tics, 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 me