A Completely Inadequate Bibliography of the History of Mathematics

Caveat: This is neither exhaustive nor unbiased. Primarily, it consists of books that I have read that I will vouch have some value (even if the value is entertainment, as in Bell's book). Most of the following books are aimed at the professional non-mathematician (i.e., someone to whom the land of mathematics is an interesting place to visit, but you wouldn't want to live there).

A note about the ratings: Again, no claim of objectivity is made. However, the following rating system is fairly objective, as it indicates the amount of mathematical background necessary to truly understand the information contained:

+
Comprehensible with even the most basic mathematics background.
!
Some mathematics, but nothing too extreme; if you can recognize the "!" symbol, you should be OK.
!!
A bit more difficult, though a year's worth of calculus or so should give you enough background to appreciate it; you can "blip" over the scarier mathematics.
!!!
Tough stuff. Best suited for math majors, or people who are really interested in the subject.

Topics:

General

If I had to pick one...: Kline's is probably the best single volume history of mathematics that I know of for those who want to truly understand the subject. While he is slightly guilty of modernizing the notation, he is faithful to the original source material as well, and his extensive bibliography is a good jumping off point for further reserach.

A history of mathematics, Carl Boyer. Rating: +
Excellent introduction to the history of mathematics, especially for a non-mathematician. Unlike many other introductions, Boyer does not spend all his time on the Greeks and Renaissance Italians, but tries to cover the entire history of mathematics from ancient Egypt to the 20th century.
Mathematical thought from ancient to modern times, Morris Kline. Rating: !
Very good introduction to the history of mathematics, though Kline does spend most of his time in analysis (calculus and its spinoffs). Every chapter contains an excellent bibliography.
Number Words and Number Symbols, Karl Menninger. Rating:+
A fascinating introduction to numbers and how different cultures represent them and compute with them. A bonus: Menninger includes the quotes both original and translated quotes.

Arithmetic

Capitalism and arithmetic, Frank Swetz.
Includes Smith's translation of The Treviso Arithmetic, the first printed mathematics text. A fascinating portrait of mathematics at the beginning of the 16th century.

Algebra

A history of algebra, from al-Khwarizmi to Emmy Noether, B. L. van der Waerden. Rating: !
A good account of its subject matter. However, van der Waerden takes great liberties in "modernizing" the notation used by the older authors, which makes for easier reading by a modern audience, though it loses some of the historical flavor (as much of the development of mathematics is the development of mathematical notation).

Geometry

If I had to pick one...It would be Heath's Euclid, hands down.

The Elements of Euclid, Thomas L. Heath. Rating: +
If I was a medieval scholar, I would call this Heath's commentary on Euclid, for that is what it is: it is Euclid's Elements, translated and extensively footnoted. Heath traces many of the connections between Euclid's work and the works of later mathematicians.

Calculus

The history of the calculus and its conceptual development, Carl Boyer. Rating: !
Boyer spends a great deal of time discussing the earlier development of calculus, up to Newton and Leibniz, and only the very last section dealing with developments since then. However, you might want to read:
The higher calculus: a history of real and complex analysis from Euler to Weierstrass, Umberto Bottazzini. Rating: !!
Just what it says. To continue the study, read:
Lebesgue's theory of integration: its origins and development, Thomas Hawkins. Rating: !!!
A continuation of the study of the development of analysis. Its !!! rating is mainly due to the fact that its subject matter is usually reserved for graduate students; the book itself is very good (speaking from my totally objective viewpoint of being one of Dr. Hawkins' graduate students...) However, anyone with an !! background can certainly appreciate the problem whose history Hawkins addresses.

Analysis

Note: to make the historiography a bit more clear, I'm going to differentiate this section from the section on calculus. (And if you expect me to apologize for that awful pun, you little know me...

Zermelo's Axiom of Choice, Gregory H. Moore. Rating: !!
The axiom of choice and its ramifications are discussed, as well as the various paradoxes to arise out of it (such as Banach-Tarski and other pathologies). While the mathematics itself is not too difficult, some of the concepts might be a bit hard to take for the uninitiated, so it gets the !! rating.

Number Theory

Number Theory and Its History, Oystein Ore. Rating: !
This is more of a book on number theory than its history, though it incorporates elements of both. Interesting for the one who wants to know a little about the historical background to the solutions to certain problems in number theory.

Applications of mathematics

A history of the stability problem in celestial mechanics, Jeff Suzuki. Rating: !!
In my totally unbiased opinion, this is something no home should be without. Its length alone makes it valuable as a paperweight.

Sources of Mathematics

The World of Mathematics, James R. Newman. Rating: +
Excerpts from the writings of mathematicians, historians of mathematics, and philosophers of mathematics, from ancient Egypt to the modern age. (Overall the rating is +, though sometimes it becomes !.)
A sourcebook in mathematics, David Eugene Smith.
A compilation of mathematical writings. Smith's selections are mainly modern (only about a dozen date to before 1600), and not always representative. A better sourcebook is:
A source book in mathematics, 1200-1800, D. J. Struik.
Just what it says; Struik's selections are much more representative of mathematical development than Smith's.

Biographies of Mathematicians

Cardano: The Gambling Scholar, Oystein Ore. Rating: +
A readable biography of Cardano; includes a sizable discussion on the algebra of the 16th century.
Men of mathematics, Eric Temple Bell. Rating: +
Contains biographies of a few dozen of the "great mathematicians" of history. Warning: Bell's history is suspect (and, in the case of Galois, almost entirely fictional), and this work suffers especially from a lack of referencing. The best use of this book is to point out interesting things for future study.
Georg Cantor, Joseph Warren Dauben. Rating: !
The history of the infinities developed by Cantor, and a biography of Cantor as well.
The Mathematics of Sonya Kovalevskaya, Roger Cooke. Rating: !
Sonya Kovalevskaya was one of the few women mathematicians of the 19th century, and as such would be a heroine to many groups regardless of her abilities as a mathematician. However, Cooke examines her mathematics as well, showing that she was also a first-class mathematician.

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