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 same position that algebra stands in to arithmetic.

To briefly illustrate their respective powers—­arithmetic can express in one formula the value of a particular tangent to a particular curve; algebra can express in one formula the values of all tangents to a particular curve; transcendental analysis can express in one formula the values of all tangents to all curves.  Just as arithmetic deals with the common properties of lines, areas, bulks, forces, periods; so does algebra deal with the common properties of the numbers which arithmetic presents; so does transcendental analysis deal with the common properties of the equations exhibited by algebra.  Thus, the generality of the higher branches of the calculus, when compared with the lower, is the same kind of generality as that of the lower branches when compared with geometry or mechanics.  And on examination it will be found that the like relation exists in the various other cases above given.