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THE theory of maxima and minima contains pitfalls into which have fallen such well-known mathematicians as Lagrange, Bertrand, Serret, and Todhunter. A peculiar interest, therefore, is attached to the subject, and the reader will find Prof. Hancock's book well worth his study. Except that there is no reference to calculus of variations, the author has succeeded in covering the ground fairly thoroughly, and that without allowing the argument to be anywhere tedious. He gives many references, and a few quite interesting examples.After a short investigation of maxima and minima of functions of a single variable, he gives in some detail the methods of Scheeffer and von Dantscher, which introduced rigour into the discussion of functions of two or three variables. The theory here is intimately connected with the theory of quadratic forms and singularities of higher plane curves. The author seems not to have read such books as Bromwich's Quadratic Forms, Hilton's Linear Substitutions, or Muth's Elementartheilar, which would have enabled him in places to simplify his treatment of quadratic forms. In tracing a plane curve near a singularity, the author should have made use of Newton's diagram. He should also have avoided such a phrase as cusps of the first and second kind, which implies that the cusps in question are comparable. whereas the latter is a singularity of much higher complexity than the former.The chapter on relative maxima and minima is especially interesting. The discussion usually given in the text-books is very scanty, and the fuller treatment here given is very welcome. A valuable point is made in 98-107. The usual proof that the maximum triangle inscribed in a given circle is equilateral runs as follows: If not, suppose ABC to be the greatest triangle. If AB+AC, let D bisect the arc BAC. Then the triangle BDC>BAC, etc. Is this argument admissible? The reader may compare the following reasoning, due to an Italian author: Unity is the greatest integer. For, if not, suppose n ( x) the greatest. Then n2 > n, etc. The proofs run parallel, but the tacit assumption (a greatest triangle or integer exists) is lawful in one case and not in the other.-Nature, Vol. 102

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