital’s book, Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes, was translated and distributed throughout Europe, earning its place as the pioneer in the study mathematical change. This book entitled Analyse des Infiniment petits pour L’Intelligence des Lingues Courbes spread throughout Europe and was the first to introduce calculus. Euler’s work and treatises took over this title fifty years after. Despite this remarkable record, there was still controversy surrounding this book after L’Hopital’s 1704 death. Johann Bernoulli claimed intellectual property of a significant portion of the ideas in this textbook. Bernoulli said that the ideas in the textbook were “borrowed” and that he never spoke out about it because of an agreement between L’Hopital’s publisher and himself. Although this claim was initially questioned, further research revealed that L’Hopital’s book followed Bernoulli’s notes on the French nobleman’s lectures almost exactly. L’Hopital’s name-bearing rule and his work may be rooted in the work of a different mathematician. L’Hopital’s work and the rule that bears his name to this day, which results in his presence in calculus classrooms across the world, may have been based on another mathematician entirely.
