Mathematics treats of the relations of all sorts of things considered as
quantities, namely, as equal to, or greater or less than, one another.
Things may be quantitatively equal or unequal in _degree_, as in
comparing the temperature of bodies; or in _duration_; or in _spatial
magnitude_, as with lines, superficies, solids; or in _number_. And it
is assumed that the equality or inequality of things that cannot be
directly compared, may be proved indirectly on the assumption that
'things equal to the same thing are equal,' etc.

Logic also treats of the relations of all sorts of things, but not as to
their quantity. It considers (i) that one thing may be like or unlike
another in certain attributes, as that iron is in many ways like tin or
lead, and in many ways unlike carbon or sulphur: (ii) that attributes
co-exist or coinhere (or do not) in the same subject, as metallic
lustre, hardness, a certain atomic weight and a certain specific gravity
coinhere in iron: and (iii) that one event follows another (or is the
effect of it), as that the placing of iron in water causes it to rust.
The relations of likeness and of coinherence are the ground of
Classification; for it is by resemblance of coinhering attributes that
things form classes: coinherence is the ground of judgments concerning
Substance and Attribute, as that iron is metallic; and the relation of
succession, in the mode of Causation, is the chief subject of the
department of Induction. It is usual to group together these relations
of attributes and of order in time, and call them qualitative, in order
to contrast them with the quantitative relations which belong to
Mathematics. And it is assumed that qualitative relations of things,
when they cannot be directly perceived, may be proved indirectly by
assuming the axiom of the Syllogism (chap. ix.) and the law of Causation
(chap. xiv.).