INTRODUCTION
In an address to the Committee of the Cayley Portrait Fund in 1874 Clerk Maxwell, after referring in humorous terms to the work of Arthur Cayley in higher algebra and algebraical geometry, concluded his eulogium with the lines—
March on, symbolic host! with step sublime,
Up to the flaming bounds of Space and Time!
There pause, until by Dickenson depicted,
In two dimensions, we the form may trace
Of him whose soul, too large for vulgar space
In n dimensions flourished unrestricted.
In those days any conception of "dimensions" beyond length, breadth and height was confined to advanced mathematicians; and even among them, with very few exceptions, the fourth and higher dimensions afforded only a field for the practice of algebraical analysis with four or more variables instead of the three which sufficiently describe the space to which our foot-rules are applicable. Any geometrical conclusions reached were regarded only as analogies to the corresponding results in geometry of three dimensions and not as having any bearing on the system of Nature. As an illustration, reference may be made to the "more divine offspring of the divine Cube in the Land of Four Dimensions" mentioned on p. 94 infra which has for its faces eight three-dimensional cubes and possesses sixteen four-dimensional angular points or comers.
During the present century the work of Einstein, Lorentz, Larmor, Whitehead and others has shewn that at least four
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