*The Calculus Gallery: Masterpieces from Newton to Lebesgue* by William Dunham

As in stated in the introduction, this book is not strictly a history of calculus nor a biography of its inventors (discoverers?) and practitioners, though it has much of both. Nor is it a calculus textbook, though one can learn a thing or two. It is best treated like a museum gallery where selected works of the great masters are on display, with the author as commentator and guide.

The book is divided into three sections separated by two interludes, which Dunham calls:

- the Early Wing, comprising Newton, Leibniz, Jacob and Johann Bernoulli, and Euler;
- the Classical Wing, featuring Cauchy, Riemann, Liouville, and Weierstrass;
- the Modern Wing, with Cantor, Volterra, Baire, and Lebesgue.

Each mathematician (with the Bernoulli brothers combined) gets a chapter with selected pieces from their oeuvres on display. The early chapters are easier to follow (for me at least) as they are based more on geometry and intuition, whereas the later chapters get comparatively more hairy as the logical basis is rigorized, geometry gives way to algebra, and the subject area is generalized. For example, early on we play around with infinite series that sum to surprising values, later we deal with beasts that are everywhere continuous but nowhere differentiable.

That brings up a warning though: This book requires some background in calculus, such as an introductory college-level year’s worth, and the more the better. I think anyone with some algebra training can get through the first few chapters fine, but if you’ve never seen Weierstrass’ delta-epsilon formulation of limits the latter chapters will be tough going (I definitely noticed myself having to reread some of the theorems and proofs of the latter few chapters). This isn’t to impugn Dunham, who I think does an exemplary job as museum guide (and a decent lecturer on this kind of thing to boot) but rather is caused by the necessary rise in the sophistication of the mathematics (if it wasn’t so much more sophisticated, the founding fathers of calculus would have done it themselves).

If you do have such training, though, *The Calculus Gallery* is a joy and is the kind of thing that I wish were a lot more popular: mathematics on par with art as something to be sampled and enjoyed. There can certainly be as much creativity and cleverness in a theorem as there is in a painting or a piece of music, and the mental exertion to extract the mathematical jewels is a great reward. With the significantly higher barrier to entry, though, it’s likely inevitable that only a small subset of the population will be able to sit back, crack open a beer, and say “Damn, that was a good proof.”

Dunham did write an excellent book, Journeys Through Genius I believe was the title, that seems to be along the lines of this one: just traipsing through a selection of interesting proofs, from showing how to construct a square with the same area as a given rectangle (or other shapes) using straightedge and compass up to, well, the proofs of countable and uncountable infinities (which are neat, but, every pop mathematician writes that). I’m glad to know he’s writing more in the same vein.

Another is Euler, Master of Us All which is similar but focusing just on Leonhard Euler. I have that one and Journey Through Genius on my shortlist.