Poleni’s geometrical instruments to trace transcendental curves, by Pietro Milici, University of Insubria (Italy)

Figure 1. Poleni’s machines for the tractrix (left) and the logarithmic curve (right) in the Letter to Hermann.

During the 17th century, mathematicians radically modified their idea of a curve, transitioning from the trace of a geometrical mechanism to the solution of an equation. With that in mind, Descartes and Leibniz introduced theoretical machines (i.e., sketches to be considered mentally but not practically realized) to legitimate the geometrical status of algebraic and transcendental curves obtained as solutions of analytical equations. In the first half of the 18th century, a few scholars converted some theoretical machines for transcendental curves into material instruments. We focus on the geometric instruments designed by Giovanni Poleni (1683-1761), a polymath and Professor at the University of Padua in Italy. These instruments had been described with an engineer’s precision both in textual and graphical representations (as visible in Figure 1) in a letter to Jacob Hermann who was Poleni’s predecessor in the chair of Mathematics. The letter was included in Poleni’s Epistolarum Mathematicarum Fasciculus of 1729.

In 1739, Poleni inaugurated a unanimously praised laboratory of experimental physics (“Teatro di Filosofia Sperimentale“) that grew also with his successors. Such a Paduan museum still exists and is dedicated to its founder (today it’s named “Giovanni Poleni museum”). Between the machines present in the collection, Poleni listed the ones for the tractrix and the logarithmic curve: also the modern catalogue (Gian Antonio Salandin & Maria Pancino, Il “teatro” di filosofia sperimentale di Giovanni Poleni, LINT, 1987) confirms that there is a sample of a geometrical machine for our transcendental curves. Thanks to an SIS grant, I got the opportunity to observe, manipulate and film such a machine (see Figure 2).

Figure 2. The machine of the Giovanni Poleni Museum classified as the one for the tractrix and the logarithmic curve.

Indeed, differences between the artefact in the collection and the designs of the Letter are visible at the first glance. Furthermore, reading the description in Poleni’s index, the machine also differs in materials. In any case, also because of missing pieces, we have no idea of the purposes of such an artefact.

To our knowledge, no copy of Poleni’s geometrical machines is available: therefore, together with Frédérique Plantevin (University of Brest, France), we decided to reconstruct the machines meticulously described in the Letter to Hermann. After a 3D digital modelling of the machine, we contacted the artisan Uri Tuchman to realize a working model in the materials described by Poleni. We are still working to complete an informative video and a paper on these topics.