Reflecting on the bases of geometry: Construction with the tractrix

Abstract

Construction is, historically, the first form of geometry, and from early days the subtleties of how constructions are done have been considered important. In this paper, we examine geometric construction using a straightedge and a device for drawing tractrices. The tractrix was the first curve traced by the mechanical solution of an inverse tangent problem, the geometrical issue at the basis of Leibniz's conception of infinitesimal analysis. This nonalgebraic curve cannot be axiomatized simply, as the circle can. We show that some important constructions can be done based on a weak axiomatization that does not fully specify the curve, and that more may be done using its Cartesian representation.