II MODERN PURE GEOMETRY By THOMAS F. HOLGATE I. INTRODUCTION 1. In Analytical Geometry conclusions are reached through the application of algebraic processes to geometric properties and relations. By making use of certain conventions the given relations are expressed in algebraic language, then certain algebraic operations are performed and the results are reinterpreted as geometric propositions. During the process the geometric concept may be entirely lost sight of and the resulting statement may bear no apparent relation to the premises from which it was derived. In Pure Geometry, on the other hand, the geometric concept is kept continually in mind throughout the reasoning process, and the steps by which a conclusion is reached from given conditions are readily traceable. 2. Pure geometry was cultivated by peoples of the earliest times. By them many important theorems were discovered on the relations of triangles and other rectilinear forms, on the properties of circles and spheres, and on areas, ratios, and the equality and similarity of geometric figures. The investigations of the ancient geometers were carried so far as to include the conic sections and certain curves of higher order whose principal properties were discovered, but the methods used were fragmentary and the results for the most part were disconnected. The ancient geometry is typified most clearly by Euclid's Elements, which was in fact a collation and systematic arrangement of the geometric knowledge of his time. 55 In it properties and relations are demonstrated each by itself, and little attention is paid to relations common to all forms of the same class. The method of Euclid has come to be known as the method of Elementary Geometry, and the subject-matter of his elements has prescribed the field of elementary geometry. 3. The methods of the ancient geometers were not materially modified till the period of the revival of learning early in the sixteenth century, when with the introduction of certain new concepts, and the application of well-known older ones, as, for example, infinitely distant elements, the harmonic division of a line segment, the principle of continuity, and the theory of imaginary intersections, the science began to take on a more generalized form. The renewed activity in geometric research resulted in the invention by Descartes of the analytical geometry, and for two and a half centuries investigations by purely geometric methods were for the most part pushed aside. Happily interest in pure geometry was revived toward the close of the eighteenth century through the publications of Monge, and during the first half of the nineteenth century it reached its highest development at the hands of Poncelet, Steiner, Von Staudt, and Chasles. 4. Modern pure geometry differs from the geometry of earlier times not so much in the subjects dealt with as in the processes employed and the generality of the results obtained. Much of the material is old, but by utilizing the principle of projection and the theory of transversals, facts which were thought of as in no way related, prove to be simply different aspects of the same general truth. This generalizing tendency is the chief characteristic of modern geometry, and while it may perhaps be attributed largely to the influence of the analytic method, still it is true that some progress had been made in this direction before the analytic method was invented, and pure geometry has done much in recent times to enliven and heighten the interest in analysis. II. SIMPLE ELEMENTS IN GEOMETRY 5. Points, straight lines, and planes are the simple undefined elements of pure geometry. Each of these may be thought of as having an existence independent of the others; a plane may be thought of without considering the lines and points which lie in it; we may think of a line without considering the points which lie on it or the planes which pass through it, and of a point without considering either the lines or the planes which pass through it. In fact each of these simple elements may be the base on which rest an indefinite number of elements of either of the other kinds. III. THE PRINCIPLE OF DUALITY 6. Duality in space. Two points will fix the identity of a straight line and three points will in general determine a plane. So also two planes intersect in a straight line and three planes in general have one point in common. If three points lie in a specialized relative position, namely, in a straight line, then many planes pass through them. Similarly, if three planes be in a specialized relative position, namely, with one line in common, then many points lie in all three. But apart from such special cases the following statements may be made: al. Three points determine a plane. a2. Three planes determine a point. b1. Two lines which have a common point determine a plane. 62. Two lines which have a common plane determine a point. c1. A line and a point determine a plane. c2. A line and a plane determine a point. In these statements taken two and two there will be noted an interchangeable relation between the elements point and plane, and between line and line. This is spoken of as a dual relation, and in accordance with it any geometric form will yield another by replacing every point in one by a plane in the |