The results are 'relations' which are all expressed by judgments of subsumption and of comparison.
The judgments of subsumption are themselves subsumed under the laws of logic.
Those of comparison are expressed in classifications, and in the sciences of arithmetic and geometry.
Mr. Spencer's opinion that our consciousness of classificatory, logical, and mathematical relations between ideas is due to the frequency with which the corresponding 'outer relations' have impressed our minds, is unintelligible.
Our consciousness of these relations, no doubt, has a natural genesis. But it is to be sought rather in the inner forces which have made the brain grow, than in any mere paths of 'frequent' association which outer stimuli may have ploughed in that organ.
But let our sense for these relations have arisen as it may, the relations themselves form a fixed system of lines of cleavage, so to speak, in the mind, by which we naturally pass from one object to another; and the objects connected by these lines of cleavage are often not connected by any regular time- and space-associations. We distinguish, therefore, between the empirical order of things, and this their rational order of comparison; and, so far as possible, we seek to translate the former into the latter, as being the more congenial of the two to our intellect.
Any classification of things into kinds (especially if the kinds form series, or if they successively involve each other) is a more rational way of conceiving the things than is that mere juxtaposition or separation of them as individuals in time and space which is the order of their crude perception. Any assimilation of things to terms between which such classificatory relations, with their remote and mediate transactions, obtain, is a way of bringing the things into a more rational scheme.
Solids in motion are such terms; and the mechanical philosophy is only a way of conceiving nature so as to arrange its items along some of the more natural lines of cleavage of our mental structure.
Other natural lines are the moral and æsthetic relations. Philosophy is still seeking to conceive things so that these relations also may seem to obtain between them.
As long as things have not successfully been so conceived, the moral and æsthetic relations obtain only between entia rationis, terms in the mind; and the moral and æsthetic principles remain but postulates, not propositions, with regard to the real world outside.