1. If A, B, C are points of a line and B is between A and C, then B is between C and A.
2. If A and C are points of a line a, then there exists on a at least one point B between A and C, and at least one point D such that C is between A and D.
3. Of any three collinear points, one and is between the other two.
hese three axioms deal with the line, while the fourth deals with the plane.
4. If A, B, C are any three non-collinear points, and a is a line in their plane passing through a point of the segment AB, but not through A or B or C, then a contains a point of either the segment AC or the segment BC.
III. Axioms of Congruence.— The first five axioms of this group relate to congruent seg ments and congruent angles. For example, a segment AB is congruent to itself and to the reversed segment BA; and segments con gruent to the same segments are congruent to each other. Finally, the sixth is a metrical axiom concerning triangles: if two sides and the included angle of one triangle are con gruent respectively to two sides and the in cluded angle of another triangle, then the re maining angles are also congruent.
The fact that the remaining sides are con gruent is not included as a part of the axiom be cause it may be proved. The other cases of congruent triangles are theorems. In Euclid the above statement is a theorem, but this is possible, as already observed, merely on account of unstated assumptions relating to displace ment. Euclid's axiom that all right' angles are congruent, in Hilbert's system becomes a theorem.
IV. The Axiom of Parallels.—This contains only the so-called Euclidean axiom, in the form: Given a line a and a point A not on a, then in the plane a determined by a and a there is only one line through A which does not intersect a.
V. Axioms of Continuity.— The continuity notion is analyzed into two parts of which the first (1) is stated in the axiom Archimedes. 1. On a straight line consider any two points A, B and a point Al between them; construct the points A,, A,, ...., in order, so that A, is between A and AI, A. between A. and A,,
etc., and so that the segments AA,, A,A,, A.A.. ...., are congruent; then among the points so constructed there exists a point An such that B is between A and An. That is, by repeatedly laying off a given segment however small any assigned point of the line will be passed after a finite number of steps.
This axiom is sufficient for the development of the usual theorems of geometry. However the space to which the theorems apply would not be continuous in ordinary sense. It would in fact contain only those points of the space considered in analytical geometry whose co.or dinates are rational or expressible by• radicals of the second order. To identify with con tinuous space it is necessary to add a final axiom (2) relating to convergent. point sets, or else the so-called axiom of completeness which states that the system of elements (points, lines, planes) cannot be enlarged by adjoining other elements in such a way that all the previous axioms are preserved.
The fact that this set of axioms is suffi cient is shown by actually deducing the usual body of theorems. This is done in Hil bert's Diagrams are here used, it is true, but only for convenience; the proofs can be given without any reference to the diagrams. Often the deduction of those re sults which are evident to the intuition is long and complicated. This is the case, for ex ample, in showing that a triangle (or any simple polygon) has the properties expressed by the terms inside and outside. It must be shown from the axioms that the given triangle brings about a division of the points in its plane into three classes, namely, points P, points 1, and points 0, such that any two I points or any two 0 points may be connected by a broken line not containing any P point, while any broken line from an I point to an 0 point necessarily contains P points. To the intui tion, of course, the P points are the points on the perimeter of the triangle, the I points are those inside, and the P points are those out side.