HYPER-GEOMETRY. Generalization has led geome ters to imagine other spaces than that in which we live, and to seek the properties of figures ex isting in space of more than three dimensions. The result has been the building up of a geometry of hyper-space or of n dimensions. Reasoning in this geometry is possible only by the use of symbols. Since a line segment, i.e. a figure of one dimension, is represented by an algebraic quantity of degree 1, such as a; since a square having two dimensions is represented by the algebraic expression and, finally, since a cube, having three dimensions, is represented by the algebraic expression a'—the idea naturally sug gests itself that some figure of four dimensions corresponds to the symbol a', and that, in general, some figure of n dimensions corresponds to the symbol an. The fact that four dimensions can not be represented in the three-dimensional space in which we live has little bearing upon the idea itself ; a three-dimensional figure (a solid) can not completely be represented on a plane, and yet mathematical thought involving the concept of three-dimensional space would remain logical and useful even if all actual figures were only two-dimensional.
The idea of the fourth dimension thrusts it self upon the mind even more prominently in studying rectangular coordinates in analytic geometry; ax = b represents a point, one axis being necessary; ax + by = c represents a line, two axes being necessary; and ax + by + ez = d represents a plane, three axes being necessary. This suggests that ax + by + cz dw = e may represent a three-dimensional figure in a four dimensional space. It is evident that just as we can draw in a plane the nets of the five regular bodies, we ought to be able, by analogy, to model in three-dimensional space the solid nets of all the six structures of four-dimensional space cor responding to the five regular bodies. This has been done by Schlegel, the models being made by Brill of Darmstadt. The figure corresponding to the square and cube may be described as follows: It is hounded by 8 cubes, just as the cube is hounded by 6 squares; it has 16 corners, 24 squares, and 32 edges, so that from every corner 4 edges, 6 squares, and 4 cubes proceed, and from every edge 3 squares and 3 cubes. Thus, reason ing by analogies, mathematicians have gradually developed higher geometric systems, and have suc ceeded in greatly extending the scope of geometry. The idea of higher dimensions has been brought somewhat into disrepute owing to the efforts of the followers of Professor 'Miner, of Leipzig, to explain the phenomena of spiritualism by making the fourth-dimensional world the abode of spirits. Nevertheless, mathematicians agree as to the great practical value of the idea, inas much as it leads to important simplifications of mathematical language, and especially inasmuch as by its perfect generality it gives remarkable clearness to the concepts of real geometry. A reasonable mathematical treatment of the subject may be found in Schubert's essay on the "Fourth Dimension," in his Mathematical Essays and Recreations (Chicago, 1898).
The phases of modern geometry are closely in terwoven in their historic as well as in their logical development. Monge, the father of mod ern geometry, published his Geometrie descrip tive in 1800, five years after the work of his pupil, Lacroix, appeared. Following his works were those of Hachette (1812, 1818, 1821), and later Leroy (1842), Olivier (1845), de la Gour nerie (1860). In Germany leading contributors have been Ziegler (1843), Anger (1858), Fiedler (3d ed. 1883-88), and Wiener (1884-87). Monge did not confine his labors to descriptive geometry; he set forth the fundamental theorem of recip rocal polars, though not in modern language, gave some treatment of ruled surfaces, and ex tended the theory of polars to 'quadrics. Monge and his school concerned themselves especially with the theory of form, but Desargues, Pascal, and Carnot treated chiefly the metrical relations of figures. Carnot investigated those relations in particular connected with the theory of trans versals, in his works, Geometrie de position ( 1803 ), Theorie des transversales ( 1808 ) . The present geometry of position (Geometric der Lage) has little in common with Carrot's Geome trie de position.
Although Newton had discovered that all curves of the third order can be derived by cen tral projection from five fundamental types, the origin of projective geometry is generally at tributed to Poncelet (1822). He first made prominent the power of the projective relations, and the principle of continuity in research. Mains followed Poncelet, making much use of anharmonic ratios in his Barycentrischer Calcul (1827). The anharmonic point and line prop erties of conics have been further elaborated by Brianchon, Chasles, Steiner, and Von Staudt. Plucker applied the theory of transversals to curves, and Salmon discovered the so-called cir cular points at infinity. Brianchon (1806) ex tended the application of Desargues's theory of polars. To Gergonne (1825-26) is due the prin ciple of duality, the most important after that of continuity in modern geometry. Gergonne was the first to use the word 'class,' and explicitly defined class and degree (order), showing their dual relation. He and Chasles were the first to study scientifically surfaces of higher order. Steiner (1832) gave the first complete discussion of the projective relations between ranges and pencils, and laid the foundation for modern pure geometry. In 1848 Steiner showed that the theory of polars can serve as a foundation for the study of plane curves, independent of the use of co6rdinates. He introduced the noteworthy curves which now bear the names of himself, Hesse, and Cayley. Chasles, in his Apergu his torique (1837), popularized the new geometry and introduced the name homographic, and ex tended the homographic theory. Von Staudt (1847, 1856-60) set forth a complete pure geo metric system in which metric geometry finds no place. Cremona (1862, 1875), Townsend (1863). and Clifford did much to extend the knowledge of modern geometry.