Leonhard Euler's Books in the Analysis and Algebra 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.    
 A: I love this phrase from the famous mathematician, Paul Halmos, given at the beggining of a series of lectures on linear operators:

"A thing that happens very often in mathematics is that you start with something concrete(...), and out of this concrete concept grows an abstract axiomatic notion (...), then you suddenly discover, with a pleasant surprise (only it shouldn't have been all that surprising), that every one of these abstract objects has a concrete representation by one of those things that you started out with." 

My recommendation for a general historical overview on mathematics is Stillwell's "Mathematics and it's history". This is not a theorem-proof kind of book, since he is not interested in developing theory, but gives many examples of concrete objects, abstract notions and theorems that have been studied by mathematicians throughout history, which at later stage became the prototypical examples of more general theories. Also, he is much more focused on the mathematics than on the biographies of the great mathematicians. Another nice thing is that he uses modern notation, so you don't have to struggle to decipher what he means. 
I find this last point very important; I do not doubt there is much to be gained by reading the old texts, and by any means I'm not arguing not to do so. Reading Euclid or Newton's arguments makes you appreciate the benefits of our modern notation and rigour, for example. Nevertheless, I would recommend modern books focused on the historical development of the theory.
P.S. Halmos lectures can be seen here: http://av.cah.utexas.edu/index.php/Category:P.R._Halmos_Lecture_Film_Series 
