If it is useful for the geometer to have studied the other fundamental sciences, it is not less indispensable for other learned men to have gone through the study of mathematics. As an “initial” discipline, this science can be neglected by no one. It is the common school of positivity for all minds. It is therefore to be regretted that the scientific education of future physiologists should be mainly made up of literary studies and of a few notions of physics and chemistry. The more complex the phenomena whose laws they will have to seek, the more necessary will it be for them to have become familiarised in mathematics and in astronomy, with the precise idea of scientific truth. And, as a matter of fact, until this century, the study of the exact sciences had always been regarded as a preliminary condition for that of the natural sciences. Buffon and Lamarck in their day had still received this discipline. If it has been so difficult to constitute social science, it comes, among other reasons, from the lack of scientific education among those who, up to the present time, have wished to study social phenomena. Where, for instance, could economists have found the scientific idea of what constitutes natural laws, ignoring as most of them did not only biology which was being formed beside them, but even the sciences which had already reached a positive state? The exclusive cultivation of a single science is always a danger for the intellect. Nevertheless, so long as the chief task of the positive spirit was to disorganise the system of beliefs which constituted theological and metaphysical philosophy, the speciality of the works and of the methods was an inconvenience of secondary importance. It mattered little that the discoveries of the astronomers, the physicians, the biologists should be more or less co-ordinated and directed by a universal positive method, so long as they did their work and prepared the future. But, when the positive spirit had to become organic instead of critical, when it had to substitute a new philosophy to the one which it had overthrown, then it was obliged to subordinate the special processes which it had made use of until then to a single universal method. Should the “scientific anarchy” have lasted, the progress of the positive spirit would undoubtedly have led to the discrediting of the metaphysical régime, but without replacing it, and consequently without having done with it. By rejecting any new general discipline, modern scientific men would unknowingly tend to re-establish the system which they seemed to have shattered for ever. In a word, the triumph of the positive method, to be final, presupposes the acceptance of the positive philosophy by all men of learning. The old logic was bound by the narrowest ties to the metaphysical doctrines which were then dominant. In the same way positive logic is bound up with positive philosophy. Speaking more precisely, it is an expression of this very philosophy. III.