chanics--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.

It may be well to mention, parenthetically, that in defending his
alleged law of progression from the general to the special, M. Comte
somewhere comments upon the two meanings of the word _general_, and the
resulting liability to confusion. Without now discussing whether the
asserted distinction can be maintained in other cases, it is manifest
that it does not exist here. In sundry of the instances above quoted,
the endeavours made by M. Comte himself to disguise, or to explain away,
the precedence of the special over the general, clearly indicate that
the generality spoken of is of the kind meant by his formula. And it
needs but a brief consideration of the matter to show that, even did he
attempt it, he could not distinguish this generality, which, as above
proved, frequently comes last, from the generality which he says always
comes first. For what is the nature of that mental process by which
objects, dimensions, weights, times, and the rest, are found capable of
having their relations expressed numerically? It is the formation of
certain abstract conceptions of unity, duality and multiplicity, which
are applicable to all things alike. It is the invention of general
symbols serving to express the numerical relations of entities, whatever
be their special characters. And what is the nature of the mental
process by which numbers are found capable of having their relations
expressed algebraically? It is just the same. It is the formation of
certain abstract conceptions of numerical functions which are the same
whatever be the magnitudes of the numbers. It is the invention of
general symbols serving to express the relations between numbers, as
numbers express the relations between things. And transcendental
analysis stands to algebra in the s