Mathematical Math History books?

Question

I'd like to compile a list of books that aren't only good books on the history of mathematics—but books that are themselves mathematical explanations. For example, imagine a biography on Cantor. This biography would be a good historical book—but also provide (maybe even walk-through) all of the relevant Cantor proofs. It'd teach you both history and mathematics.

Unfortunately, I don't have any ideas. My experience with mathbooks is strictly textbooks with, at best, a "Did You Know It?" esque insert at the top of a page or something.

My targets are any and all topics, honestly.

Answer 1

Below are a few books I pulled off my bookshelves just now that might qualify. I’ve grouped them into three categories.

Math history books that include a lot of mathematical development for the reader.

The Historical Development of the Calculus by Charles Henry Edwards

Classical and Modern Integration Theories by Ivan Nikolaevich Pesin

Journey Through Genius. The Great Theorems of Mathematics by William Wade Dunham

Euler. The Master of Us All by William Wade Dunham

Algebra in Ancient and Modern Times by Veeravalli Seshadri Varadarajan

Specific topic books that include a lot of history.

A Treatise on the Binomial Theorem by Craig Alan Smorynski (see also Phill's answer)

Solving Kepler’s Equation Over Three Centuries by Peter Colwell

Space-Filling Curves by Hans Sagan

The Ellipse. A Historical and Mathematical Journey by Arthur Mazer

Proofs of the Cantor-Bernstein Theorem by Arie Hinkis

Textbooks that include a lot of history.

Analysis by Its History by Ernst Hairer and Gerhard Wanner

A Radical Approach to Real Analysis by David Marius Bressoud

A Radical Approach to Lebesgue’s Theory of Integration by David Marius Bressoud

Classical Algebra. Its Nature, Origins, and Uses by Roger Lee Cooke

Answer 2

I highly recommend MVT: A Most Valuable Theorem by Craig Smorynski. It is not only a careful comparison of proofs of the Mean Value Theorem, but it is also a history of Calculus, from the Greeks to Non--standard Analysis.

Answer 3

I think the books by Thomas Hawkins deserve to be mentioned:

• Lebesgue's Theory of Integration: Its Origins and Development
• Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926
• The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics

They are not ordinary textbooks aiming to teach a mathematical subject to a beginner, but if you already know the subject a bit and want to get a deeper understanding by looking at it from the historical perspective, they are very nice.