ity; and as the science of quantity is, so far as a science can
be, quite deductive, the theory of that special kind of qualities
becomes so likewise. It was thus that Descartes and Clairaut made
geometry, which was already partially deductive, still more so, by
pointing out the correspondence between geometrical and algebraical
properties.

CHAPTERS V. AND VI.

DEMONSTRATION AND NECESSARY TRUTHS.

All sciences are based on induction; yet some, e.g. mathematics, and
commonly also those branches of natural philosophy which have been made
deductive through mathematics, are called Exact Sciences, and systems of
Necessary Truth. Now, their necessity, and even their alleged certainty,
are illusions. For the conclusions, e.g. of geometry, flow only
seemingly from the definitions (since from definitions, as such, only
propositions about the meaning of words can be deduced): really, they
flow from an implied assumption of the existence of real things
corresponding to the definitions. But, besides that the existence of
such things is not actual or possible consistently with the constitution
of the earth, neither can they even be _conceived_ as existing. In fact,
geometrical points, lines, circles, and squares, are simply copies of
those in nature, to a part alone of which we choose to _attend_; and the
definitions are merely some of our first generalisations about these
natural objects, which being, though equally true of all, not exactly
true of any one, must, actually, when extended to cases where the error
would be appreciable (e.g. to lines of perceptible breadth), be
corrected by the joining to them of new propositions about the
aberration. The exact correspondence, then, between the facts and those
first principles of geometry which are involved in the so-called
definitions, is a fiction, and is merely _supposed_. Geometry has,
indeed (what Dugald Stewart did not perceive), some first principles
which are true without any mixture of hypothesis, viz. the axioms, as
well those which are indemonstrable (e.g. Two straight lines cannot
enclose a space) as also the demonstrable ones; and so have all sciences
some exactly true general propositions: e.g. Mechanics has the first law
of motion. But, generally, the necessity of the conclusions in geometry
consists only in their following necessarily from certain _hypotheses_,
for which same reason the ancients styled the conclusions of all
deductive sciences _necessary_. T