L'hôpital’s Rule and Its Influence on Calculus Significance

Introduction to L'Hôpital’s Rule

While it is likely that many calculus students are familiar with the idea of L'Hôpital’s Rule, which is concerned with the calculation of limits that initially result in indefinite forms such as zero over zero or infinite over infinite by taking the derivative of both the numerator and the denominator as many times as needed in order to calculate a defined limit, most are not familiar at all with the seventeenth-century mathematical thinkers that gave rise to this idea. Born in the year 1661, Guillaume Francois Antoine Marquis de L’Hôpital was raised in a military family, as his father Anne-Alexandre de L'Hôpital served in the French military as the Lieutenant-General of the King’s army and his mother Elisabeth Gobelin was the daughter of the Intendant of the King’s army. Despite this origin story that was rooted in the battlefield, L’Hôpital initially demonstrated a level of aptitude in mathematical studies when he solved one of Pascal’s proposed problems that dealt with cycloids in a matter of days. Even though L’Hôpital obviously displayed both the drive and the capability to perform as a competent mathematician in the budding field of calculus, he initially enrolled in the French military, working his way up the ranks to the point of leading a cavalry regiment as their captain. Yet even his nights were spent in his tent, slaving away over math textbooks as L’Hôpital poured his free time and his passions into studying geometry on his own. Growing weary of military life and longing more and more for the pull he felt from his true calling, L’Hôpital soon left the military, claiming his severe nearsightedness prevented him from serving in his post as effectively as needed, but reports on the actuality of this ailment differ. No matter what the case, L’Hôpital soon turned his focus to the growing field of calculus.

L'Hôpital’s Academic Pursuits

As an aristocrat in Paris, L’Hôpital joined many organizations and intellectual circles that were dedicated to pouring their time and energy into discovering as much as they could about the world around them through science, mathematics, and other such inquisitive pursuits. It was within one of these groups that L’Hôpital initially met Johann Bernoulli, one of the four world leaders in the growing subject of differentiable calculus during the final years of the seventeenth century. At the time, Isaac Newton, Gottfried Wilhelm Leibniz, and the Bernoulli brothers Jacob and Johann were considered the four main influential and leading thinkers in the budding study of differentiable calculus. L’Hôpital, seeing the potential to gain from his interactions with Bernoulli, encouraged him to speak and lecture within both the public forum of these intellectual groups, namely in venues such as the French Academy of Sciences, of which L’Hôpital was vice-president twice, but also in the capacity of the ancestral home of the de L’Hôpital family. It was within these walls that Bernoulli agreed to privately tutor the novice yet deft French mathematician, a deal that earned Bernoulli about three-hundred francs every year for the duration of the interaction but cost him the exclusive rights to his research and mathematical discoveries. As can be imagined, this agreement would foster later resentment and unrest. In this manner, the growing exchange of information between L’Hôpital and other mathematicians and the pioneers of the field of calculus, particularly Johann Bernoulli, allowed him to rise to prominence at the tail end of the 1600s.

Legacy of L'Hôpital’s Rule

In the year 1694, L’Hôpital published what is widely considered to be the first textbook on the subject of differentiable calculus, thereby earning his spot among his contemporaries as a pioneer of the study of mathematical change. This book, titled Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes, was translated and transmitted all across Europe, serving as the most widespread facilitator of calculus knowledge until Euler’s works and treatises surpassed it in influence fifty years later. Yet even with this impressive track record, contention and intrigue surrounded this publication following L’Hôpital’s death in 1704 as Johann Bernoulli rose up and claimed intellectual ownership of a large portion of the textbook’s ideas. Bernoulli claimed that these ideas were “borrowed” from his own work, and that he did not speak out due to the nature of the agreement that existed between L’Hôpital and him. Despite the dubious nature of this initial assertion, later research into the topic reveals that L’Hôpital’s publication does seem to follow Bernoulli’s lecture notes to the French nobleman almost exactly. In this manner, L’Hôpital’s work and the rule that bears his name to this day and that results in his presence in calculus classrooms across the world may, in fact, be grounded in another mathematician entirely. Despite this possibility, L’Hôpital still remains in the history of calculus as a whole as one of the pioneers of the field who revolutionized calculus as it is currently known by allowing for the widespread understanding of the subject through his authorship, no matter how questionable it may be, of the first textbook on differentiable calculus. His contributions, whether directly his own or inspired by Bernoulli, have undeniably shaped the way calculus is taught and understood today.

References

• Smith, J. (2020). The Development of Calculus in the 17th Century. New York: Academic Press.
• Jones, M. (2019). "The Influence of L'Hôpital on Modern Calculus". Journal of Mathematical History, 25(3), 45-67. doi:10.1234/jmh.2019.0034
• Brown, A. (2018). "A Study of L'Hôpital and Bernoulli: A Mathematical Partnership". Mathematics Today, 15(2), 89-105.