When one leaves school and enters university, one can find that leaps are required which can overtax the brain. I am always pleased to find books in maths and the sciences which allow one to make these leaps. In Mathematician’s Delight (Pelican 1943), W. W. Sawyer wrote:
Too often, boys at school fail to realize how much mathematics there is to know. Talented boys find themselves ahead of their fellows, and begin to think they have a mastery of mathematics. As a result, they waste their last year at school. The first year at college (for those who are able to go there) they meet the best boys from other schools, and experience a tremendous shock.Now I would not describe myself as having been an especially talented boy, but I nevertheless experienced the shock. So for my “book of last year”, I would like to introduce
Euler: The Master of Us All by William Dunham
This is his third book (see footnote for the first two). Now the Mathematical Association of America review and Amazon customer reviews will tell you plenty enough and more about what is in the book. But of the eight main chapters, the one that was of particular importance to myself was 6: Euler and Complex Variables.
For the last four years I have been presenting lectures on the history of mathematics, but what is available in even the best books tends to be a bit scrappy on this particular area. I had previously obtained material from the Euler Archive – The works of Leonhard Euler online at Dartmouth College, Hanover, New Hampshire. However, this book brings it all together.
In fact, there exist three different papers with three different ways in which Euler derived the logarithms of complex numbers, thereby showing that eiπ = –1, and all are presented in this book. In regard to complex algebra, previous mathematicians including Euler’s own teacher Johann Bernoulli had, so to speak, started to “settle West of the Appalachians”, but it was Euler who single-handedly “crossed the continent” and reached the Pacific..
He achieved this in many other fields also. The other chapters of this book have been real eye-openers for me, though I will have to re-read the appropriate one to remind myself exactly what is analytic number theory. If I were starting a university course on mathematics today, this book would certainly be a welcome bit of hanging vegetation for this leopard cub.
Some other books by William Dunham
Journey Through Genius; here I learned how it was Archimedes who proved that the π in πr2 and the π in 2πr are the same π!
The Mathematical Universe: in chapters A to Z, he shows what mathematicians from Euclid to Bertrand Russell have been on about.