Angles of contact

 

In 1686, Leibniz published “Meditatio nova de natura anguli contactus et osculi” (Leibniz, 1686). This article is the fruit of investigations conducted in the previous years in order to resolve new geometrical and optics problems, concerning the characterization of contacts between two curves. To do so, Leibniz revisited the figure of the angle of contact – that is to say the angle between a circle, or more generally a curve, and one of its tangents – figure which had provoked many discussions during the Renaissance specially between Christoph Clavius and Jacques Pelletier. In order to characterize different degrees of contact between two curves, Leibniz generalized the notion of angle of contact in an original way by introducing the idea of angles of osculation — namely, angles which follow each other by degrees of infinity, and which are incomparable from one degree to another.

This section presents several texts – dating from 1681 to 1686 – containing research that was either successful or not, but which led to the results published in the article Meditatio. Leibniz’s investigations about angles of contact emerged in a context in which he also questioned the foundations of geometry and thought on new grounds basic geometrical notions – such as homogeneity, similarity and measure – (De Risi, 2007), (Leibniz 2018). One of the main interests of these texts is that, perhaps as a way of legitimising his geometrical reformulations, Leibniz tested his new notions of angle of contact and angles and osculation by means of the former to see if they can be considered as magnitudes, and whether they can be measured directly – as in the case of magnitudes consisting partes extra partes which are measured by a common measure – or indirectly – as in the case of rectilinear angles, which are measured as the ratio of the arc to the lengh of the circumference –. His attempts are particularly noteworthy in “De Angulus Linearum plane nova” (See here) and “De Angulus Contactus et Curvedine et de natura quantitatis” (See here). Leibniz introduced the osculating circle and tried to characterise it by determining the formula for its radius. Most of these texts show that Leibniz knew the identity between the locus of the centres of the osculating circles of a curve and its evolute, but he did not succeed in finding a general formula for the radius of curvature that would provide a measure of the angle between two tangent curves. Jacques Bernoulli's manuscripts show that he knew the formulas at least as early as 1691, and he published them in 1694 (Bernoulli Jacob, 1694), (Radelet-de-Grave, 2004).