- Length: 5 pages
- Subject: Education - Mathematics
- Type: Term Paper
- Paper: #56354020

The seventeenth century marks the dawn of a number of truly great and influential French scientists, mathematicians, and philosophers. Early in the century Rene Descartes invented analytical geometry, and "is commonly said to be the founder of modern philosophy." (Feinberg 149). Another influential Frenchman of the time was Blaise Pascal who is credited with inventing the first digital calculator, the second mechanical calculator, being the first to study the Pascal triangle and binomial coefficients; he also helped to lay the theory of probability (O'Connor 1996). Clearly, L'Hopital was born in a country that was ripe for his contributions.

L'Hopital was fortunate enough, also, to live during the conception of modern calculus. Two mathematicians claimed to simultaneously and independently invent the calculus. Although today, Newton is generally regarded as the father of calculus, during his time there was a significant dispute between him and Leibniz. A friend of Leibniz and contemporary of the two, Jacques Bernoulli, learned the calculus from the German mathematician and brought his knowledge back to France. "However, the real introduction of the calculus in France is due to J. Bernoulli's visit to Paris. When he arrived in 1691, he went directly to Malebranche. This move was decisive, for in Malebranche's room, he met the Marquis de L'Hopital, whom he taught the calculus during the winter of 1691-1692. The result of this tuition was the Analyse des infiniments petits, which became the French reference book in the calculus for a century. Malebranche played an essential role in all of the above. He was a catalyst in the process of the 'conversion' of French mathematicians to the calculus, although he did not contribute to it in any way." (Goggin 2002). Importantly, Bernoulli sided with his friend, Leibniz, in the dispute over calculus. As a consequence, Leibniz's form of calculus was favored on continental Europe for many years to come.

L'Hopital, on the other hand, having impressed Bernoulli so much convinced him to do something unprecedented: "While tutoring L'Hopital, Johann signed a pact saying that he would send all of his discoveries to L'Hopital to do with as he wished, in return for a regular salary. This pact resulted in one of Johann's biggest contributions to the calculus being known as L'Hopital's rule on indeterminate forms." (Struick 1987). This came about due to the fact that L'Hopital was the first to recognize precisely what it was that Bernoulli had discovered, and because of Bernoulli's own reluctance to publish his work before it was fully completed.

Nevertheless, L'Hopital's name is guaranteed to survive in the memories of thousands of mathematicians to come thanks to the rule he was the first to recognize and bears his name. Mathematically, this famous rule can be stated: "If f (z) and g (z) are differentiable at z?, with f (z?) = g (z?) = 0 and g'(z? ) ? 0, then lim f (z) = f'(z?)

z-z? g (z) g'(z?)." (Greenberg 1998).

This rule is extremely useful when dealing with indeterminate forms. "If functions f and g are continuous at x = a but f (a) = g (a) = 0, the limit lim f (x)

x -- a g (x)

cannot be evaluated by substituting x = a, since this produces 0/0, a meaningless expression known as an indeterminate form." (Addison 1994). The power of L'Hopital's rule is that the limit of these types of functions can be determined under specific guidelines.

Additionally, L'Hopital "contributed solutions to several problems posed by Johann Bernoulli, among them the problem of the brachistochrone, which was solved at the same time by three others -- Newton, Leibniz, and Jacques Bernoulli." (Robinson 2002). This established him as one of the most talented mathematicians of his time, and gave him license to begin publishing works in his own right.

Although today, people remember him primarily in association with his rule, his most meaningful contribution to mathematics was the publication of his aforementioned book, Analyse des infiniment petis pour l'intelligence des lingnes courbes. The first edition was published in 1696 and is significant because it was "the first introductory differential calculus text." (Addison 1994). "Following the classical custom, the book starts with a set of definitions and axioms. Thus, a variable quantity is defined as one that increases or decreases continuously while a constant quantity remains the same while others change." (Robinson 2002). Accordingly, a differential is formally defined for the first time "as the infinitely small portion by which a variable quantity increases or decreases continuously." (Robinson 2002). Together, these two definitions lay the foundation for the notion that functions that only differ from one another by infinitesimally small amounts may be regarded as the same function, and consequently, may be substituted for one another.

Like many introductory calculus textbooks of today, after L'Hopital established the necessary definitions in his first chapter he then moved on to the idea of the tangent. From the idea of the tangent he proceeded to establish that a formula can be derived to describe the slope at any given point of many functions. Thus, in his second chapter L'Hopital introduced the derivative. Like thousands of books that would follow, he then provided the reader with examples and applications of the first derivative -- and then moved on to higher order differentials (Robinson 2002). Essentially, L'Hopital created the template by which all calculus texts would be modeled and measured against for the next three hundred years.

'The Analyse des infiniment petits was the first textbook of differential calculus. The existence of several commentaries on it -- one by Varignon (1725) -- attests to its popularity." (Robinson 2002). However, much debate has surrounded the intellectual ownership of the ideas therein. Like the rule that bears his name, many of the principles published in L'Hopital's Analyse des infiniment petits could easily be attributed to Bernoulli's work. It should be noted that, "L'Hopital himself, in the introduction to his books, freely mentions his indebtedness to Leibniz and to the Bernoulli brothers. On the other hand he states that he regards the foundations provided by him as his own idea, although they have also been credited by some to Jean Bernoulli." (Robinson 2002).

It may be significant, though, that Bernoulli waited to dispute the content of L'Hopital's work until after his death. This may indicate either his level of respect for the mathematician, or the acknowledgement of L'Hopital's accomplishments as genuinely unique.

L'Hopital's second manuscript, Traite analytique des sections coniques et de leur usage pour la resolution des equations dans les problemes tant determines pu'intermines, was published in 1720 after his death. This book examined analytic techniques of calculus more in depth than his previous work -- and was also well received by his peers.

A natural progression from his two first works on the topic of calculus would have been a serious examination of the integral calculus. Indeed, this was a project that L'Hopital was capable of and actually began to write before his death. However, one of his contemporaries -- Leibniz -- made it known to L'Hopital that he also endeavored to…