presupposed in measurement, and the process of
measurement is merely a procedure to extend the recognition of
congruence to cases where these immediate judgments are not available.
Thus we cannot define congruence by measurement.

In modern expositions of the axioms of geometry certain conditions are
laid down which the relation of congruence between segments is to
satisfy. It is supposed that we have a complete theory of points,
straight lines, planes, and the order of points on planes--in fact, a
complete theory of non-metrical geometry. We then enquire about
congruence and lay down the set of conditions--or axioms as they are
called--which this relation satisfies. It has then been proved that
there are alternative relations which satisfy these conditions equally
well and that there is nothing intrinsic in the theory of space to lead
us to adopt any one of these relations in preference to any other as the
relation of congruence which we adopt. In other words there are
alternative metrical geometries which all exist by an equal right so far
as the intrinsic theory of space is concerned.

Poincare, the great French mathematician, held that our actual choice
among these geometries is guided purely by convention, and that the
effect of a change of choice would be simply to alter our expression of
the physical laws of nature. By 'convention' I understand Poincare to
mean that there is nothing inherent in nature itself giving any peculiar
_role_ to one of these congruence relations, and that the choice of one
particular relation is guided by the volitions of the mind at the other
end of the sense-awareness. The principle of guidance is intellectual
convenience and not natural fact.

This position has been misunderstood by many of Poincare's expositors.
They have muddled it up with another question, namely that owing to the
inexactitude of observation it is impossible to make an exact statement
in the comparison of measures. It follows that a certain subset of
closely allied congruence relations can be assigned of which each member
equally well agrees with that statement of observed congruence when the
statement is properly qualified with its limits of error.

This is an entirely different question and it presupposes a rejection of
Poincare's position. The absolute indetermination of nature in respect
of all the relations of congruence is replaced by the indetermination of
observation with respect to a small subgroup of t