## Letters to the Editor: János Bolyai and Mathematical Bombshells

**January 1, 2005**

To the Editor:

I read with pleasure Phil Davis's book review in the December 2004 issue of *SIAM News*
("Running Counter to Inert Crystallized Opinion"), but I wish to make
some corrections. Davis writes that Transylvania, the birthplace of the
Bolyais, has been oscillating between Hungary and Romania for ages. Not
so: Romania as a state was created by the Treaty of Versailles in 1919;
Transylvania had been part of Hungary for a thousand years, except for
a period in the 16th and 17th centuries, when Hungary was under Turkish
occupation and Transylvania became an independent principality. Under
enlightened rulers it had its renaissance flowering; it was in
Transylvania that the Unitarian faith grew into a sect.

Davis is not quite right that Bolyai's Nachlass contains little of
mathematical interest. Bolyai had studied the arithmetic of Gaussian
integers, and used them to give an elegant proof that primes of form 4*n*
+ 1 can be written as a sum of two squares. This material can be found
in Elemér Kiss's Mathematical Gems from the Bolyai Chests (Akadémiai
Kiadó, 1999).

Davis classes the discovery of non-Euclidean geometry among four
mathematical bombshells that have seriously altered the philosophy of
mathematics, the other three being Newton's Principia, Cantor's set
theory, and Gödel's incompleteness theorem. I would add to this list
John Milnor's discovery of differential topology, which has altered our
notion of how space hangs together.---Peter D. Lax, Courant Institute
of Mathematical Sciences.