Best books on Calculus

[carey]

Matt,

Since you like Calculus Made Easy by Sylvanus Thompson (originally published in 1910), to find similar books, it may be helpful to consider what makes Thompson's book different from other calculus books.

All calculus books agree on the rules of differentiation, but not on how to get there! It all boils down to deciding which terms to neglect. Thompson describes the differential dx as a "little bit of x." While a "little bit of x" can't be neglected, its square, cube, etc. can, so Thompson's approach is neglect of higher order (squared, cubed, etc.) terms. A modern book that employs the same approach as Thompson (neglect of higher order terms), but derived differently (from mathematical logic and Category Theory) is A Primer of Infinitesimal Analysis by J. L. Bell, 2nd edition.

The alternative (more popular) approach for deciding which terms to neglect is to use limits. Limits allow neglect of dx's directly (i.e., without requiring higher order terms) but their rigorous definition (using deltas & epsilons) can be a turn off. A nice little book (written a decade before Thompson's) uses a brief and intuitive approach to limits and was the first Calculus book written by a woman in America. It is Calculus With Applications (An Introduction to the Mathematical Treatment of Science) by Ellen Hayes.

For a full, traditional treatment of calculus that is old school, yet not overly stuffy, there is Differential and Integral Calculus by Harold Maile Bacon.