L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli (Science Networks. Historical Studies Book 50) by Robert E Bradley (PDF)

Description

This monograph is an annotated translation of what is considered to be the world’s first calculus textbook, originally published in French in 1696. That anonymously published textbook on differential calculus was based on lectures given to the Marquis de l’Hôpital in 1691-2 by the great Swiss mathematician, Johann Bernoulli. In the 1920s, a copy of Bernoulli’s lecture notes was discovered in a library in Basel, which presented the opportunity to compare Bernoulli’s notes, in Latin, to l’Hôpital’s text in French. The similarities are remarkable, but there is also much in l’Hôpital’s book that is original and innovative.This book offers the first English translation of Bernoulli’s notes, along with the first faithful English translation of l’Hôpital’s text, complete with annotations and commentary. Additionally, a significant portion of the correspondence between l’Hôpital and Bernoulli has been included, also for the fi rst time in English translation.This translation will provide students and researchers with direct access to Bernoulli’s ideas and l’Hôpital’s innovations. Both enthusiasts and scholars of the history of science and the history of mathematics will fi nd food for thought in the texts and notes of the Marquis de l’Hôpital and his teacher, Johann Bernoulli.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It is very nice to have here a complete English translation of the first published textbook on the differential calculus. At least as valuable are the extensive appendices translating documents relevant to its historical context. There is also a useful and quite thorough introduction by the editors. Praise be to them for making all these things available to us in such a convenient form.This work will be a very useful reference for anyone studying this part of the history of mathematics. For general-interest mathematical readers, however, it will be less compelling. L’Hospital was never much of a mathematician, and his book is somewhat tedious. In particular, anyone hoping to find some compelling motivation or fascinating application of L’Hospital’s rule will be bitterly disappointed: the rule is indeed here, but it is used for nothing but pointless algebra problems.I shall spend the rest of this review giving some personal reflections on a few particulars that happened to be of interest to me. Basically it will amount to nitpicking about details that are probably of little significance or interest to anyone, but here it goes.First some comments on the translations. I had previously had occasion to translate a few sentences of the texts appearing here in my own work. I therefore decided to compare my translations with those of the present edition.The passages in question are from Johann Bernoulli’s lectures on the differential calculus, which are translated here in full as Appendix A. Both passages concern the fact that the differential calculus is superior to the tangent method of Descartes since it can differentiate any rational expression whereas Descartes’s method requires the problem to be reduced to a polynomial equation. Commenting on one problem of this type, Bernoulli writes that it:”… aequationem dabit … quae sex dimensionum erit, et ultra triginta terminos continebit, adeo problema per Methodum Cartesianem solutu tantum non plane impossibile sit.”This is translated here as: the problem “will produce an equation … that will have six dimensions, and in the end will contain more than thirty, to such a degree that solving the problem by the Cartesian Method is clearly not possible.” (p. 214)What does this mean? More than thirty what? Dimensions? But how can something have six dimensions and then “in the end” more than 30? What does that even mean?No, the translators are mistaken. “Terminos” should not be translated as “in the end” but rather as “terms”: the equation “is of the sixth degree, and has more than thirty terms” is what Bernoulli is saying.In another passage on the same subject Bernoulli says that in order to use the method of Descartes:”oportet primo, cujus generis sit haec linea Curva, invenire, ut et aequationem ex puris rationalibus constantem”Which is here translated as: “it would first be necessary, in order to bring forth that curved line, to find the equation in purely rational terms” (p. 198)I disagree again. “Generis” is not referring to the “bringing forth” of the line, but rather the class/type/family of it. “It would be necessary first of all to find of what class the curve is [i.e., what degree],” Bernoulli says.These were the only two passages I checked in this whole 45-page appendix. Probably it was bad luck that I had something to complain about in both.In any case, those were translations from the Latin, whereas the rest of the translations in this book are from the French. So I felt that I should have a look at some French translations instead, as they would be more indicative of the overal quality of these translations.Since I was interested in this particular point regarding the method of Descartes, I looked to see if L’Hospital also expresses the same point somewhere. Indeed he does: in fact, he does so in the very last paragraph of his entire work, making this point a sort of grand conclusion. Here it is in translation:”We clearly see from what we have just explained in this chapter, the way in which one should use the Method of Messrs. Descartes and Hudde to solve these kinds of questions when the Curves are Geometric. However, we also see at the same time that it is not comparable to that of Mr. Leibniz, which I have tried to explain thoroughly in this Treatise, because this latter gives general solutions, where the other gives only particular ones, that it extends to Transcendental lines, and that it is not necessary to remove incommensurables, which is very often impractical.” (pp. 184-185)It gets the point across, but again I am not satisfied. The way I read the grammar of this translation it seems to be saying:”We see [three things, namely:] [1] that … [the method of Descartes] is not comparable to that of Mr. Leibniz …, [2] that it extends to Transcendental lines, and [3] that it is not necessary to remove incommensurables …”But this is course completely wrong. [2] and [3] obviously refer to the method of Leibniz, not the method of Descartes. The translators understand this full well, as they make perfectly clear in their introduction (p. xli), and yet I see no other way of parsing the grammar of their translation.What the passage is really saying is:”… [the method of Descartes] is not comparable to that of Mr. Leibniz … because [of the following three reasons:] [1] this latter gives general solutions, where the other gives only particular ones, [2] it extends to Transcendental lines, and [3] it is not necessary to remove incommensurables …”Indeed, this agrees with the French original, which has a colon before “puisque,” signalling that the three points being enumerated have to do with the reasons why Leibniz’s method is superior, not the three things we see from the above.There are also a few slips in the editorial material. “The differential of x^n is nx^n-1” (p. xviii) is missing a “dx” at the end. The Bernoulli who studied the equiangular spiral and chose it for his tombstone was Jacob, not Johann as it says on p. xxxiii. I also think it would be useful for readers to know that the construction of a hyperbola given by L’Hospital, which the editors discuss at some length on pp. xxiv-xxv, is nothing but the canonical construction method of Descartes, which he gave pride of place in his Géométrie.

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