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).
Overviews
Overview of De Angulis Curvarum
"De Angulis Curvarum", as well as "De Angulo contactus et curvedine et de natura quantitatis", was most probably written after 1682 – since the conceptual elaborations on curvature are more advanced than in “Multa et mira de Angulo contactus”, but a little before June 1683 – since we find in "De Angulis Linearum plane nova" a reprise of the issues dealt with in the three manuscripts.
viewOverview of De Angulis Linearum plane nova
viewOverview of De Angulo contactus et curvedine et de natura quantitatis
Among the texts preceding the “Meditatio nova de natura anguli contactus et osculi” (Leibniz, 1686), the “De angulo contactus et curvedine et de natura quantitatis” holds a special place. Indeed, it is the only text in which Leibniz attempted to legitimise circular angles as magnitudes (on the grounds of his own reformulations of the basic Euclidean concepts, also expressed in other writings from this period). This is significant because in other texts, in particular in the Meditatio, he states that the angle between the two osculating circles of two tangent curves could be used to estimate the angle between these curves. This is made possible thanks to the introduction of the angle of osculation, which Leibniz defined as the angle between the curve and its osculating circle. The angle of osculation is presented as the smallest angle between the curve and one of its tangent circles. It is infinitely small in comparison to the angle of contact.
viewOverview of Multa et mira de Angulo contactus
Leibniz wrote this manuscript on December 3/13, 1681. As far as we know, this is the first manuscript in which he explains how he found "a clear measure of the angle of contact".
According to Leibniz, the angle of contact between a curve and its tangent is measured by the curvature, which is itself measured by a circle. But why? Leibniz claims that the circle is the only plane figure that has the same angle of contact with either of its tangents, and this on both sides of the tangent - which is not the case, he stressed, for the other curves. Moreover, because all circles are similar and their curvature is produced similarly, it follows that - sunt effectus in ratione causarum - their curvatures are like their radii.
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