Excerpt from The Fourth Dimension

TO the general reader, the name of the fourth dimension brings reminiscences of Flatland and The Time Machine. On hearing that to the mathematician the extension from three dimensions to four or five is trivial, he thinks he is being told that a study of mathematics, if reasonably intense, creates physical faculties or powers of visualisation with which the uninitiated are not endowed. Learning that Minkowski and Einstein combine space and time into a single continuum, he tries to believe in the existence of a state of mind in which the sensations of space and time are con fused, and naturally he fails.

The position of students of mathematical physics, and of all but a fortunate few of the students of pure mathematics, is little better. Accustomed to regard a Cartesian frame of axes as a scaffolding erected in the real space around them, they can attach no meaning to a fourth coordinate, but having used complex electromotive forces with success in the theory of alternating currents, and having treated a symbol of differentiation as a detachable algebraic variable even to the extent of resolving operators into partial fractions for the solution of differential equations, these students are prepared to give pragmatical sanction to the most fantastic language.

The pure mathematician makes no attempt to imagine a space of four dimensions; he lays no claim to visualising a world that is inconceivable to other men. Only he finds that certain notions. In algebra are discussed most readily in terms adopted from geometry and given a meaning entirely algebraic, and since it is to the mathematician alone that algebraic problems are of concern in themselves, fear lest the man in the street should mistake the very subject of a mathematical conversation he might overhear has not prevented the mathematician from using the vocabulary he finds best suited to his own needs. Now it has happened that the talk of a few mathematicians has suddenly become of universal and absorbing interest, and a dictionary explaining the meanings they are in the habit of giving to some familiar words is required.

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