**De L'Hôpital** was a French mathematician who wrote the first textbook on calculus, which consisted of the lectures of his teacher Johann Bernoulli.

- There are various spellings of the name Hôpital, the earlier versions being l'Hospital or Lhospital with l'Hôpital being a relatively modern form of the name.
- A few days later l'Hôpital had solved the problem.
- 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.
- 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.
- 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.
- 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.
- In 1696 L'Hôpital's famous book Analyse des infiniment petits pour l'intelligence des lignes courbes Ⓣ(Infinitesimal analysis to understand curved lines) 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.
- 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 many occasions.
- Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.
- The infinitely small part by which a variable quantity increases or decreases continuously is called the differential of that quantity.
- 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.
- 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.
- 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.
- 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 function giiven as a fraction whose numerator and denominator tend to zero at a point.
- 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.
- On the other hand, L'Hôpital's personality and deep understanding of the concepts led colleagues to take his side.
- 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.
- Bernoulli had not been in a position to complain when L'Hôpital's book was published because of the agreement between them.
- 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 we have to give L'Hôpital credit for understanding quickly the novel mathematics that was being presented to him.
- It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachistochrone 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.
- In this respect it is worth noting the fact that l'Hôpital was wondering whether the notion of a fractional derivative makes sense.
- 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 Ⓣ(Analytic treatise on conic sections) in 1707.

Born 1661, Paris, France. Died 2 February 1704, Paris, France.

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Algebra, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive