The prominent association which Anne-Alexandre de l'Hôpital had with the house of Orléans and the trust which Gaston d'Orléans placed in him conferred a brilliant lustre on the reputation of the l'Hôpitals; it also afforded them a protection second only to that of the king himself.As a child, l'Hôpital had no talent for subjects like Latin, but he developed strong mathematical abilities and a real passion for the subject. Bernard de Fontenelle, in , recounts that when l'Hôpital was fifteen years old he was, on one occasion, discussing mathematics with the Duke of Roannès and a Mr Arnaud. They told him about a very difficult problem on the cycloid that Blaise Pascal had proposed. A few days later l'Hôpital had solved the problem. Given his family background it is not surprising that Guillaume de l'Hôpital followed a military career and served as a captain in a cavalry regiment. He certainly did not give up his interest in mathematics, as Fontenelle explains :-
He entered into the service, but without giving up his dearest passion. He studied geometry even in his tent. It is not just that he retired there to study, it was also to hide his application to study. 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.However he resigned from the army because of nearsightedness being unable to see beyond ten paces. Julian Coolidge writes :-
One is inclined to believe that it was l'Hôpital's love of mathematics rather than the imperfection of his vision that led him to abandon a military career in favour of a scientific one.Certainly from that time on he directed his attention to mathematics. It is almost certain that l'Hôpital would be totally unknown in the world of mathematics today but for a chance meeting between him and Johann Bernoulli towards the end of 1691. Bernoulli at this time was 24 years old and he had just arrived in Paris after giving lectures on the latest development in mathematics, namely Leibniz's differential calculus. l'Hôpital was at the time a member of Nicolas Malebranche's circle at the Congregation of the Oratory which contained the leading mathematicians and scientists of Paris. It was an obvious place for Bernoulli to go to meet the leading French mathematicians and he soon discovered that l'Hôpital was the most enthusiastic. l'Hôpital, for his part, was intrigued to meet Bernoulli, for it quickly became clear to him that he was much more knowledgeable about the new developments in infinitesimal methods than anyone else in Paris. Bernoulli told l'Hôpital, and others in Malebranche's circle, that he knew the general formula for the radius of curvature of a curve. Although others such as Huygens, Leibniz and Newton knew this, it was thought in Paris to be an important open question so l'Hôpital, although probably one of the best mathematicians in France, realised he could learn much from Bernoulli. Bernoulli agreed to give four lectures a week to Malebranche's circle and he did so for the next six months. l'Hôpital attended these lectures but then moved from Paris to his estate at Ouques where he employed Bernoulli to give him private lessons. By November 1692 Bernoulli had left Ouques and returned to Basel from where he carried out a correspondence with l'Hôpital.
L'Hôpital had published a few brief mathematical notes, but in 1692, while Bernoulli was giving him lessons at Ouques, l'Hôpital sent a solution of de Beaune's problem to Huygens. Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hôpital. L'Hôpital did not claim that the solution he sent Huygens was his own but Huygens made the reasonable assumption that it was. Shortly after this l'Hôpital published the solution under a pseudonym. By the time Bernoulli saw the published solution he was back in Basel and, naturally enough, he was highly displeased. For about six months the correspondence between l'Hôpital and Bernoulli ceased then, on 17 March 1694, l'Hôpital sent a letter to Bernoulli with a remarkable proposition :-
I will be happy to give you a retainer of 300 pounds, beginning with the first of January of this year. ... I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out. ... I am not so unreasonable as to demand in return all of your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you not to disclose any of them to others. I ask you even not to send here to Mr Varignon or to others any copies of the writings you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this ...Although no copy of Johann Bernoulli's reply has been found, we know from l'Hôpital's next letter that Bernoulli rapidly accepted the proposition. The correspondence continued and in one of the letters Bernoulli sent to l'Hôpital in 1695 he made the following remarkable promise :-
You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.On 1 September 1695 Johann Bernoulli left Basel to take up a new appointment as professor of mathematics at Groningen. In 1696 L'Hôpital's famous book Analyse des infiniment petits pour l'intelligence des lignes courbes Ⓣ was published; it was the first text-book to be written on the differential calculus. In the introduction l'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but l'Hôpital regarded the foundations provided by him as his own ideas. He wrote:-
I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groningen. I have made free use of their discoveries, as well as those of Mr Leibniz, so that I frankly return to them whatever they please to claim as their own.He was also aware of Newton's contributions, writing:-
I must here in justice own (as Mr Leibniz himself has done in 'Journal des Sçavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ... Bit the method of Mr Leibniz is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on may occasions.The book begins with two definitions:
Definition 1. Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.There follow two axioms:
Definition 2. The infinitely small part by which a variable quantity increases or decreases continuously is called the differential of that quantity.
Axiom 1. Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (what is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.In the second chapter of the work L'Hôpital went on to determine tangents to a curve. Given his definition of a curve as a polygon with an infinite number of sides each of infinitely small length, he can define the tangent at a point on the curve as being the straight line produced from the infinitely small straight line at that point. He gives many examples of computing tangents as well as a general method. In the third chapter he considers maximum and minimum problems giving examples from mechanics and geography. In later chapters he goes on to consider points of inflection, cusps, curvature, evolutes, involutes, and higher order derivatives. In Chapter 9 is found the rule, now known as L'Hôpital's rule, for finding the limit of a rational function whose numerator and denominator tend to zero at a point.
Axiom 2. Grant that a curved line may be considered as the assemblage of an infinite number of infinitely small straight lines; or (what is the same thing) as a polygon with an infinite number of sides, each of infinitely small length such that the angle between adjacent lines determines the curvature of the curve.
This book was an extremely important contribution. It was used for a long time, with new editions produced until 1781, and it was also a model for the next generation of calculus books. But, of course, we have to examine the question of how dependent the work was on Johann Bernoulli. Well, he did not complain too vigorously when the book appeared and only after L'Hôpital's death did he become more forceful in saying that the book was essentially his. Johann Bernoulli, although having an outstanding reputation as a mathematician, had been involved in various priority disputes, particularly with his brother Jacob Bernoulli, and his claims were not taken too seriously. On the other hand, L'Hôpital's personality and deep understanding of the concepts led colleagues to take his side. For example, Robinson writes in :-
L'Hôpital possessed a very attractive personality, being, among other things, modest and generous, two qualities which were not widespread among mathematicians of his time.Only in 1921 did a manuscript copy of the course given by Johann Bernoulli to L'Hôpital come to light and it was seen how closely the book followed the course notes. Also when the agreement between the two men was seen, more understanding of the events became possible. In fact Bernoulli had not been in a position to complain when L'Hôpital's book was published because of the agreement between them. In  Truesdell tries to put the arrangement into perspective:-
We should not judge L'Hôpital's procedure too harshly. While perhaps financial necessity compelled Bernoulli to accept the arrangement initially, it continued after he had settled in his professorship at Groningen in 1695. L'Hôpital, being a noblemen, was accustomed to pay for the services of others, and what he did would not have been considered wrong had Bernoulli been a politician, a lawyer, perhaps even an architect. Certainly it was nothing for L'Hôpital to be proud of. Careful examination of the letters in which L'Hôpital reported his mathematical progress to Leibniz and Huygens shows that with one or two possible exceptions L'Hôpital did not lie, but rather referred to Bernoulli in a condescending tone without acknowledging any debt whatever to him and in matters of provenance wrote in such a way as to suggest without actually asserting.So why did Johann Bernoulli agree so quickly to L'Hôpital's proposal? Although as Truesdell suggests, money may have played a part, it does not seem to be the whole reason. Certainly social status was far more significant at this time than in the modern world and Bernoulli would, to quite a large extent, feel obliged to be subservient to a nobleman. Towards the end of his life Bernoulli boasted of the money he had received from L'Hôpital, exaggerating the amount he had received. But it is also possible that it may have been more important to the youthful Bernoulli that his mathematical discoveries became well known than that the world knew him to be the author.
But we have to give L'Hôpital credit for understanding quickly the novel mathematics that was being presented to him. He was a very competent mathematician and certainly was able to write with great clarity. It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachystochrone problem was probably not his own. The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli would put l'Hôpital in very good company indeed if the solution was indeed due to him. As well as the problems a mathematician solves, it is often the quality of the questions they ask which single them out for greatness. In this respect it is worth noting the fact that l'Hôpital was wondering whether the notion of a fractional derivative makes sense. There is a letter from him to Leibniz dated 1695 in which he asks whether dny/dxn has a meaning for n = 1/2.
It might be thought that being from the nobility, L'Hôpital would have found it easy to be elected to the Académie des Sciences. However, his nobility made election under the normal processes impossible but after the reorganisation of the Academy in 1699 he was given honorary status.
L'Hôpital considered publishing a work on integration but, on learning that Leibniz was going to publish a work on the topic, he dropped his plans. A manuscript of a book was discovered following his death and this was published under the title Traité anlytique des sections coniques Ⓣ in 1707. A second edition of this text appeared in 1720.
L'Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.
Let us end by giving Coolidge's assessment of L'Hôpital :-
He was a shining example of a man of the highest social distinction whose love of learning drove him to devote much of his short life to scientific writing.
Article by: J J O'Connor and E F Robertson