APPENDIX.
To illustrate how the analogy of the relation between two and three-dimensional space enables us to determine some of the properties of four-dimensional figures:
(1)
"Any figure in a space of a given dimensionality generates a corresponding figure in the next higher space, by moving in a direction at right angles to any direction that can be drawn within itself.[7] Or, in general, space of any dimensionality generates, by such a movement, the next higher space."
Thus, the lowest sort of space is space of zero dimensions, i.e., a mathematical point. If it moves a distance of one inch, it traces out a Line one inch long—that is to say a one space "figure." If this moves at right angles to itself for a distance of one inch, it traces out a two space figure, viz., a square of side one inch. If this again moves a distance of one inch in a direction at right angles to every direction that can be drawn within it, that is, in a direction perpendicular to itself, it traces out a cube of side one inch, i.e., a three space figure or "solid."
We must, therefore, conclude, from analogy, that if the cube were itself to move, a distance of one inch, in a direction at right angles to every direction that can be drawn in our space—in the unknown direction, that is, of the fourth dimension—it would generate a "higher solid" of side one inch. The higher solid thus generated is called a "Tesseract" and its properties are quite well known.
(2)
"Every figure, in a space of a given dimensionality, contains an infinite number of the 'corresponding' figures—see (1)—in the next lower space."