Overview 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.
To measure the angle of contact at any point of a curve, Leibniz then reasons by analogy. In the same way that the tangent measures the direction because it is the straight line that extends from each side of the curve to the side of the convexity - which corresponds to two equal roots -, the largest of the tangent circles inside the concavity of the curve will be the circle that gives the curvature. In this case, the problem corresponds to - Leibniz emphasizes - at least three equal roots.
However, for Leibniz, who based his reasoning mainly on diagrammatic considerations, it is not clear whether the circle measuring the curvature - Leibniz has not yet called it "osculator" - should remain strictly within the concavity of the curve before and after the contact. In almost all points of a curve the osculating circle touches the curve by crossing it - which corresponds to exactly three equal roots - but in the case of certain vertices - that of the parabola or other conics - on which Leibniz precisely focuses his investigation here and in other of these manuscripts - "De Angulis Curvarum" and “De Angulis Linearum plane nova" - the circle remains inside the curve and the problem corresponds to four equal roots. Later, Leibniz will insist on making this particular case the general case, ad that, until Johann Bernoulli persuaded him to the contrary in 1695.
Even if the method for determining the circle of curvature was not yet established, the introduction of this concept lead Leibniz to make sense of the angle of contact between a curve and its tangent: it is the same as the angle of contact between the circle and the tangent, and is measured by the radius of the circle.