Page:Mind (New Series) Volume 12.djvu/204

 190 B. RUSSELL : founded" (p. 422). He took distance to be independent of the straight line and anterior to it (p. 417) ; but he was unable to- deduce the fundamental properties of the straight line. He failed to make a Geometrical Calculus, and merely introduced a new and less convenient system of co-ordinates, the system of bipolars or tripolars ; and his failure was due to his remaining metrical. This metrical bias is attributed by M. Couturat (pp. 438-439) to respect for the " narrow, poor and stunted principles of Euclid's Geometry ' '. Doubtless respect for Euclid was one cause of failure ; but it appears to me highly probable that the relational theory of space was a more potent cause. When I formerly held this theory, I made almost exactly the same attempts to base Geometry on distance ; and if the relational theory were true, such a basis would be alone correct. The straight line, it is true, is generated by a relation, but this relation holds, for a given straight line, between only some points and some others, whereas a given relation of distance holds between every point and some others. Thus the generating relation of a straight line picks out some points of space as inherently peculiar, so that the straight line, if taken as funda- mental, is fatal to thorough-going relativity. Nevertheless, geom- etry imperatively requires that the straight line should be made fundamental, though distance can be introduced with advantage as a late and derivative notion. A mere mathematician might have been unaffected by this consequence of the relational theory, but not so a philosopher such as Leibniz ; and in the discussions with Clarke, the necessarily fundamental nature of distance, in any such theory, often very plainly appears (e.g., Gerh., vii., 400, 404). In a short conclusion, M. Couturat sums up his results, and ends with an impressive warning against too great respect for authority. Leibniz, he says, was not the autodidact that he boasted himself to be, and erudition interfered with his originality. "We shall never know the price that the human mind has had to pay for over-perfect works such as the Organon of Aristotle and the Elements of Euclid, nor by how many centuries they have re- tarded the progress of the sciences by discouraging innovators " (p. 440). An admirable remark for readers ! As for authors, the danger of producing over-perfect works is one which is by no means pressing, and need scarcely disturb their equanimity. The work ends with five appendices and a number of notes, in which much useful information will be found. In the article on Leibniz's metaphysic already referred to, which should be read in connexion with the book, the main outlines of his doctrine of monads are deduced, in his own words, from his logical principles It is also shown that his Dynamics had very little influence on his philosophy, though his philosophy had much influence on his Dynamics (p. 21 ff.). This is established beyond question by a MS. of 1676, in which most of his metaphysical theories are already to be found, in combination with a belief in atoms (p. 24). The general conclusion, that Leibniz's logic was the true founda-