ANALYTICAL METRICS, or NON EUCLIDIAN GEOMETRY INTERPRET ED ANALYTICALLY. For a historico critical account of the notion and developments of Non-Euclidian geometry from the standpoint of pure geometry-- from the point of view of Lobatschewsky and Bolyai— the reader is re ferred to the article 'Non-Euclidian Geometry,' in this work. The aim of the following para graphs is to present a readily intelligible ac count of the epoch-making concept by means of ideas familiar in analytical projective geometry, the point of departure and method being those of Cayley as modified and im proved by Klein.
General Considerations Concerning Geo metric Measurement.— Every problem of geometric measurement is reducible to one or the other or both of two fundamental prob lems: (1) to find the distance between two points; (2) to find the angle between two (in tersecting) lines. Any two points belong to a range; any two (intersecting) lines belong to a pencil ; a range and a pencil are one-dimensional spaces; so it is seen that the fundamental prob lems constitute the problem of measurement in one-dimensional spaces. In the projective ge ometry of the plane (see GEOMETRY, PROJECTIVE, and GEOMETRY, MODERN ANALYTICAL) we learn that the range and the pencil are related by the principle of duality, or reciprocity. A range and a pencil if rendered projective, i.e., if the range can be obtained by the repeated projec tion of the range established by the pencil on one of its transversals, are so related that the anharmonic ratio of any four points (or lines) is equal to that of the corresponding four lines (or points). Countless, examples are met of the fact that in general to any point (or line) proposition there corresponds a line (or point) proposition; such pairs of propositions being so related that either proposition of a pair of reciprocals being given, the other can be found by merely exchanging the notions of point and line. This reciprocity does not, however, at first appear to be universal and all-pervasive. For example, the distance, + V (xi— x2)*-1- (Yi between two points (x,, (x., y,), or xi yis + 1=0, X. e y,'/ +1 =0, is an algebraic function of the co-ordinates; while the angle, tan el'!.): (66 +sm.) of the lines ( ), ((:rh ) or 6 x + y + 1 = 0, 6x + th y+ 1=-0, is a transcendental function of the co-ordinates. Sign being disregarded, a segment of a range is uniquely determined, while an angle of a pencil may have any one of an in finite series of values differing from each other by multiples of a period rr, or 180°. Again a
given segment is divisible by ruler and com passes into any chosen number of equal parts, while, by the same means, such division is possible in case of an arbitrarily given angle only if the division is to be bisection, quadri section, etc., but not if it is to be trisection, for example.
It is such discrepancies in the general scheme of reciprocity that furnish the motive for seeking to generalize the ordinary concep tion of distance and angle measurement in such a way that the discrepancies shall disappear and that the principle of duality shall apply without exception. Such being the motive, on the one hand, the possibility, on the other, of making the needed generalization lies in the fact that, despite the differences indicated, the ordinary notion of distance and the ordinary notion of angular measurement have two fundamental properties in common. These are: (1) That the distances between points of a range or the angles between lines of a pencil are added in such a way that, if 1, 2, 3 denote three such points or three such lines, we have in the first case seg. 12 + seg. 23= seg. 13, and, in the sec ond case, 1 12 -F Z 23 = L 13. In particular, seg. 11 = 0 1 11=0, whence it follows that seg. 12= — seg. 21, and 1 12=— 1 21; (2) the second of the common properties is that distance and angle are not altered in magnitude by displacement, where displacement of the pen cil consists in rotating it as a rigid figure about its centre or vertex, and displacement of the range consists in moving it as a rigid figure along its base. These properties come clearly to light if we scrutinize the way in which dis tance and angle are not altered in magnitude. For this purpose we use a scale, which in case of the range consists of a segment divided into equal intervals by points, and in case of the pencil consists of an angle divided into equal intervals by lines. Each interval is a unit of distance or angle. Then to measure the dis tance (say) between two given points, we place any division of the scale on one of the points, and then count the number of intervals to the other given point. Analogously for angles. The first property above mentioned is used in determining the distance by counting the in tervals. The displacement property is em ployed in that we are indifferent as to which division of the scale we start with; that is, the distance or angle is found to have the same size if we measure, then displace the scale and measure again.