The authors point out a relation between the exponential curve and a tractrix, which means that, if one can trace one of these curves, then, by a very simple linkage device, one can also trace the other. The relationship is the following. Let \(B\) be a point on an exponential curve \(y=ae^{bx}\), let \(A\) be the point perpendicularly below it on the \(x\)-axis, and let \(C\) be the point where the tangent at \(B\) cuts the \(x\)-axis. Exponential functions have constant subnormal \(CA\), so this segment can be physically realised by a rigid rod of fixed length, from which the perpendicular \(AB\) extends like a carpenter’s square. As this device is moved along the axis, and \(B\) along the exponential curve, put another ruler along \(BC\) and mark the point \(D\) on it such that \(AD=AC\) (mechanically realised by a rigid rod of this length attached at \(A\) with its other end \(D\) constrained to move only along \(BC\)). The point \(D\) traces a tractrix. The same device could also be used to, conversely, trace the exponential curve if the tractrix is given.
The authors note that curve-tracing devices have played a significant role in the history of mathematics, both for constructive-foundational reasons and as a way of drawing solutions to differential equations without the need to solve them analytically. Exponential and tractional curves have figured prominently in this tradition, and good devices for tracing them were constructed in the 18th century. The authors’ result shows that a machine for tracing one of these curves can be readily repurposed into a machine for tracing the other. The historical devices were not based on such interconstructibility.