THESE two books at once invite comparison with A Maurice Kendall's “Advanced Theory of Statistics” and A. C. Aitken's “Statistical Mathematics”. All four are mathematically flawless, and they all require from the reader some mathematical training and ability, but not to the same extent, nor are they all addressed to quite the same category of scientific workers. Kendall, in his preface, called his “a book on statistics, not on statistical mathematics” and declared his intention “to keep the mathematics to heel”. Cramer's book (based on his Stockholm lectures since 1930) is very definitely for the mathematician, and its avowed purpose is to join the two lines of development due on one hand to the British and American schools of statisticians, and on the other to the rigorous work of French and Russian mathematicians in developing the calculus of probability. The first twelve chapters are a purely mathematical introduction, dealing with the properties of sets, Lebesque integrals, Fourier integrals, matrices and quadratic forms, and the like, preparatory to the later chapters on random variables, probability distributions, and statistical inference. The author advises the reader to jump to Chapter 13 after reading the first three chapters and a few other paragraphs, and thereafter to use this earlier part only as needed. The advice is good, and for my part I would say jump to Chapter 13 at once. Many who ought to read the book would be highly discouraged by the apparent irrelevance of those first three chapters, yet might turn to them without dismay later after seeing their use and necessity. Mathematical Methods of Statistics By Prof. Harald Cramer. (Princeton Mathematical Series, 8.) Pp. xvi + 575. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1946.) 33s. 6d. net. A First Course in Mathematical Statistics By Prof. C. E. Weatherburn. Pp. xv + 272, (Cambridge: At the University Press, 1946.) 15s. net.