This volume is a record of an individual attempt to construct numerical models of the cubic surface, founded on the lines of the surface; it is carried through with great earnestness, and so far as possible with the simplest materials; its obvious sincerity cannot fail to be inspiring to anyone who will be at pains to understand it. It would be a mistake to criticise the earlier half of the book as if it were a treatise on the cubic surface; it is the author's assembling of his materials for the constructions which follow, and the very want of elaboration which it occasionally exhibits is a proof, if an incidental one, of the independence with which the author has carried out his research. Perhaps the analytical investigation of the doublesix theorem, which occupies pp. 16 to 19, is an extreme case; a geometrical proof might have been given, though the author's is simple and selfcontained.The book is accompanied by reproductions of elaborate diagrams, carefully drawn to scale. They would have been more interesting if not so much reduced in size. There is a long bibliography of the general literature in regard to the cubic surface, which is likely to be useful. Under 1902 there should certainly be the entry, "Reziehungen der allgemeinen Flache dritter Ordnung zu einer covarienten Flache dritter Classe," by Th. Reye, Math. Annalen, Bd. lv. Also the paper of G. Kohn, "Ueber einige Eigenschaften der allgemeinen Flache dritter Ordnung," Wiener Sitzungsberichte, Bd. cxvii., p. 66, should be referred to, and a recent paper by Prof. W. Burnside in the Camb. Phil. Proceedings, on double-sixes with projective transformations.--Nature, Vol. 90.