The Construction of the Straight Line joining Two Given Points
William R. Burnside · Proceedings of the London Mathematical Society · 1897
Euclid's postulate: " Let it be granted that a straight line may be drawn from any one point to any other point " implies the use of a ruler or straight-edge of any required finite length.The postulate is clearly not intended to apply to the case in which the distance between the two points is infinite, i.e., in which no finite number of repetitions of a finite operation will lead from one point to the other.In fact, Prop, xxxi, Book I, gives a compass and ruler construction for the line when one of the points can be reached while the other cannot.The other exceptional case, when neither point can be reached, i.e., when the two given points are the pointB at infinity on two non-parallel lines, is not dealt with by Euclid.It is, however, not difficult to show by elementary geometrical considerations that in this case no point of the joining line can be reached by a finite number of finite operations.In elliptic space any one point can be reached from any other by a finite number of finite operations.The line joining two given points can therefore be always constructed with the ruler alone.In hyperbolic space as in Euclidean, there are, so long as we deal with elementary geometry, three cases of the construction to consider.These are (i) that in which the two points to be joined are points which can bo reached from any assigned point by a finite number of finite operations, or say " finite points " ; (ii) that in which one of the points is a finite point, and the other a point at infinity on a given straight line ; (iii) that in which both the points are points at infinity.If, however, we deal with projective geometry, we must assume that every two straight lines in a plane determine a point.When• Math.Ann., Vol.vn., 1873, "Ueberdie algebraischen Funotionen nnd ihre Anwendung in der Geometrie."t This is, of course, only true if we assume no knowledge of the application of the equations, and hence no connexion at all among the unknown integers.