so absolutely that no possible doubt could be entertained as to the justice of this award. Helmholtz had already originated the idea of studying the essential characteristics of space by a consideration of the movements possible in it. Klein called the attention of Lie to this problem of Helmholtz and encouraged him to undertake an investigation of it by means of his Theory of Groups. We can but meagerly indicate the outcome of this investigation. As stated by Lie his problem is: "To determine all finite continuous groups of transformations in three dimensional space in which two points have a single invariant and more than two points have no essential invariant"; meaning by invariant the distance, D, between the two points and by the statement that more than two points have no essential invariant, no invariant which is not expressible in terms of D. He finds that under the conditions of the problem the group must be six-parametered and transitive and cannot contain two infinitesimal transformations whose path curves coincide.37 Two solutions of the problem are given. He first investigates a group in space, possessing free mobility in the infinitesimal, in the sense that if a point and any line element through it be fixed, continuous motion shall still be possible; but if, in addition, any surface element through this point and the line

37 Transformations-Gruppen, Vol. III., p. 405 f., 1893.

element be fixed, no continuous motion shall be possible. The groups which in tri-dimensional space harmonize with these conditions Lie finds to be only those which are characteristic of the Euclidean and non-Euclidean geometries, but strange to say he also discovers that for the apparently analogous but simpler case of the plane or two dimensional space there are, besides these, certain other groups where the paths of the infinitesimal transformations are spirals. In his second demonstration, starting from transformation-equations with Helmholtz's first three postulates he proves that for a space of three dimensions the fourth postulate is entirely superfluous.

38

From an analytical point of view Professor Hilbert in a recent article in the Mathematische Annalen,39 which may be mentioned here, has advanced beyond these results of Lie by showing that it is possible to do away with the differentiability of functions which Lie's discussion requires. From the intuitional standpoint his article offers no improvement and is open to certain criticisms which Dr. Wilson 40 has pointed out.

38 For a brief statement of what is essentially Lie's method in English, see Halsted's “Columbus Report,” Proc. A. A. A. S., Vol. 48, 1899. Also Poincaré's Art. in Nature previously cited.

39 Bd. 56, Heft 3, pp. 381-422, October, 1902.

40 Archiv der Mathematik und Physik III. Reihe VI. 1. u. 2. Heft, Jan. 1903.

41

In his famous Festschrift, however, Professor Hilbert has done more perhaps than any one else except certain Italians to determine the precise number, meaning and relations of the postulates essential to geometry. In this older work Hilbert followed essentially the Euclidean method with a logic so keen and pure and a result so simple that many have even expressed the opinion that it will ultimately supersede Euclid in the elementary schools. Certain defects, however, have been pointed out by Schur,42 Moore, 43 and others, showing that Hilbert's postulates are not independent as he had supposed they were, and also illustrating how difficult it is to satisfy logic when one seeks to determine the foundations of geometry by the intuitional method. The discovery of this fact has led Hilbert in his recent article to abandon this method for the more strictly logical one. He starts from the ideas of Manifoldnesses and Groups as Lie had done, but uses the new conception of Manifoldnesses introduced by Georg Cantor thus dispensing with any special reference to a system of co-ordinates in a geometric space.

When the Lobatchewsky prize was awarded to Lie, the thesis of M. L. Gérard, of Lyons, also re

41 Grundlagen der Geometrie, Leipzig 1899.

42 Mathematische Annalen Bd. 55, p. 265 ff.

43 Transactions of the American Mathematical Society, Vol. III., pp. 142 ff.

ceived honorable mention. In this thesis Gérard endeavors to establish the fundamental propositions of non-Euclidean geometry without any hypothetical constructions except the two which are assumed by Euclid.44 (1) Through any two points a straight line can be drawn. (2) A circle may be described about any center with any given sect as radius. But in order to establish the relations between the elements of a triangle in a thoroughgoing manner he adds to these, two other assumptions as follows: (1) A straight line which intersects the perimeter of a polygon in some other point than one of its vertices intersects it again, and (2) two straight lines, or two circles, or a straight line and a circle, intersect if there are points of one on both sides of the other. One of the most important considerations for the advocates of non-Euclidean geometry is the requirement that all its figures shall be rigorously constructed. It was to meet this requirement that Gérard's investigation was undertaken and it is in this fact that its significance mainly lies.

When the Commission of the Physico-Mathematical Society of Kazan met in 1900 for the purpose of awarding again the Lobatchewsky prize, they found before them two new treatises on non

44 This idea was suggested and partially developed by G. Battaglini in his "Sulla Geometria Imaginaria di Lobatchewsky," Giornale di Mat. Anno V., pp. 217-231, 1867.

Euclidean geometry, the merits of which were so nearly equal that the decision between them was finally made by the casting of lots. These were A. N. Whitehead's investigations in his "Universal Algebra " 45 and Wilhelm Killing's "Grundlagen der Geometrie." 46

In the opinion of Sir Robert Ball, Whitehead's investigation excels anything previously done in two important particulars. In the first place he can treat n-dimensions by practically the same formulæ as those used for two or three dimensions; and secondly, the various kinds of space, parabolic, hyperbolic and elliptic, present themselves in Whitehead's methods quite naturally in the course of the work, where they appear as the only alternatives under certain definite assumptions. Perhaps the most significant portion of Killing's effort is his treatment of the "Clifford-Klein space-forms," whose importance lies in the fact that they show what a difference it makes whether we assume the validity of our fundamental axioms for space as a whole or only for a completely bounded portion of space. The first assumption yields the Euclidean and three non-Euclidean space-forms already mentioned, but the second gives a "manifoldness, at

45 Cambridge, England, 1898.

46 Paderborn, 1898.