With regard to mathematics, it is to be observed, in the first place, that M. Comte mixes up under that head the pure relations of space and of quantity, which are properly included under the name, with rational mechanics and statics, which are mathematical developments of the most general conceptions of physics, namely, the notions of force and of motion.  Relegating these to their proper place in physics, we have left pure mathematics, which can stand neither at the head, nor at the tail, of any hierarchy of the sciences, since, like logic, it is equally related to all; though the enormous practical difficulty of applying mathematics to the more complex phaenomena of nature removes them, for the present, out of its sphere.

On this subject of mathematics, again, M. Comte indulges in assertions which can only be accounted for by his total ignorance of physical science practically.  As for example:—­

“C’est donc par l’etude des mathematiques, et seulement par elle, que l’on peut se faire une idee juste et approfondie de ce que c’est qu’une science.  C’est la uniquement qu’on doit chercher a connaitre avec precision la methode generale que l’esprit humain emploie constamment dans toutes ses recherches positives, parce que nulle part ailleurs les questions ne sont resolues d’une maniere aussi complete et les deductions prolongees aussi loin avec une severite rigoureuse.  C’est la egalement que notre entendement a donne les plus grandes preuves de sa force, parce que les idees qu’il y considere sont du plus haut degre d’abstraction possible dans l’ordre positif. Toute education scientifique qui ne commence point par une telle etude peche donc necessairement par sa base."[24]

That is to say, the only study which can confer “a just and comprehensive idea of what is meant by science,” and, at the same time, furnish an exact conception of the general method of scientific investigation, is that which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation!  And education, the whole secret of which consists in proceeding from the easy to the difficult, the concrete to the abstract, ought to be turned the other way, and pass from the abstract to the concrete.

M. Comte puts a second argument in favour of his hierarchy of the sciences thus:—­

“Un second caractere tres-essentiel de notre classification, c’est d’etre necessairement conforme a l’ordre effectif du developpement de la philosophie naturelle.  C’est ce que verifie tout ce qu’on sait de l’histoire des sciences."[25]

But Mr. Spencer has so thoroughly and completely demonstrated the absence of any correspondence between the historical development of the sciences, and their position in the Comtean hierarchy, in his essay on the “Genesis of Science,” that I shall not waste time in repeating his refutation.

A third proposition in support of the Comtean classification of the sciences stands as follows:—­