Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometry at this stage The theory of distance will be considered after the principles of descriptive geometry have been developed.
Descriptive Geometry.
Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch.[36] An improved version is due to G. Peano.[37] Both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert’s[38] formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered[39] as the science of a class of relations, each relation being a two-termed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold are the points. But a combination of these two points of view yields[40] the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be “in the [linear] order ABC.”
O. Veblen’s axioms and definitions, slightly modified, are as follows:—
1. If the points A, B, C are in the order ABC, they are in the order CBA.
2. If the points A, B, C are in the order ABC, they are not in the order BCA.
3. If the points A, B, C are in the order ABC, A is distinct from C.
4. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC.
Definition.—The line AB (A ≠ B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute the segment AB.
5. If points C and D (C ≠ D) lie on the line AB, then A lies on the line CD.