This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

Similarly, for angles at a point one could talk of a circular ordering. But when a hypothesis of relative position of points and lines in one part of a diagram implies a relationship for other parts of the figure far away, it seems clear that something important is happening, and it may be dangerous to rely on intuition. For example, how do you know that the angle bisector at a vertex A of a triangle ABC meets the opposite side BC between the points BC and not outside? Of course, it is obvious

7.14 (Linear ordering) Given a finite set of distinct points on a line, it is possible to label them Aj ,Az, ... ,An in such a way that Ai * Aj * 4k if and only if either i < j < k or k < j < i. 7.15 Suppose that lines a, b, c through the vertices A, E, C of a triangle meet at three points inside the triangle. Label them X=a·c, Y=a·b, Z=b·c. Show that one of the two following arrangements must occur: (i) A * X * Y and E * Y * Z and C * Z * X (shown in diagram), or (ii) A * Y * X and E 8 A c.

hypothesis we have DA ~ D'A'. So a third application of (C6) shows that the triangles BDA and B' D' A' are congruent. In particular, LBAD ~ LB'A'D' , which was to be proved. Note: We may think of this result as a replacement for (I.13), which says that the angles made by a ray standing on a line are either right angles or are equal to two right angles. We cannot use Euclid's statement directly, because in our terminology, the sum of two right angles is not an angle. However, in applications,

definition of ordered field. Note again that ee' is not a field. But if rp E ee', rp> 0, then -..ftP E ee' also. Now we take K to be the set of all elements of ee' that can be obtained from lR(t) by a finite number of operations +, -, ., -:-, and rp > 0 t----> -..ftP. The proof that K' is a field can be carried out exactly as in the proof of (18.2). Clearly, K' is Euclidean, and taking p' = K' n Pcc' makes K' into an ordered field. Example 18.4.1 Let IT be the Cartesian plane over the field Q'

the radical axis in that case.) 20.5 If three circles each meet the other two in two points, and their centers are not collinear, show that the three radical axes of the circles, taken two at a time, meet in a single point. (We will see later (Exercise 39.20) that this result also holds in the Poincare model of non-Euclidean geometry. So we can ask, is it true in any Hilbert plane?) 20.6 In a Hilbert plane with (P), given two circles by their centers and one point each, but without being given