Prior to this double volume by Constance Reid, the best mathematical biographies I had read were R. S. Westfall’s mammoth work on the life and times of Isaac Newton, and *The **Volterra Chronicles* by J.R. Goodstein. Like them, Reid’s biographies of Hilbert and Courant convincingly encapsulate the personalities of their subjects and the social and political circumstances from which they emerged. Of no less importance is the fact that all four biographies provide comprehensive non-technical accounts of the mathematical achievements of the mathematicians concerned.

The life of Vito Volterra (1860–1940) coincided almost exactly with that of David Hilbert (1862–1943) and the political conditions in their respective nations were depressingly similar. Due to fascism in both Italy and Germany, people were persecuted on the basis of religion, race or political affiliation. Consequently, because of his refusal to swear an oath of allegiance to the Mussolini regime, Volterra was forced to resign his post and was stripped of all honours. Courant, along with many other German mathematicians who were Jewish, was also deprived of his post, and avoided the further victimisation by the life-changing process of emigration to the USA.

Another common factor between Volterra and Hilbert was that each became a focal point of mathematical activity in his own country. Hilbert spent most of his life Göttingen, which consequently became the Mecca of European mathematics in the early 20^{th} century. Among his colleagues were Felix Klein Emmy Noether and Hermann Minkowski, and his many notable students included John von Neumann, Hermann Weyl, Ernst Zemelo and, of course, Richard Courant.

I first came across the name of Richard Courant in the 1960s, when I discovered ‘his’ marvellous book *What is Mathematics?* And, although I’d heard mention of the Courant Institute, it was only by reading this account of his life by Constance Reid that I learned it was modelled it upon the principles envisioned by Felix Klein, which came to fruition via the Mathematical Institute of Göttingen. Less well known is the fact that the existence of the Springer ‘yellow series’ is solely due to the influence brought to bear upon Ferdinand Springer by Richard Courant. (The first ever volume of the series appeared in 1921.)

In Reid’s biography, Courant is portrayed as a first class mathematician who was also an extremely effective facilitator who encouraged and supported emerging talent. In that respect, Hilbert was similar, but his achievements established him as one of the world’s greatest mathematicians. By way of illustration, one only has to think of the Hilbert cube, Hilbert norm, Hilbert basis theorem, Hilbert infinite hotel paradox, Hilbert’s 23 problems, Hilbert’s axioms for Euclidean geometry, Hilbert’s formalism, Hilbert computability, Hilbert space etc.

Although Courant’s mathematical originality was considerably less than that of Hilbert, I found the account of his life more interesting than the description of Hilbert’s relatively secure upbringing. For one thing, Courant endured life in the trenches in WWI, and he was eventually assigned to a non-combative role due to the wounds he received. At one point, in the 1930s, when the Nazis were in power, he was denied entry to the lecture theatre by former students in Nazi regalia who told him that, being Jewish, he was no longer allowed to teach.

The task of the biographer is to portray individuals in non-judgmental terms, and Constance Reid has carried out her task with great success. Courant is seen as a person who was widely, but not universally, admired. Many examples of his magnanimity are mentioned throughout the book, but one unsavoury incident was his great reluctance to accept Herbert Robbins’ rightful claim to be named as co-author of *What is Mathematics?*

Hilbert, on the other hand, is conveyed as a man of enormous ability and great personal integrity. He was also a bon vivant and something of a flirt. But when his son developed mental illness in his late teens, Hilbert more or less declared him persona non grata (I seem to recall that Bertrand Russell did a similar thing).

Both biographies were written on the basis of extensive conversations with a host of mathematicians who had worked with Hilbert and Courant. In fact, Reid frequently met with Courant in his later years and gained much insight into his experience as one of Hilbert’s students and his connection with other mathematicians. So, although there are no references or bibliographies in either of the two books, they are so well constructed that it feels to be of no importance.

Peter Ruane spent most of his working life in mathematics education for primary and secondary teachers.