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## Main Question or Discussion Point

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.

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.