CONGRUENCE AXIOMS FCR ABSOLUTE GEOMETRY J. F. Rigby (communicated by H. G. Forder, 1 March, 1974) Introducti on The modem axiomatic foundations of Euclidean geometry were laid by Pasch (1882), Peano (18S9), Hilbert (1899), Veblen (1904), Moore (1908), etc, There are four groups of axioms: axioms of order, axioms of congruence, the axiom of continuity, and the Euclidean parallel axiom [5], (Hilbert’s extra axioms of incidence can be deduced from properly formulated axioms of order.) If we omit the parallel axiom, the remaining axioms give either Euclidean or hyperbolic geometry. Many important theorems can be proved using only the axioms of order and congruence, and the name absolute geometry is given to geometry in which we assume only these axioms. Over the years, some of the original modem axioms of congruence have been shown to be unnecessary, and other axioms have been weakened (see §3). Between 1967 and 1969, I wrote two papers in which the process of weakening was continued [12, 13] and two others giving counterexamples necessary for the discussion of the independence of the axioms [14, 15]. More recently I have realized that the slightly unsatisfactory concept of isometric points [12,§1] can be replaced by a more natural concept of related points, and as a result I have succeeded in replacing axiom C3* [12] by a special case of the axiom; I have discarded part of axiom C5a [12] which was purely negative in the sense that it applied only to a geometrical configuration which was later shown not to exist; also I have found an elementary way of proving thatevery segment has a mid-point, using only the weak axioms (§8); this new proof replaces my proof in 12[ ,§6], which used rather more sophisticated concepts and theorems from Bachmann's account of metric planes [l]. (I was previously unable to find a simple proof of this theorem even assuming Math. Chronicle 4(1975), 13-44 axiom Cl instead of Cl* [12].) I have incorporated these new developments in the present paper, which consists of a self-contained account of the proofs of the basic theorems of absolute geometry starting from the weak axioms of congruence Cl1, C2', C3', C4', C4a , C5‘, C5", which are stated below. We shall also consider the construction of congruent segments, perpendiculars and mid-points, using weak methods of construction. Some of the proofs from [12, 13] are repeated here for the sake of completeness with occasional modifications and improvements, and a few other parts of the text of [12, 13] are used here verbatim; other proofs, and the order of presentation of certain theorems, have had to be changed. We shall assume without comment various results of ordered geometry. For the axioms and theorems of ordered geometry see [5]. The symbol 1 [.ABC]' means ' 3 and C lie in the geometrical order ABC ' . 1. Initial axioms of congruence An absolute space is an ordered space [5, Chapter 2] together with a relation of congruence satisfying certain axioms of congruence. Congruence is a relation between ordered pairs of distinct points. (The term 'ordered pair' is used in its set-theoretical sense, and has no connection with geometrical order.) Intuitively, two segments are congruent if one can be picked up and placed exactly on the other, and the axioms of congruence are based on this idea; it is more con­ venient to use ordered pairs rather than segments in the axioms, but we must then bear in mind that 04>B) is not the same as (BjA). Euclid's use of 'equal' where we now use 'congruent' confuses the significance of his postulates and common notions', in m o d e m terminology two ordered pairs 04,3) and (C>D) are equal if and only if A coincides with C and B with D, so two ordered pairs can be congruent without being equal. If 04,3) is congruent to (CSD) we shall write AB CD. As we introduce the axioms to be used in the present paper, we shall compare them with my previous axioms [12], Forder's axioms [5] and Euclid’s postulates and common notions [7]. 14 Euclid's Postulate 3 states that 'a circle can be drawn with any centre and distance1 [7, pp. 154, 199], In other words, we can draw a circle with centre A and any radius AB . He does not assume that we can draw a circle with centre and radius congruent to a given segment OC ; this is a consequence of his Proposition 2: To place at a given point (as an extremity) a straight line equal to a given straight line [7, p. 244]. To prove Proposition 2 he assumes that certain circles intersect. Modern geometers have preferred not to mention circles or their properties in the axioms, so they have taken Proposition 2 as an axiom. Thus we have Forder’s axiom Cl: If A , B are distinct points and 0 is any point, then on any ray from 0 (i.e. any ray with origin ) 0 there exists just one point C such that AB -*■ OC . We shall assume a weaker axiom Cl 1 [see 12, Cl*], similar to Postulate 3. Cl'. If A ,B are distinct points, then on any ray from A there exists just one point C such that AB -> AC , From an intuitive point of view we can regard postulate 3 as providing a means of physically constructing the circle with centre A and radius AB , The physical implement with which we do this has been described as ’collapsing compasses': we cannot place the point of these compasses on 0 and the pencil on C , then transfer the point to A , as they will then collapse. In Cl' we use these collapsing compasses; but not to draw a circle, only to mark off C on the given ray such that A3 -> AC . The subsequent axioms, defini­ tions and proofs in this section and the next are based on an intuitive investigation of what we can construct by using the collapsing compasses in this restricted way and also using a straight edge to join any two given points. We shall discuss later what constructibility means in abstract geometry C§4). Our axiom C2' is the same as Forder's C2 and corresponds to Euclid’s Common notion 1: Things which are equal to the same thing are also equal to one another [7, p. 155]. 15 C 2 '. If AB + CD and CD + EF then AB + EF Forder's axiom C3 is: If [ABC] and [A'B'C'] } and if AB -*■ A'B' , BC -»•B'C' , then AC -*■ A'C' . This corresponds to Euclid's Common notion 2: If equals be added to equals, the wholes are equal [7, p. 155]. In [12] I used C3*, which, with slight variation, stated: If B ., C lie on a ray from A , if B ' , C' lie on a ray from A' , and if [ABC] , AB •+ A'B' , AC A'C' , then [A'B'C'] and BC -* B'C' . This corresponds to Euclid's Common notion 3: If equals be subtracted from equals, the remainders are equal. The extra conclusion 1[A'B'C'] ' does not appear to be explicitly stated by Euclid, but if it had been it might have appeared in Heath's translation as: That which is equal to the greater is greater than that which is equal to the less. Here we shall use only a special case of C3*, in which A = A' . C3* (Figure 1A). If [ABC] , if B' , C' lie on a ray from A t and if AB + AB' , AC + AC' , then [AB'C'] and BC -* B'C' . These three axioms will suffice for the present. The usual proof of our first result below requires Forder's axiom C4: AB -> BA . Our C 4 1 (§3) would be of no use here. 1.1 [12, 1.2]. If A * B , then AB -* AB . Proof. On ray AB , there exist B' and B~ such that AB -> AB' and AB' -*■ AB" (Cl'). Hence AB AB~ (C2‘). Since AB + AB' also, we have B' = B~ (Cl1). Suppose B f B' and [ABB'] . Since AB + AB' and AB' + AB' , we have [AB'B'] (C31), a contradiction. Similarly if B t B' and [AB'B] . Hence B = B' , so AB -*■ AB . 1.2. If B , B' lie on the same ray from A and if AB + AB' then B = B' . (Use 1.1 and the uniqueness in Cl'.) 1.3. If AB + AC then AC AB . ifi. Proof. There exists B' € ray AB such that AC -»■AB' (Cl1); then AB -> AB' (C21). Hence B = B' (1.2), so AC + AB . 2. Related points We now define related -points inductively. The term 'related points’ was suggested to me by Professor Forder as a preferable alternative to my 'isometric points' in 12[ ]; I have used it here in a similar but somewhat different sense. Definition. If there exists a point P such that PA PB , then A is 1-related to B ; if A is n-related to C and C is 1-related to B , then A is (n+1)-related to B ; A is 0-related to A ; if A is n-related to B for some integer n > 0 , then A is related to B and we write A ~ B . Note that it is possible for A to be both m-related and n-related to B for m f n \ for instance A is 2-related to A . 2.1. (i) If A is m-related to C and C is n-related to B , then A is (m+n)-related to C .