*L’Imagination et les mathématiques selon Descartes*. Par P. Boutroux, licencié ès lettres. Bibliothèque de la Faculté des Lettres de l’Université de Paris, No. x. Paris: Alcan, 1900. Pp. 45.

This volume contains a careful exposition of Descartes’ doctrine on the subject dealt with, but abstains from all criticisms; the many objections to the doctrine are not mentioned, and some, at least, seem not to be perceived. The difficult questions as to the Cartesian meaning of imagination are left untouched. The work has as motto a quotation from the *Regulæ* to the effect that the intellect alone can perceive truth, but that it is well to assist it by means of imagination, senses and memory. This thesis is amplified in the text. Descartes aimed at restricting the use of imagination in mathematics, but regarded it, nevertheless, as in some degree an indispensable auxiliary. M. Boutroux divides his discussion into two parts, the first on the principles of mathematical knowledge, the second on mathematical demonstration. In the first part, it is pointed out that, though knowledge requires ideas, not images, yet imagination is useful, not only in Geometry, but also in Algebra, from which Descartes excluded every notion not capable of representation by an image. In the second part, it is pointed out, to begin with, that Descartes asserts not only that the triangle can be conceived, but also that its properties can be proved, without the help of imagination or the senses (p. 13). But demonstration, being regarded as a practical method of arriving at new truths, may be pursued by whatever method is most convenient, and practically it is easier to employ the imagination to some extent. M. Boutroux proceeds to remark (p. 15) that imagination always intervenes in deduction, since this operation takes time. This view seems irreconcilable with the previous view as to the demonstrability by the pure understanding of the properties of the triangle. It seems also scarcely possible to hold, as he does, that imagination is, essentially to be distinguished from the understanding by the fact that the former, but not the latter, acts in time. For the imagination is a part of the body, situated in the brain (*Regulæ*, xii.), which is surely part of its essential difference from the understanding. M. Boutroux points out that Algebra, for Descartes, has to borrow its definitions and axioms from Geometry, and in this way makes use of imagination; and that the practical utility of symbols depends upon their being imaginable. Descartes’ universal mathematics is regarded as a youthful dream, which he afterwards abandoned. Demonstration, we are told, is not properly an affair of the understanding, for, from the point of view of the understanding, one proposition does not precede another or give its reason. This view, by the way, though probably Cartesian, is certainly false. The volume ends with two appendices, one on Vieta, pointing out that he was more dependent on imagination than Descartes, the other on the differences between the *Regulæ* and later works.

Though many of Descartes’ remarks on mathematics are excellent, his theory of the imagination appears thoroughly erroneous – so much so, as to possess nothing but a historical interest. But such as it is, the theory has been clearly, and, I think, correctly, set forth by M. Boutroux.

B. RUSSELL

* Bertrand Russell, Review of Boutroux, *L’Imagination et les mathématiques selon Descartes*, *Mind*, n.s. 10, no. 38 (Apr 1901), 274