Non-Euclidean Geometry

NON-EUCLIDEAN GEOMETRY The various metrical geometries are concerned with the prop erties of the various types of congruence-groups, which are defined in the study of the axioms of geometry and of their immediate consequences. But this point of view of the subject is the out come of recent research, and historically the subject has a differ ent origin. Non-Euclidean geometry arose from the discussion, extending from the Greek period to the present day, of the vari ous assumptions which are implicit in the traditional Euclidean system of geometry. In the course of these investigations it became evident that metrical geometries, each internally con sistent but inconsistent in many respects with each other and with the Euclidean system, could be developed. A short historical sketch will explain this origin of the subject, and describe the famous and interesting progress of thought on the subject.

History.

In 1621 Sir Henry Savile called attention to the existence of two blemishes (duo naevi) in geometry, namely, the theory of parallels and the theory of proportion. In both respects the work of later scholars has given rise to important branches of mathematics, while at the same time showing that Euclid is in these respects more free from blemish than had been previ ously credible. It was from endeavours to improve the theory of parallels that non–Euclidean geometry arose ; and though it has now acquired a far wider scope, its historical origin remains instructive and interesting. Euclid's "axiom of parallels" appears as Postulate V. to the first book of his Elements, and is stated thus, "And that, if a straight line falling on two straight lines make the angles, internal and on the same side, less than two right angles, the two straight lines, being produced indefinitely, meet on the side on which are the angles less than two right angles." To Euclid's successors this axiom had signally failed to appear self-evident, and had failed equally to appear indemonstrable. Without the use of the postulate its converse is proved in Euclid's i8th proposition, and it was hoped that by further efforts the postulate itself could be also proved. The first step consisted in the discovery of equivalent axioms. Christopher Clavius in deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight. John Wallis in 1663 showed that the postulate follows from the possi bility of similar triangles on different scales. Girolamo Saccheri showed that it is sufficient to have a single triangle, the sum of whose angles is two right angles. Other equivalent f may be obtained, but none shows any essential superiority to Euclid's. Indeed plausibility, which is chiefly aimed at, becomes a positive demerit where it conceals a real assumption.

Saccheri.

A new method, which, though it failed to lead to the desired goal, proved in the end immensely fruitful, was in vented by Saccheri, in a work entitled Euclides ab omni naevo vindicatus (Milan, 1733). If the postulate of parallels is involved in Euclid's other assumptions, contradictions must emerge when it is denied while the others are maintained. This led Saccheri to attempt a reductio ad absurdum, in which he mistakenly be lieved himself to have succeeded. What is interesting, however, is not his fallacious conclusion, but the non-Euclidean results which he obtains in the process. Saccheri distinguishes three hypotheses (corresponding to what are now known as Euclidean or parabolic, elliptic and hyperbolic geometry), and proves that some one of the three must be universally true. His three hy potheses are thus obtained : equal perpendiculars AC, BD are drawn from a straight line AB, and CD are joined. It is shown that the angles ACD, BDC are equal. The first hypothesis is that these are both right angles ; the second, that they are both obtuse; and the third, that they are both acute. Many of the results after wards obtained by Lobachevski and Bolyai are here developed. Saccheri fails to be the founder of non-Euclidean geometry only because he does not perceive the possible truth of his non Euclidean hypotheses.

Lambert.

Some advance is made by Johann Heinrich Lam bert in his Theorie der Parallellinien (written 1766; posthumously published 1786). Though he still believed in the necessary truth of Euclidean geometry, he confessed that, in all his attempted proofs, something remained undemonstrated. He deals with the same three hypotheses as Saccheri, showing that the second holds on a sphere, while the third would hold on a sphere of purely imaginary radius. The second hypothesis he succeeds in con demning, since, like all who preceded Bernhard Riemann, he is unable to conceive of the straight line as finite and closed. But the third hypothesis, which is the same as Lobachevski's, is not even professedly refuted.

Three Periods of non-Euclidean Geometry.—Non-Euclid ean geometry proper begins with Carl Friedrich Gauss. The advance which he made was rather philosophical than mathemati cal. It was he (probably) who first recognized that the postulate of parallels is possibly false, and should be empirically tested .by measuring the angles of large triangles. The history of non Euclidean geometry has been aptly divided by Felix Klein into three very distinct periods. The first—which contains only Gauss, Lobachevski and Bolyai—is characterized by its synthetic method and by its close relation to Euclid. The attempt at indirect proof of the disputed postulate would seem to have been the source of these three men's discoveries; but when the postulate had been denied, they found that the results, so far from showing contra dictions, were just as self-consistent as Euclid. They inferred that the postulate, if true at all, can only be proved by observa tions and measurements. Only one kind of non-Euclidean space is known to them, namely, that which is now called hyperbolic. The second period is analytical, and is characterized by a close relation to the theory of surfaces. It begins with Riemann's inaugural dissertation, which regards space as a particular case of a manifold (see MANIFOLDS) ; but the characteristic standpoint of the period is chiefly emphasized by Eugenio Beltrami. The conception of measure of curvature is extended by Riemann from surfaces to spaces, and a new kind of space, finite but unbounded (corresponding to the second hypothesis of Saccheri and Lam bert), is shown to be possible. As opposed to the second period, which is purely metrical, the third period is essentially projective in its method. It begins with Arthur Cayley, who showed that metrical properties are projective properties relative to a cer tain fundamental quadric, and that different geometries arise according as this quadric is real, imaginary or degenerate. Klein, to whom the development of Cayley's work is due, showed fur ther that there are two forms of Riemann's space, called by him the elliptic and the spherical. Finally, it has been shown by Sophus Lie, that if figures are to be freely movable throughout all space in co' ways, no other three-dimensional spaces than the above four are possible.

Gauss.—Gauss published nothing on the theory of parallels, and it was not generally known until after his death that he had interested himself in that theory from a very early date. In he announces that Euclidean geometry would follow from the assumption that a triangle can be drawn greater than any given triangle. Though unwilling to assume this, we find him in 1804 still hoping to prove the postulate of parallels. In 183o he an nounces his conviction that geometry is not an a priori science; in the following year he explains that non-Euclidean geometry is free from contradictions, and that, in this system, the angles of a triangle diminish without limit when all the sides are increased. He also gives for the circumference of a circle of radius r the formula irk(erhk — where k is a constant depending upon the nature of the space. In 1832, in reply to the receipt of Bolyai's Appendix, he gives an elegant proof that the amount by which the sum of the angles of a triangle falls short of two right angles is proportional to the area of the triangle. From these and a few other remarks it appears that Gauss possessed the foundations of hyperbolic geometry, which he was probably the first to regard as perhaps true. It is not known with certainty whether he influ enced Lobachevski and Bolyai, but the evidence we possess is against such a view.

Lobachevski.—The first to publish a non-Euclidean geometry was Nicholas Lobachevski, professor of mathematics in the new university of Kazan. In the place of the disputed postulate he puts the following : "All straight lines which, in a plane, radiate from a given point, can, with respect to any other straight line in the same plane, be divided into two classes, the intersecting and the non-intersecting. The boundary line of the one and the other class is called parallel to the given line." It follows that there are two parallels to the given line through any point, each meeting the line at infinity, like a Euclidean parallel. Hence a line has two distinct points at infinity, and not one only as in ordinary geom etry. The two parallels to a line through a point make equal acute angles with the perpendicular to the line through the point. If p be the length of the perpendicular, either of these angles is denoted by The determination of H (p) is the chief prob lem; it appears finally that, with a suitable choice of the unit of length, Before obtaining this result it is shown that spherical trigonom etry is unchanged, and that the normals to a circle or a sphere still pass through its centre. When the radius of the circle or sphere becomes infinite all these normals become parallel, but the circle or sphere does not become a straight line or plane. It be comes what Lobachevski calls a limit-line or limit-surface. The geometry on such a surface is shown to be Euclidean, limit-lines replacing Euclidean straight lines. It is, in fact, a surface of zero measure of curvature. By the help of these propositions Loba chevski obtains the above value of H (p), and thence the solution of triangles. He points out that his formulae result from those of spherical trigonometry by substituting ia, ib, ic, for the sides a, b, c.

Bolyai.—John Bolyai, a Hungarian, obtained results closely corresponding to those of Lobachevski. These he published in an appendix to a work by his father, entitled Appendix Scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori lzaud unquam decidenda) independentem: adjecta ad casum falsitatis, quadratura circuli geornetrica. This work was published in 1831, but its conception dates from 1823. It reveals a profounder appreciation of the importance of the new ideas, but otherwise differs little from Lobachevski's. Both men point out that Euclidean geometry is a limiting case of their own more general system, that the geometry of very small spaces is always approximately Euclidean, that no a priori grounds exist for a decision, and that observation can only give an approximate answer. Bolyai gives also, as his title indicates, a geometrical construction, in hyperbolic space, for the quadrature of the circle, and shows that the area of the greatest possible triangle, which has all its sides parallel and all its angles zero, is where i is what we should now call the space-constant.

Riemann.—The works of Lobachevski and Bolyai, though known and valued by Gauss, remained obscure and ineffective un til, in 1866, they were translated into French by J. Hovel. But at this time Riemann's dissertation, Ober die Hypothesen, welche der Geometrie zu Grunde liegen, was already about to be published. In this work Riemann, without any knowledge of his predecessors in the same field, inaugurated a far more profound discussion, based on a far more general standpoint ; and by its (posthumous) publication in 1867 the attention of mathematicians and philoso phers was at last secured.

Riemann's work contains two fundamental conceptions, that of a manifold and that of the measure of curvature of a continuous manifold possessed of what he calls flatness in the smallest parts..

There are four points in which this profound and epoch-making work is open to criticism or development—(r) the idea of a mani fold requires more precise determination ; the introduction of coordinates is entirely unexplained and the requisite presupposi tions are unanalysed; (3) the assumption that ds is the square root of a quadratic function of dxl, dx_, . . . is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connection with the general subject of the axioms of geometry.

Helmholtz.—The publication of Riemann's dissertation was closely followed by two works of Hermann von Helmholtz, again undertaken in ignorance of the work of predecessors. In these a proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction. Helm holtz's remaining works on the subject are of almost exclusively philosophical interest.

Beltrami.—The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann's work was brought into connection with that of Lobachevski and Bolyai. As he gave a convenient Euclidean interpretation of hyperbolic plane geom etry, his results will be stated at some length. The Saggio shows that Lobachevski's plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. Such surfaces are capable of a conformal representation (q.v.) on a plane, by which geodesics are repre sented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations.

Transition to the Projective Method.

The Saggio gives a Euclidean interpretation confined to two dimensions. But a con sideration of the auxiliary plane suggests a different interpreta tion, which may be extended to any number of dimensions. If, in stead of referring to the pseudosphere, we merely define distance and angle, in the Euclidean plane, as those functions of the coor dinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobachevski's. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. If we give our circle an imaginary radius the geometry on the plane become elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimen sions. Instead of a circle or sphere we may take any conic or quadric. With this definition, if the fundamental quadric be 2xx= o, and if 'xx' be the polar form of 'xx, the distance p be tween x and x' is given by the projective formula cos =Zxx'/ That this formula is projective is rendered evident by observing that e-2ip/k is the enharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric. With this we are brought to the third or projective period. The method of this period is due to Cayley; its application to previous non-Euclidean geometry is due to Klein. The projective method contains a generalization of discoveries al ready made by Laguerre in 1853 as regards Euclidean geometry. The arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by Lie's theory of con gruence. We then arrive at the stage of thought which finds its expression in the modern treatment of the axioms of geometry.

The Two Kinds of Elliptic Space.

The projective method leads to a discrimination, first made by Klein, of two varieties of Riemann's space; Klein calls these elliptic and spherical. They are also called the polar and antipodal forms of elliptic space. The latter names will here be used. The difference is strictly analogous to that between the diameters and the points of a sphere. In the polar form two straight lines in a plane always intersect in one and only one point ; in the antipodal form they intersect always in two points, which are antipodes. The antipodal form may be called a "quasi-geometry." Similarly in the antipodal form two diameters always determine a plane, but two points on a sphere do not de termine a great circle when they are antipodes, and two great circles always intersect in two points. Again, a plane does not form a boundary among lines through a point : we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a plane does not divide space, and does not, like a Euclidean plane, have two sides. But, in the antipodal form, a plane is, in these respects, like a Euclidean plane.

Finally, it is of interest to note that, though it is theoretically possible to prove, by scientific methods, that our geometry is non Euclidean, it is wholly impossible to prove by such methods that it is accurately Euclidean. For the unavoidable errors of observa tion must always leave a slight margin in our measurements. A triangle might be found whose angles were certainly greater, or certainly less, than two right angles; but to prove them exactly equal to right angles must always be beyond our powers. If, therefore, any man cherishes a hope of proving the exact truth of Euclid, such a hope must be based, not upon scientific, but upon philosophical considerations.

BIBLIOGRAPHY.-For

Lobachevski's writings, see F. Engel and P.Bibliography.-For Lobachevski's writings, see F. Engel and P.

Steckel, "Nikolaj Iwanowitsch Lobatschefsky," Urkunden zur Ge schichte der nichteuklidischen Geometrie (Leipzig, 1898) . For John Bolyai's Appendix, see J. Frischauf, Absolute Geometrie nach Johann Bolyai (Leipzig, 1872) ; the edition of his father, Tentamen . . . , vol. ii., published by the Mathematical Society of Budapest; J. Frischauf, Elemente der absoluten Geometrie (Leipzig, 1876) ; M. L. Gerard, Sur la geometrie non-Euclidienne (Paris, 1892).

For expositions of the whole subject, see F. Klein, Nicht-Euklidische Geornetrie (Gottingen, 1893) ; R. Bonola, La Geometria non-Euclidea (Bologna, 1906) ; P. Barbarin, La Geometrie non-Euclidienne (Paris, 1902) ; W. Killing, Die nicht-Euklidischen Raumformen in analytischer Behandlung (Leipzig, 1885). The last-named work also deals with geometry of more than three dimensions; in this connection see also G. Veronese, Fondamenti di geometria a piu dimensioni ed a piis specie di unita rettilinee . .. (Padua, 1891, German trans., Leipzig, G. Fontene, L'Hyperespace a (n-1) dimensions (Paris, 1892) ; A. N. Whitehead, Universal Algebra (Cambridge, 1898) ; E. Study, "L)ber nicht-Euklidische and Liniengeometrie," Jahr. d. Deutsch. Math. Ver. (1906) ; W. Burnside, "On the Kinematics of non Euclidean Space," Proc. Lond. Math. Soc. (1894) . A bibliography on the subject up to 1878 was published by G. B. Halsted, Amer. Journ. of Math. vols. i. and ii.; and one up to 1900 by R. Bonola, Index operum ad geometriam absolutam spectantium . . . (Leipzig, 1903) .