accurately defined and contrasted them, and even enunciated the law according to which extension varies inversely with intension (p. 362).
The final chapter (IX) deals with Leibniz's geometrical calculus, in which he attempts to free geometry from the consideration of magnitude, and, by an analysis of position (Analysis situs), express position directly (pp. 406, 427). What he succeeded in producing, however, was but a system of bipolar and tripolar coordinates (p. 428); and his lack of success was due to his failure to free himself from metrical considerations (pp. 412, 428, 438). That his idea of the geometrical calculus was, nevertheless, neither chimerical nor sterile, as so many philosophers and mathematicians have held, is shown, M. Couturat thinks, by the fact that Grassmann in his Ausdehnungslehre (1844) has successfully built upon Leibniz's foundations. As Boole rediscovered and realized one part, so Grassmann did another part of Leibniz's Universal Characteristic, and the two have revealed the fact that Leibniz's most daring conceptions were no idle dreams, but prophetic intuitions, anticipating by nearly two centuries the progress of science and of the human mind.
The brief Conclusion (pp. 431-441) brings together the more important intrinsic difficulties and defects in Leibniz's logical labors, which the exposition in the body of the book reveals, and declares that, however powerful and original Leibniz's mind may have been, he was in no sense the autodidact that he boasted of being; and that his great erudition and his reverence for the authority of Aristotle in logic and Euclid in mathematics, from whose influence he never fully freed himself, were the fundamental causes of his failure successfully to perfect himself in his logical calculus and his geometrical calculus. "It will never," remarks M. Couturat (p. 440), "be known how much such overperfect works as Aristotle's Organon and Euclid's Elements have cost the human mind, nor how many centuries they have retarded the progress of the sciences by discouraging innovators through their appearance of definitiveness."
This is not the place to discuss the merits of Leibniz's logical labors. As for M. Couturat's work, the scholarship and temper displayed throughout this book are worthy of the highest admiration; in these respects, it is a model of what such a book should be. Furthermore, the book is a contribution of the very greatest importance toward the correct understanding and the evaluation of Leibniz's philosophy. Certain it is that no one hereafter who undertakes to discuss the philosophy of Leibniz can ignore it. M. Couturat has conclusively proved that Leibniz's logical labors were far more extensive and in-
