Analysis Leonard Euler's Books in Analysis and Algebra

[bacte2013]

Dear Physics Forums friends,

I am an aspiring mathematician who is deeply interested in the analysis, topology, and their applications to the microbiology. Recently, I started to become very curious about why concepts and theorems in the real analysis and topics come as they are; the legendary books in topology like Engelking and Kelley have been guiding me to answer the questions such as "Why do we care?" or "What motivates such theorems, definitions, axioms?", but I was not able to answer such questions from the analysis books like Rudin, which actually resulted in shallow understanding of the analysis (somehow I forced myself to memorize the contents from Rudin)...That is a reason why I decided to read some analysis books over the rest of this Summer, such as Euler, Hairer/Wanner, Bressoud, to understand the historical foundations of the concepts in basic analysis.

I am particularly interested in Euler's books: "Introduction to Analysis of the Infinite, I-II", "Foundations of Differential Calculus", and "Elements of Algebra". For those who have experience or read those books, could you tell me how they inspired or benefited you? Also, are those books fairly independent of each other? Are they better than books like Hairer/Wanner to learn about the historical background of real analysis?

I also might try Gauss' Disquisitiones Arithmaticae to learn about the number theory in details.

 

[The Bill]

I've gotten a lot of millage out of Umberto Bottazzini's The higher calculus: A history of real and complex analysis from Euler to Weierstrass. I've never studied Euler's written works directly, but I agree that studying the history of foundations has helped me understand and apply the mathematics I've learned.

 

[Ssnow]

Hi @bacte2013, as said before studying the history of mathematics can help a lot, especially when you study in the same time on modern books (basic or advanced). This is a good practice because sometimes looking to the history of the concept you can understand much better a topic in its completeness. In other hand study directly from the author books as Euler, Newton, or Lagrange (if you have time) can be very helpful and advantageous but it requires a lot of energy and sometimes the texts are difficult to understand (especially in the language). I suggest to use modern texts and support these with history or epistemology books.

Ssnow

 

[mathwonk]

i of course have a huge respect for euler, and own his analysis book but have actually learned something from his algebra book.

 

[bacte2013]

i of course have a huge respect for euler, and own his analysis book but have actually learned something from his algebra book.

How is his algebra book? What does he cover? Regarding to his analysis books, will it not be a good idea (I might choose Hairer/Wanner) to learn analysis? Despite the name "analysis", I feel like his books cover only the infinite series and some pre-calculus topics (but I do not know the details).

 

[mathwonk]

the algebra book covers everything from the beginning up through and beyond solutions formulas for cubics, usually taiught in grad school. indeed the "analysis" book mainly covers power series and pre calc, meaning of course that for eular precalc means infinite series. i don't recommend the analysis book as a place to learn what we today call analysis, but it is interesting, and he makes formidable calculations, such as evaluating the series SUM (over n) of 1/n^k, for various even values of k.

 

[bacte2013]

the algebra book covers everything from the beginning up through and beyond solutions formulas for cubics, usually taiught in grad school. indeed the "analysis" book mainly covers power series and pre calc, meaning of course that for eular precalc means infinite series. i don't recommend the analysis book as a place to learn what we today call analysis, but it is interesting, and he makes formidable calculations, such as evaluating the series SUM (over n) of 1/n^k, for various even values of k.

That is interesting. I am actually very interested in the series and sequences, and I was thinking of picking up relevant books like Knopp. I assume that the first volume deals with the infinite series, and the second volume deals with the geometry and algebra. I think I will read Volume I of Euler alongside Knopp. Does his algebra book focuses only on the solutions to polynomials? From what I remembered, the algebra at the time of Euler was mainly focused on the polynomials.

Also have you read "Analysis by Its History" by Hairer/Wanner and Euler's Differential Calculus? I could not find copy to the latter, but the former is very well-written. Very, very motivating and enriching like Korner (another great book in the analysis), unlike Rudin and Apostol.

 

[mathwonk]

here is a free searchable copy of euler's algebra, and a table of contents, so you can see for yourself what it covers: or just google euler's elements of algebra. note he ahs covered up through the cubic formula of "cardano" already in just part I.

https://books.google.com/books?id=X8yv0sj4_1YC

 

[jonjacson]

You can find euler books translated into english here:

http://www.17centurymaths.com/

The books are all related, Euler said he wrote that Algebra book because he thought the reason people struggled with analysis is they did not have the proper background. Think that he does not use limits, he use differentials, and manipulates them as other finite quantities, obivously neglecting the ones at lower orders, basically taking appropriate precautionary measures. That was not rigorous until the 1960s, you can search nonstandard analysis and Robinson and you will find info.

The good thing from Euler is that you see the complete path (algebra, expansion in series, differencial calculus, integral calculus and variational calculus) in a different way as modern textbooks, which is always nice.

For example, I loved to see his investigations in variational calculus before Lagrange. What is the minium difference between a curve and a different one? Well, you could use epsilons and so on, but he said they could just differ from one point, and that was enough to calculate everything, just amazing.

I forgot to mention that he follows Newton presenting differential calculus before integral calculus because you can always compute the differential, that is not the case in integral calculus where you don't know if there is a function satisfying the equation.