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." Since a point is defined as having "position but no magnitude," it follows that it would require an infinite number of points to make up a line. Similarly a line has length, but no breadth or thickness, and it would therefore require an infinite number of lines laid side by side to make up a surface. Again a surface has, theoretically, no thickness, and it would therefore require an infinite number of surfaces superimposed on one another to make up a solid. We must therefore conclude, by analogy, that it would require an infinite number of solids to make up a "higher solid." In particular, a Tesseract must be supposed to contain an infinite number of cubes, and, in general, four space must be conceived of as containing an infinite number of three spaces. (3) "The Boundaries of a figure in a space of any dimensionality are themselves figures in the next lower space." Thus a Line (one space) is bounded by Points (zero space). A surface (two space) is bounded by Lines (one space). A solid (three space) is bounded by Surfaces (two space). We must conclude therefore that "higher solids" (four space) are bounded by Solids (three space). [Illustration: _Fig. 10_]