Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

240 michael hallettbetween two circles. The power of a point O with respect to a circle C in thesame plane can be defined thus: Consider any straight line OPQ through Owhich cuts C in the two (not necessarily distinct) points P and Q. The powerof O with respect to C is now the product OP · OQ. This is a constant for Oand C, since this ratio is the same wherever P (respectively Q) happens to lieon the circle. (See Euclid’s Elements, Book III, Proposition 35.) If O is insidethe circle, the power is negative; if it is outside, and P and Q coincide, thenOPQ is a tangent, and the power is OP 2 . Given two circles, we can considerthe points which have the same power with respect to both circles; these lie ona straight line perpendicular to the line joining the centres of the circles, a linewhich is called the radical axis or the line of equal power (in German, a ‘Chordal’)between the two circles. (See Coolidge, 1916, Theorem 167.) What was oftencalled the Hauptsatz of the theory of these lines is: Given three circles in aplane, intersecting or not, the three lines of equal power that the three pairs ofcircles generate must intersect at a single point. (Ibid., Theorem 168.) Clearly,the TCT is just the case where each pair of circles intersect each other. There isanother interesting special case. Consider a triangle, and consider three circlesin the plane of the triangle centred respectively on the three vertices. As theradii of the three circles tend to zero, the three lines of equal power will tendto the perpendicular bisectors of the three bases. It follows that these threebisectors have a common point of intersection.Hilbert gives a direct proof of the TCT in his 1898/1899 notes. The proof,which is clearer in the Ausarbeitung, pp. 61–64 (Hallett and Majer, 2004,pp. 335–337), assumes that the three circles in question, all in the plane α, arethe equators of three spheres which have just two points P and Q of mutualintersection; it is then shown fairly easily (loc. cit., p.61) that the three chordsof the circles, which lie, like those circles, in α, must all intersect the line PQ.But since neither P nor Q lies in α, PQ has only one point in common with α,and this point is therefore common to all three chords. Thus the simple proofdepends on the assumption that three mutually intersecting spheres intersectin exactly two points, in other words, that these points exist. Hilbert’s focus isthen on this assumption, no longer on the TCT itself.Hilbert’s next step is to connect this with the theorem that a trianglecan be constructed from any three line segments which are such that anytwo of the segments taken together are greater than the third. Call this theTriangle Inequality Property. In Euclid, this is proved in I, 22 (see Heath,1925, Volume 1, pp. 292–293); call this the Triangle Inequality Theorem. Thequestion Hilbert asks is: On what assumption is the proof of this propositionbased? In his lecture notes (p. 64), Hilbert states this triangle property in the