1.1. 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...

2.1. Definitions and axioms. In any geometry, logically developed, each definition of an entity or relation involves other entities and relations; therefore certain particular entities and relations must remain undefined. Similarly, the proof of each proposition uses other propositions; therefore certain particular propositions must remain unproved; these are theaxioms. We take for granted the machinery of logical deduction, and the primitive concept of aclass(or “set of all”).

Unless the contrary is stated, the wordcorrespondencewill be used in the sense ofone-to-onecorrespondence. Thus a set of entities is said to correspond to another set if...

3.1. Two-dimensional projectivities. The history of conics begins about 430 b.c., when Hippocrates of Chios expressed the “duplication of the cube” as a problem which his followers could solve by means of intersecting curves. Some seventy years later, Menaechmus showed that these curves can be defined as sections of a right circular cone by a plane perpendicular to a generator. Their metrical properties (such as the theorem regarding the ratio of the distances to focus and directrix) were described in great detail by Aristaeus, Euclid, and Apollonius.* Apollonius introduced the names ellipse, parabola, and hyperbola, and discovered the harmonic property...

4.1. The von Staudt-Hessenberg calculus of points. Protective geometry might well be described as “What we can do with an ungraduated straight edge or ruler, without compasses.” It is hoped that Chapters II and III have shown what a wealth of elegant theorems can be obtained without any appeal to measurement. It is one of von Staudt’s greatest discoveries that even such apparently metrical notions as coordinates and cross ratios can be introduced non-metrically. The following is a brief outline of his method, as revised by Hessenberg,* and a summary of the standard results in analytical projective geometry.

We saw,...

5.1. Elliptic geometry in general. As we saw in §1.7, Klein’s elliptic geometry is a metrical geometry in which two coplanar lines always have a single point of intersection. One method of approaching this geometry is to introduce an undefined relation of congruence, satisfying certain axioms such as the following:

5.11.From any pointDon a given line, we can lay off two segments, CDandDE,each congruent to a given segmentAB.

We can then develop a chain of propositions similar to Euclid I, 1-15, and conclude* that all lines are finite and equal. A line being...

6.1. 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 throughπabout the perpendicular diameter is the central inversion...

7.1. 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 theirone-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 areparallelin...

8.1. Klein’s projective model for hyperbolic geometry. The two chief ways of approaching non-Euclidean geometry are that of Gauss, Lobatschewsky, Bolyai, and Riemann, who began with Euclidean geometry and modified the postulates, and that of Cayley and Klein, who began with projective geometry and singled out a polarity.

In Klein’s treatment, two lines areperpendicularif they are conjugate in the absolute polarity, and the geometry is elliptic or hyperbolic according to the nature of this polarity. We have considered the elliptic case exhaustively in the preceding three chapters; the null polarity is easily seen to be unsuitable. Setting these...

9.1. 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...

10.1. Ideal elements. As a sufficient set of axioms for plane hyperbolic geometry (based onpoint, intermediacy, andcongruence) 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 ofpoints at infinityhas 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...

11.1. 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...

12.1. Formulae for distance and angle. For many purposes, such as the development of trigonometry, it is desirable to take the absolute polarity in its general form 4.52, namely

12.11.X_{\mu }=c_{\mu \nu }x_{\nu },\; x_{\mu }=C_{\mu \nu }X_{\nu }(μ=0, 1, 2),

wherec_{\mu \nu }=C_{\nu \mu }andc_{\lambda \nu }C_{\mu \nu }=\delta _{\lambda \mu }.

We shall find it convenient to make the abbreviations*

(xy)=c_{\mu \nu }x_{\mu }y_{\nu },\; [XY]=C_{\mu \nu }X_{\mu }Y_{\nu },

so that the hyperbolic Absolute is the conic (xx) = 0 or [X X] = 0. Since the polarity is not changed by reversing the signs of all thec_{\mu \nu }andC_{\mu \nu }, there is no loss of generality in assuming that (xx)>0 for one ordinary point, and therefore (by...

13.1. 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...

14.1. The meaning of “elliptic” and “hyperbolic.” In ordinary Euclidean geometry, a central conic may be either anellipseor ahyperbola. 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 hasself-con jugate diameters (viz. its two asymptotes). Accordingly, any involution (and so, conveniently, any one-dimensional projectivity) is said to behyperbolicif it has two self-corresponding elements, andellipticif it has none. Analogously, apolarityis said to be hyperbolic or elliptic according as it does or does not contain...

15.1. Hjelmslev’s mid-line. If AB and A′B′ are congruent point-pairs in a plane, we can find a congruent transformation that takes AB to A′B′ (see pp. 113, 201-203). In fact, since A′ and B′ are invariant by reflection in the line A′B′, there are two such transformations.

If the geometry is Euclidean or hyperbolic, one of these two transformations is direct and the other opposite. The latter, being the product of three reflections, cannot be anything but aglide-reflection(including, as a possible special case, a simple reflection). Let o denote the axis of the glide-reflection. Since A, A′ are...