640 Book Reviews

and of Theorem 7.2 on p. 206) Lang requires essentially the same result about finitely generated abelian groups. Every time he gives a proof at (unusual) length. At the first occurrence he attributes it to me, but it is a commonplace I probably learnt from Mordell. The original Diophantine geometry expected the reader to have at his fingertips that part of algebraic geometry, in the formulation of Weil's Foundations, which is given in the author's Introduction to algebraic geometry together with the contents of his Abelian varieties. Otherwise the prerequisites were comparatively modest. The present book is still written for the most part in the language of Foundations but from time to time the reader is expected to switch to that of schemes, even in the early and fairly basic §4 of Chapter 2. The previous knowledge required for some of the new material is sometimes rather wide. Diophantine geometry does not appear to have been reviewed by the London Mathematical Society. There was a review, however, by Mordell in the Bulletin of the American Mathematical Society and Lang reproduces it here in an appendix. Some of Mordell's comments, about the use of terms such as "injective" and "finitely generated field", for example, merely reflect his frustration, as an old man, at seeing important advances in an area in which he was intensely interested but achieved by unfamiliar techniques expressed in a novel jargon. I hope Lang lives to have the same experience. But Mordell was a shrewd critic who had tried hard to get to grips with the book, and much of what he says remains valid. His review clearly got under Lang's skin (see his introduction and p. 37) but does not seem to have done good. Lang also reproduces his review from the AMS Bulletin of Mordell's Diophantine equations. This includes comments on Mordell's review. Together the reviews give an interesting contrast of approach both to mathematics and to mathematical exposition. Incidentally, this was Mordell's only hard-cover book and it was written when he was 80—another contrast to our author.