(1)THIS is a systematic text-book on the elements of the theory of numbers, intended for beginners. The greater part of it deals with the usual elementary topics: divisibility, the H.C.F. process, congruences, and binary quadratic forms (but without the classical theory of reduction and equivalence). The proofs of the theorems are given in full and ample detail, and there are frequent numerical illustrations, and sets of examples for solution by the student. Most of these are quite straightforward, but there are several which strike one as distinctly difficult in relation to the text. (1) Elementary Number Theory By Prof. J. V. Uspensky Prof. M. A. Heaslet. Pp. x + 484. (New York and London: McGraw-Hill Book Co., Inc., 1939.) 26s. (2) Modern Elementary Theory of Numbers By Prof. Leonard Eugene Dickson. Pp. vii + 309. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 18s. net.