h latter, though resting only on induction, _per
simplicem enumerationem_, there could never have been even any apparent
exceptions. Thus, every arithmetical calculation rests (and this is what
makes Arithmetic the type of a deductive science) on the evidence of the
axiom: The sums of equals are equals (which is coextensive with nature
itself)--combined with the definitions of the numbers, which are
severally made up of the explanation of the name, which connotes the way
in which the particular agglomeration is composed, and of the assertion
of a fact, viz. the physical property so connoted.

The propositions of Arithmetic affirm the modes of formation of given
numbers, and are true of all things under the condition of being divided
in a particular way. Algebraical propositions, on the other hand, affirm
the equivalence of different modes of formation of numbers generally,
and are true of all things under condition of being divided in _any_
way.

Though the laws of Extension are not, like those of Number, remote from
visual and tactual imagination, Geometry has not commonly been
recognised as a strictly physical science. The reason is, first, the
possibility of collecting its facts as effectually from the ideas as
from the objects; and secondly, the illusion that its ideal data are not
mere hypotheses, like those in now deductive physical sciences, but a
peculiar class of realities, and that therefore its conclusions are
_exceptionally_ demonstrative. Really, all geometrical theorems are laws
of external nature. They might have been got by generalising from actual
comparison and measurement; only, that it was found practicable to
deduce them from a few obviously true general laws, viz. The sums of
equals are equals; things equal to the same thing are equal to one
another (which two belong to the Science of Number also); and, thirdly
(what is no merely verbal definition, though it has been so called):
Lines, surfaces, solid spaces, which can be so applied to one another as
to coincide, are equal. The rest of the premisses of Geometry consist of
the so-called definitions, which assert, together with one or more
properties, the real existence of objects corresponding to the names to
be defined. The reason why the premisses are so few, and why Geometry is
thus almost entirely deductive, is, that all questions of position and
figure, that is, of quality, may be resolved into questions of quantity
or magnitude, and so Geomet