Equivalent regions. Two polygonal regions in a plane are said to be equivalent if they can be dissected into parts which are respectively congruent. For instance, in Fig. 9.6D, the triangle ABC is equivalent to the isosceles birectangle ABED, since the parts CFJ and CFI of the former are congruent to the parts ADJ and BEI of the latter. That the relation of equivalence is transitive may be seen by superposing two dissections to make a finer dissection. Regions bounded by curves can be treated similarly, by regarding them as limiting cases of polygonal regions.

This notion enables us to define the area of any region in terms of a standard unit region, as follows. A region is said to be of area 1/n (where n is a positive integer) if the unit region can be dissected into n parts each equivalent to the given region; and a region is said to be of area m/n if it is equivalent to m juxtaposed replicas of a region of area 1/n. By a natural limiting process, we obtain a real number as the area of any given region.

The meaning of “elliptic” and “hyperbolic.” In ordinary Euclidean geometry, a central conic may be either an ellipse or a hyperbola. For any central conic, the pairs of conjugate diameters belong to an involution (of lines through the centre); but it is only the hyperbola that has self-conjugate diameters (viz. its two asymptotes). Accordingly, any involution (and so, conveniently, any one-dimensional projectivity) is said to be hyperbolic if it has two self-corresponding elements, and elliptic if it has none. Analogously, a polarity is said to be hyperbolic or elliptic according as it does or does not contain self-conjugate elements. Finally, a non-Euclidean geometry is said to be hyperbolic or elliptic according to the nature of its absolute polarity.

A more direct connection with ellipses and hyperbolas will be seen in Fig. 14.2A.

Beltrami's model. In the case of two-dimensional hyperbolic geometry, we are at liberty to draw the Absolute as a circle in the Euclidean plane, provided we understand that we are then using two metrics simultaneously: the Euclidean metric by which the circle is drawn, and the hyperbolic metric defined by 10.71 and 10.73. The poles and polars are taken with respect to the circle in the ordinary sense, as the constructions involved are essentially projective. This model for hyperbolic geometry is due to Beltrami (1835-1900).

Spherical and elliptic geometry. As we saw in §1.7, a convenient model for the elliptic plane can be obtained by abstractly identifying every pair of antipodal points on an ordinary sphere. The reflections and rotations which we shall define in §§6.2 and 6.3 are represented on the sphere by reflections in diametral planes and rotations about diameters.

In elliptic plane geometry, every reflection is a rotation. This rather startling result is a consequence of the fact that the product of the reflection in any diametral plane and the rotation through π about the perpendicular diameter is the central inversion (or point-reflection in the centre of the sphere), which interchanges antipodal points and so corresponds to the identity in elliptic geometry. In any orientable space (§2.5), a reflection reverses sense, whereas a rotation preserves sense. Thus the above remarks are closely associated with the non-orientability of the real projective plane (in which elliptic geometry operates). On a sphere, corresponding rotations about antipodal points have opposite senses; so the identification of such points abolishes the distinction of sense.

Ideal elements. As a sufficient set of axioms for plane hyperbolic geometry (based on point, intermediacy, and congruence) we may take 8.311, 8.313-8.317, 8.32, 9.11-9.15, and 9.61 (along with the denial of 8.318). It is, of course, possible to prove such theorems as 8.92 and 9.69 without using ideal elements. But the advantage of points at infinity has already been seen, and the reader will find that many propositions can be handled very expeditiously with the aid of the powerful machinery of projective geometry.

By considering flat pencils of parallels (namely, lines parallel to a given ray) and flat pencils of ultra-parallels (namely, lines perpendicular to a given line) it is possible to introduce ideal points into the plane, and to distinguish certain classes of them as forming ideal lines. But the three-dimensional treatment of Chapters VIII and IX is more satisfactory, as it allows all kinds of point, ordinary and ideal, to be covered by a single definition (§§8.6, 8.7). Accordingly, we have used Axioms 8.318, 8.319, and defined the absolute polarity in terms of reciprocal pencils of planes (§9.7), obtaining an oval quadric as the locus of points at infinity. When we restrict consideration to a single ordinary plane, the points at infinity that remain form a conic (the section of the quadric by the plane). From now on, we shall reserve the name Absolute for this conic, as we shall be concerned almost entirely with two-dimensional geometry.

Euclid. Geometry, as we see from its name, began as a practical science of measurement. As such, it was used in Egypt about 2000 B.C. Thence it was brought to Greece by Thales (640-546 B.C.), who began the process of abstraction by which positions and straight edges are idealized into points and lines. Much progress was made by Pythagoras and his disciples. Among others, Hippocrates attempted a logical presentation in the form of a chain of propositions based on a few definitions and assumptions. This was greatly improved by Euclid (about 300 B.C.), whose Elements became one of the most widely read books in the world. The geometry taught in high school today is essentially a part of the Elements, with a few unimportant changes.

According to the best editions, Euclid's basic assumptions consist of five “common notions” concerning magnitudes, and the following five Postulates:

I. A straight line may be drawn from any one point to any other point.

II. A finite straight line may be produced to any length in a straight line.

III. A circle may be described with any centre at any distance from that centre.

IV. All right angles are equal.

V. If a straight line meet two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.

The introduction of congruence. In Chapters V-VII we introduced the elliptic metric into real projective geometry by means of the “absolute polarity,” and observed the equivalence of two alternative definitions for a congruent transformation: a point-to-point transformation preserving distance, and a collineation permutable with the absolute polarity. It is quite easy to introduce the hyperbolic metric similarly (see §8.1). But in order to follow the historical development more closely, we prefer to reverse the process, introducing congruence into descriptive geometry as a second undefined relation, and stating its properties in the form of axioms. The propositions of Bolyai's “absolute geometry” can then be deduced in a straightforward manner. After imbedding the descriptive space in a real projective space by the method of Chapter VIII, we shall find a definite polarity which is permutable with every congruent transformation.

The relation of congruence applies initially to point-pairs, and we write AB ≡ CD to mean that the point-pair AB is congruent to the point-pair CD. But since every point-pair determines a unique segment, no confusion will be caused by reading the same formula as “the segment AB is congruent to the segment CD.”

Congruent transformations. In the present section we describe those properties of perpendicularity, congruence, distance, angle, etc., which closely resemble their two-dimensional analogues. We mention also certain other properties which more closely resemble their one-dimensional analogues, because projective spaces of odd dimension are orientable. In contrast to 6.42, we find various kinds of congruent transformation: reflections, rotations, rotatory reflections, and double rotations. The last of these resembles the one-dimensional translation in leaving no point invariant. A special case of it, known as a Clifford translation, is intimately associated with the existence of pairs of skew lines which are parallel in the sense of 1.11. Most of this chapter is concerned with various devices for the elucidation of this fascinating idea, which is quite different from the hyperbolic parallelism described in §1.4.

As in two dimensions, so in three, we begin with real projective geometry, and introduce metrical concepts by singling out an arbitrarily chosen uniform (or elliptic) polarity, calling it the absolute polarity. Then every point A has an absolute polar plane a, every line AB has an absolute polar line (α, β), and every plane a has an absolute pole A. Every point of a is perpendicular to A, and every plane through A is perpendicular to a. The relation between two absolute polar lines is symmetrical: every point on the one is perpendicular to every point on the other, and every plane through the one is perpendicular to every plane through the other.

Various definitions for a circle. We have seen that both elliptic geometry and hyperbolic geometry can be derived from real projective geometry by singling out a polarity. In the present chapter, so far as is possible, we give the definitions and theorems in such a form as to apply equally well in either of these non-Euclidean geometries.

In §8.6 we generalized the concepts “bundle” and “axial pencil” (§2.1) in such a way that any line and plane belong to a bundle, any two planes to a pencil. Those lines of a bundle which lie in a plane of the bundle are said to form a flat pencil (§10.1). Thus any two coplanar lines determine a flat pencil. In the proof of 8.61, we saw that any flat pencil can be constructed as a plane section of an axial pencil. We now make the analogous generalization of the concept “circle.”

A circle is the class of images of a point by reflection in the lines of a flat pencil. The lines are called diameters, their common point the centre, and the absolute polar of this point the axis. In elliptic geometry this will be seen to agree with our previous definition (§6.5). But in hyperbolic geometry we have to distinguish three cases: a proper circle (the “circle” of §9.1) has an ordinary centre and an ultra-infinite axis; a horocycle has parallel diameters, so that its centre and axis are at infinity; and an equidistant curve has ultra-parallel diameters, all perpendicular to the axis, while the centre is ultra-infinite.