In the succeeding chapters treating of the concrete department of mathematics, we find similar contradictions M. Comte himself names the geometry of the ancients special geometry, and that of moderns the general geometry.  He admits that while “the ancients studied geometry with reference to the bodies under notice, or specially; the moderns study it with reference to the phenomena to be considered, or generally.”  He admits that while “the ancients extracted all they could out of one line or surface before passing to another,” “the moderns, since Descartes, employ themselves on questions which relate to any figure whatever.”  These facts are the reverse of what, according to his theory, they should be.  So, too, in mechanics.  Before dividing it into statics and dynamics, M. Comte treats of the three laws of motion, and is obliged to do so; for statics, the more general of the two divisions, though it does not involve motion, is impossible as a science until the laws of motion are ascertained.  Yet the laws of motion pertain to dynamics, the more special of the divisions.  Further on he points out that after Archimedes, who discovered the law of equilibrium of the lever, statics made no progress until the establishment of dynamics enabled us to seek “the conditions of equilibrium through the laws of the composition of forces.”  And he adds—­“At this day this is the method universally employed.  At the first glance it does not appear the most rational—­dynamics being more complicated than statics, and precedence being natural to the simpler.  It would, in fact, be more philosophical to refer dynamics to statics, as has since been done.”  Sundry discoveries are afterwards detailed, showing how completely the development of statics has been achieved by considering its problems dynamically; and before the close of the section M. Comte remarks that “before hydrostatics could be comprehended under statics, it was necessary that the abstract theory of equilibrium should be made so general as to apply directly to fluids as well as solids.  This was accomplished when Lagrange supplied, as the basis of the whole of rational mechanics, the single principle of virtual velocities.”  In which statement we have two facts directly at variance:  with M. Comte’s doctrine; first, that the simpler science, statics, reached its present development only by the aid of the principle of virtual velocities, which belongs to the more complex science, dynamics; and that this “single principle” underlying all rational mechanics—­this most general form which includes alike the relations of statical, hydro-statical, and dynamical forces—­was reached so late as the time of Lagrange.

Thus it is not true that the historical succession of the divisions of mathematics has corresponded with the order of decreasing generality.  It is not true that abstract mathematics was evolved antecedently to, and independently of concrete mathematics.  It is not true that of the subdivisions of abstract mathematics, the more general came before the more special.  And it is not true that concrete mathematics, in either of its two sections, began with the most abstract and advanced to the less abstract truths.