Geometry sets out from certain conceptions such as plane, point,
and straight line, with which we are able to associate more or less
definite ideas, and from certain simple propositions (axioms) which, in
virtue of these ideas, we are inclined to accept as true. Then, on
the basis of a logical process, the justification of which we feel
ourselves compelled to admit, all remaining propositions are shown to
follow from those axioms, _i.e._ they are proven. A proposition is then
correct ( true ) when it has been derived in the recognised manner from
the axioms. The question of truth of the individual geometrical
propositions is thus reduced to one of the truth of the axioms. Now
it has long been known that the last question is not only unanswerable
by the methods of geometry, but that it is in itself entirely without
meaning. We cannot ask whether it is true that only one straight line
goes through two points. We can only say that Euclidean geometry deals
with things called straight lines, to each of which is ascribed the
property of being uniquely determined by two points situated on it. The
concept true does not tally with the assertions of pure geometry,
because by the word true we are eventually in the habit of
designating always the correspondence with a real object; geometry,
however, is not concerned with the relation of the ideas involved in it
to objects of experience, but only with the logical connection of these
ideas among themselves.
It is not difficult to understand why, in spite of this, we feel
constrained to call the propositions of geometry true. Geometrical
ideas correspond to more or less exact objects in nature, and these
last are undoubtedly the exclusive cause of the genesis of those ideas.
Geometry ought to refrain from such a course, in order to give to its
structure the largest possible logical unity. The practice, for
example, of seeing in a distance two marked positions on a
practically rigid body is something which is lodged deeply in our habit
of thought. We are accustomed further to regard three points as being
situated on a straight line, if their apparent positions can be made to
coincide for observation with one eye, under suitable choice of our
place of observation.
If, in pursuance of our habit of thought, we now supplement the
propositions of Euclidean geometry by the single proposition that two
points on a practically rigid body always correspond to the same
5