Essai Philosophique sur les Géométries nonEuclidiennes. Par L. J. Delaporte, Docteur en Philosophie de l’Université de Fribourg (Suisse), Licencié ès Sciences Mathématiques. Paris: C. Naud, 1903. Pp. 139. This work contains a more or less popular account and criticism of nonEuclidean Geometry; the authors chiefly defended or combated are Renouvier, Delbœuf, Lechalas and Calinon. Besides an introduction and a conclusion, there are five chapters, on history (slight, but good), on mathematical considerations, on geometrical space, on the definition of the straight line, and on the fourth dimension. There is a useful appendix, setting forth side by side the elementary points of agreement and difference between Euclid, Lobachevsky and Riemann; there is also a bibliography, in which, however, no mention is made of the Italians (notably Peano and Pieri). The book suffers from an undue neglect of projective considerations, and from a certain contempt for the analysis of complex notions (such as surface or volume) which the ignorant (whom philosophers deify as Common Sense) are inclined to regard as simple. The author rather rashly identifies a great circle in a Euclidean space with a straight line in a spherical space – an identification which, if made at all, requires a far greater abandonment of “intuition” than he is prepared for. Also he errs in attributing “curvature” to nonEuclidean straight lines; and it would seem a pity to object to nonEuclid on the ground of lack of intuitiveness, without discussing what intuition is, or endeavouring to appreciate the reasons which have led mathematicians more and more to care only for logical rigour. The old error that Helmholtz’s flatland and sphereland were intended (in spite of his explicit denial) to suggest a fourth dimension, not merely other threedimensional spaces, is repeated once more in this book. But in spite of these errors, the book will be found instructive, in regard to the older phases of nonEuclidean Geometry, by those who have no previous knowledge of the subject. B. RUSSELL
