It is only through accumulated knowledge, especially the work of Gauss, Lobatschewsky, Bolyai, Riemann, and others that modern mathematicians are able to deal easily with space of more than three dimensions.
It may be noted that Kant says:
"If it be possible that there are developments of other dimensions of space, it is very probable that God has somewhere produced them. For His works have all the grandeur and glory that can be comprised."
According to Mr. G.R.S. Mead similar ideas are to be found in certain of the Gnostic cosmogonies.
(Fragments of a Faith forgotten, p. 318.)
But a detailed historical review would be out of place here and I will therefore proceed at once to a discussion of what is meant by the term "fourth dimension" and will try to explain how it is that we can determine some of the necessary properties of four-dimensional space, even although we cannot picture it to ourselves.
At this point I would urge the reader to try to believe that the subject is not one of great difficulty. As a matter of fact it is really exceptionally straightforward if only one faces it and does not allow oneself to be frightened.
I know that it is impossible to form any clear mental picture of four-dimensional conditions, but that does not matter. The ideas involved are admittedly unprecedented in our experience, but they are not contrary to reason and I do not ask more than a formal and intellectual assent to the propositions and analogies concerned.
Let me start, then, by defining what is meant by a Dimension. The best definition I can think of is to say that, in the sense in which the word is used here, a Dimension means "An independent direction in space."