LOGICAL The most prominent aspect of elementary geometry is the logical aspect: a great number of propositions, termed theorems, are deduced from a comparatively few propositions as sumed at the outset and termed axioms or postulates. In the ideal treatment of the sub ject, all the assumptions should be enumerated explicitly, so that, if the question is asked, aAre the theorems of geometry the mathe matician can answer correctly, aYes, if my postulates are true.* As to whether the postu lates are true, that is not a matter for the mathematician as such to consider, but rather comes within the province of the physicist, psychologist or philospher.
The ordinary course in geometry, modeled after Euclid, does not carry out this ideal. Assumptions are continually being made as they may be needed for the purpose of proof, in addition to those explicitly enunciated as axioms and postulates. For example, in the first proposition of Euclid, dealing with the construction of an equilateral triangle on a given segment AB, circles are drawn with centres A and B and common radius AB, and it is then assumed that these circles intersect. The only justification given is the diagram or the appeal to spatial intuition. Again, in dealing with the congruence of triangles, it is assumed that a triangle may be moved about without altering its sides or angles, though the stated axioms do not even mention displacement. In spite of himself, Euclid's treatment is (partly) physical or intuitional, instead of purely mathe matical, that is, purely logical.
It is only within the last few years that the ideal has been (practically) attained; that is, a set of explicit assumptions (termed axioms or postulates indifferently) drawn up, from which the propositions of ordinary geometry follow by purely logical processes. Geometry be comes then a branch of pure mathematics. As Poincare expresses it, in this ideal treatment aWe might put the axioms into a reasoning apparatus like the logical machine of Stanley Jevons, and see all geometry come out of it.*
Many contributors have aided in this devel opment, among whom may be mentioned Gauss, Lobachevslcv, Pasch, Veronese, and especially Peano and his coworkers in symbolic logic, Pieri and Peano. The first elaborately worked out system is that of Hilbert (1900). We give a brief account of his axioms. Since then Veblen and Huntington in this country have developed much clearer and simpler systems. Consult Veblen, O., 'A System of Axioms for Geometry' (trans. Am. Math. Soc. 1904).
Geometry deals with three systems of ob jects or elements termed points, lines (used here in sense of straight lines), and planes, connected by certain relations expressed by the words lying in, between, etc. It is not necessary for the development of the subject that these words should suggest visual images; in fact the concrete nature of the elements and relations is to be eliminated from the discussion. To emphasize this abstract aspect, it is con venient to use symbols, say capital letters for points, Roman minuscules for lines, and Greek letters for planes. The axioms are arranged in five groups as follows: I. Axioms of Association or Connection.— I: Any two different points A, B determine a line a. (Such points are then said to lie on the line.) 2. Any two different points on a line deter mine that line.
3. In any line there are at least two points, and in a plane there are at least three points not on a hne.
4. Any three non-collinear points A, B, C determine a plane a.
5. Three non-collinear points of a plane determine that plane.
6. If two points A, B of a line a are in a plane a, then every point of a is in a. (The line is then said to lie in the plane).
7. Two planes a, /3 which have a point A in common have at least a second point B in common.
8. There exist at least four points not in one plane.
II. Axioms of Order.— These deal with the relation expressed by the term between.