I've read the following quote on Wanner's Analysis by Its History:

... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than of the available modern textbooks.

(André Weil, 1979; quoted by J.D. Blanton, 1988, p. xii)

I got the mentioned book (there is a translated version published by Springer) and it seems a nice read. The translator mentions in the preface that the standard analysis courses puts low emphasis in the ordinary treatise of the elements of algebra and also that he fixes this defect.

My concern at the moment is that the book may be dated but André Weil said it's a worthy read, I'd like to know if someone already read Euler's book and some modern introduction to analysis to make a fair comparison. It's important to notice that although the book is a translation, the translator made some edits in several parts of the book, I guess that with the intention of making it a readable piece for today's needs.

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    $\begingroup$ Euler certainly was a great mathematician, but at his time analysis hadn't yet been made fully rigorous: There did not exist proper definitions of continuity and limits. My guess is that the book is an insightful reead, but that it shouldn't be replaced by a modern textbook that provides the necessary rigor. $\endgroup$ – Ulrik Jan 8 '14 at 15:04
  • $\begingroup$ Yes. That's one of the points I'm doubtful. I still don't know if the translator included such corrections. In the preface, he argues that some changes were made. Also: I guess that the non-rigorous definition could make it an good first read in analysis. $\endgroup$ – Billy Rubina Jan 8 '14 at 15:06
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    $\begingroup$ I doubt that a book where the concepts of derivative and integral are missing can be considered a good introduction to mathematical analysis. It's surely a great book, but probably less valuable than L'Hôpital's, as regards to analysis. $\endgroup$ – egreg Jan 8 '14 at 15:11
  • $\begingroup$ @egreg Do you know if there is an english translation of L'Hôpital's book? I've found only the french edition. $\endgroup$ – Billy Rubina Jan 8 '14 at 16:17
  • $\begingroup$ @GustavoBandeira I know of no translation $\endgroup$ – egreg Jan 8 '14 at 16:29

I have studied Euler's book firsthand (I suspect unlike some of the editors who left comments above) and found it to be a wonderful and illuminating book, in line with Weil's comments. You will gain from it a deeper understanding of analysis than from modern textbooks. It is true that Euler did not work with the derivative but he worked with the ratio of vanishing quantities (a.k.a. infinitesimals), which actually turns out to be a more general concept, but this is a subject for another post.

Note: we just published a detailed study of Euler that hopefully sets the record straight and vindicates Weil's hunch.

  • $\begingroup$ But that was in his two calculus volumes... OP is talking about Euler's precalculus book. $\endgroup$ – user45220 Jul 13 '15 at 22:51
  • $\begingroup$ @user45220, what you refer to as Euler's "precalculus book" contains a detailed treatment of infinite power series that would leave any modern calculus course in the dust :-) $\endgroup$ – Mikhail Katz Dec 9 '15 at 17:55
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    $\begingroup$ +1: Totally agree! Euler’s book contains more knowledge than many modern books. Reading Euler is like reading a very entertaining book. Reading a modern book is like browsing through definitions after definitions with a bunch of \ref{}’s sprayed around. Modern authors skip important steps such that you need to spend hours of understanding what they mean. There might be other modern books that are worth reading but I never encountered one book that is as clear as Euler’s books. $\endgroup$ – MrYouMath Oct 7 '17 at 6:31
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    $\begingroup$ @MrYouMath, I agree with your comment that Euler's books are a great read. A word of caution, though: if you rely on Euler exclusively to learn the subject you may be penalized in your courses because they typically expect you to learn to hop through their hoop which is not necessarily Euler's (or as good as Euler's). $\endgroup$ – Mikhail Katz Oct 10 '17 at 16:59

I was looking around the web regarding Euler's book and found the following:

The eminent historian of mathematics, Carl Boyer, in his address to the International Congress of Mathematicians in 1950, called it the greatest modern textbook in mathematics. Boyer cited Euclid’s Geometry as the greatest mathematical textbook of the classical period, perhaps of all time, appearing in over one thousand editions. For the medieval period, he chose the less well-known Al-Khowarizmi, largely devoted to algebra. But for “modern” times, Boyer made the case for Euler’s Introductio as the greatest modern textbook — and, appropriately, this time a text in analysis.

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007


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