ol, asserted that all reasoning is simply a comparison of two ideas
by means of a third, and that knowledge is only the perception of the
agreement or disagreement, that is, the resemblance or dissimilarity, of
two ideas: they did not perceive, besides erring in supposing ideas, and
not the phenomena themselves, to be the subjects of reasoning, that it
is only sometimes (as, particularly, in the sciences of Quantity and
Extension) that the agreement or disagreement of two things is the one
thing to be established. Reasonings, however, about _Resemblances_,
whenever the two things cannot be directly compared by the virtually
simultaneous application of our faculties to each, do agree with Locke's
account of reasoning; being, in fact, simply such a comparison of two
things through the medium of a third. There are laws or formulae for
guiding the comparison; but the only ones which do not come under the
principles of Induction already discussed, are the mathematical axioms
of Equality, Inequality, and Proportionality, and the theorems based on
them. For these, which are true of all phenomena, or, at least, without
distinction of origin, have no connection with laws of Causation,
whereas all other theorems asserting resemblance have, being true only
of special phenomena originating in a certain way, and the resemblances
between which phenomena must be derived from, or be identical with, the
laws of their causes.

In respect to Order in Place, as well as in respect to Resemblance, some
mathematical truths are the only general propositions which, as being
independent of Causation, require separate consideration. Such are
certain geometrical laws, through which, from the position of certain
points, lines, or spaces, we infer the position of others, without any
reference to their physical causes, or to their special nature, except
as regards position or magnitude. There is no other peculiarity as
respects Order in Place. For, the Order in Place of effects is of course
a mere consequence of the laws of their causes; and, as for primaeval
causes, in _their_ Order in Place, called their _collocation_, no
uniformities are traceable.

Hence, only the methods of Mathematics remain to be investigated; and
they are partly discussed in the Second Book. The directly inductive
truths of Mathematics are few: being, first, certain propositions about
existence, tacitly involved in the so-called definitions; and secondly,
the axioms, to whic