at the distance Y from the point of intersection will be given by the equation 2aD . 7-7T Newcomb also assumes that two straight lines intersect only in a single point. lie defines a " complete right line " as one returning into itself, as supposed in postulate 2. Any small portion of it is to be conceived as a Euclidean right line. The locus of all complete right lines passing through the same point and touching a Euclidean plane through that point will be called a " complete plane." A " region " will mean any indefinitely small portion of space iu which we are to conceive of the Euclidean geometry as holding true. Newcomb then proceeds to demonstrate the following proposi tions. I. All complete right lines are of the same length 2D. Hence D is the greatest possible distance at which the points can be situated, it being supposed that the distance is measured on the line of least absolute length. II. The complete plane is a Euclidean plane in every region of its extent. III. Every system of right lines passing through a common point A and making an indefinitely small angle with each other are parallel to each other in the region A at distance 1). IV. If a system of right lines pass in the same plane through A the locus of their most distant points will be a complete right line. Y. The locus of all the points at distance D from a fixed point A is a complete plane, and indeed a double plane if we consider as distinct the coincident surfaces in which the two opposite lines meet. VI. Conversely, all right lines perpendicular to the same com plete plane meet in a point at the distance D on each side of the plane. VII. For every complete right line there is a conjugate com plete right line such that every point of the one is at distance D from every point of the other. VIII. Any two planes in space have as a common perpendicular the right line joining their poles, and intersect each other in the conjugate to that right line. IX. If a system of right lines pass through a point, their con jugates will be in the polar plane of that point. If they also be in the same plane the conjugates will all pass through the pole of that plane. X. The relation between the sides a, &, c of a plane triangle in curved space and their opposite angles A, B, C will be the same as in a Euclidean spherical triangle of which the corresponding sides are ^ 2D 20 2D XL Space is finite, and its total volume admits of being definitely expressed by a number of Euclidean solid units which is a function of D. 8D : * XII. The total volume of space is - . IT XIII. The two sides of a complete plane are not distinct, as in a Euclidean surface. XIV. If moving along a right line we erect an indefinite series of perpendiculars each in the same Euclidean plane with the one which precedes it, then on completing the line and returning to our starting point, the perpendiculars will be found pointing in a direction the opposite of that with which we started. Newcomb concludes thus : " It may be also remarked that there is nothing within our experience which will justify a denial of the possibility that the space in which we find ourselves may be curved in the manner here supposed. It might be claimed that the dis tance of the farthest visible star is but a small fraction of the greatest distance D, but nothing more. The subjective impossibility of conceiving of the relation of the most distant points in such a space does not render its existence incredible. In fact our difficulty is not unlike that which must have been felt by the first man to whom the idea of the sphericity of the earth was suggested in con ceiving how by travelling in a constant direction he could return to the point from which he started without during his journey feel ing any sensible change in the direction of gravity." A sketch of the non-Euclidean geometry is given by Professor O. Chrystal in the Proc. Roy. Soc. Edin., vol. x., session 187_9-80. The study of this paper is recommended to all who desire to study the elements of what has been called "pangeo- metry." A more extensive work, which contains the theories of Riemann and Helmholtz, is J. Frischauf s Elementc dcr absolutcn Geometric, Leipsic, 1876. A fundamental step in the abstract theory of measure ment was taken by Professor Cayley in his " Sixth Memoir upon Quantics," Philosophical Transactions, vol. cxlix. (1859). The theory thus originated by Cayley has been 665 more fully developed by Klein in his memoir " Ueber dio nicht-Euclidische geometrie," Mathematische Annalen, vol. iv. p. 573. We shall here enter into this theory in some detail, for in it lies the true foundation of geometrical measurement. A sketch of the theory was given by the author of the present article in Hermathena, No. vi. pp. 500-541, Dublin, 1879. This theory may be regarded merely as a more general ized method of measuring distances and angles in ordinary space, but the results to which it leads are in many respects identical with those to which we are conducted by the theory just discussed. For instance, Newcomb s principle as to the length of the shortest distance between two points never exceeding a certain magnitude is common to his theory and to Cayley s. The theory of Cayley has, however, claims on our attention of a special kind. We here deal with the space with which Euclid has made us familiar, only observing that it is the measurements in that space which are to be conducted on a more general principle. We commence by assuming the existence of a certain surface called the "fundamental quadric," often called " the absolute." By the aid of this quadric and an arbitrary constant c we determine the generalized distance between the points in accordance with the following definition : The distance between two points is equal to c times the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. Let us first test this theory by a very obvious principle which any theory of distance ought to fulfil. It is plain that, if P, Q, R be three collinear points, then in ordinary measurement we ought to have but it is easy to see that this condition is fulfilled in the generalized measurement. Let the line PQ cut the fundamental quadric in the two points X, Y, then we have PQ = clog(PX-f-PY)-clog(QX-=-QY) QR = clog(QX-f-QY)-clog(KX-^RY) PR = clog (PX^PY) - clog(RX^RY) ; whence, as in the ordinary measures, PQ+QR=PR. It is also obvious that ir the generalized as in the ordinary measures (PQ)--(QP), and that the distance between the coincident points is equal to zero. From an obvious property of logarithms we also learn the im portant fact that the generalized distance between the points is indeterminate to the extent of any integral number of the periods 2ciir. The distance from any point to its harmonic conjugate with respect to the two fundamental points is equal to civ. We thus see that the distance between any two harmonic conjugates is con stant. It is usual to make the arbitrary constant c equal to - i-^-2, in which case we see that the distance between the two harmonic conjugates is equal to ir-^-2. It can also be shown that, if the two absolute points on a right line coalesce, then the generalized s} T stem of measurement degrades to the ordinary system. The two abso lute points are at an infinite distance from every other point, so that in the generalized system of measurement every right line has two points at infinity, and in general all the points in space which lie at infinity are situated on the fundamental quadric. In ordinary geometry we define a circle to be the locus of a point which is at a constant distance from a given point. In the more generalized geometry we retain the same definition of the circle, only observing that the distance to be constant must be expressed in the generalized manner. The plane of course cuts the absolute in a conic section, so that the determination of the circle whose centre is C is the following problem in conic sections : Through a fixed point a straight line OP is drawn which cuts a given conic in the points X, Y ; determine the locus of P so that the anhar monic ratio (0, P, X, Y) shall remain constant. This problem is most readily solved by projecting the conic into a circle the centre of which is the projection of O. The problem then assumes the very simple form. On the diameter of a fixed circle a point P is taken so that the anharmonic ratio of the four points consisting of P , the centre O , and the two points in which the
line O P cuts the circle remains constant. It is required to find