ve pointed out unique and definite properties in nature
which correspond to perpendicularity. We shall find that this discovery
of definite unique properties defining perpendicularity is of critical
importance in the theory of congruence which is the topic for the next
lecture.

I regret that it has been necessary for me in this lecture to administer
such a large dose of four-dimensional geometry. I do not apologise,
because I am really not responsible for the fact that nature in its most
fundamental aspect is four-dimensional. Things are what they are; and it
is useless to disguise the fact that 'what things are' is often very
difficult for our intellects to follow. It is a mere evasion of the
ultimate problems to shirk such obstacles.

CHAPTER VI

CONGRUENCE

The aim of this lecture is to establish a theory of congruence. You must
understand at once that congruence is a controversial question. It is
the theory of measurement in space and in time. The question seems
simple. In fact it is simple enough for a standard procedure to have
been settled by act of parliament; and devotion to metaphysical
subtleties is almost the only crime which has never been imputed to any
English parliament. But the procedure is one thing and its meaning is
another.

First let us fix attention on the purely mathematical question. When the
segment between two points A and B is congruent to that between the
two points C and D, the quantitative measurements of the two
segments are equal. The equality of the numerical measures and the
congruence of the two segments are not always clearly discriminated, and
are lumped together under the term equality. But the procedure of
measurement presupposes congruence. For example, a yard measure is
applied successively to measure two distances between two pairs of
points on the floor of a room. It is of the essence of the procedure of
measurement that the yard measure remains unaltered as it is transferred
from one position to another. Some objects can palpably alter as they
move--for example, an elastic thread; but a yard measure does not alter
if made of the proper material. What is this but a judgment of
congruence applied to the train of successive positions of the yard
measure? We know that it does not alter because we judge it to be
congruent to itself in various positions. In the case of the thread we
can observe the loss of self-congruence. Thus immediate judgments of
congruence are