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..