Overview 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.
According to Leibniz, the debate between Pelletier and Clavius did not lead to a consensus because of the lack of a precise notion of what a quantity was. Leibniz distinguishes two approaches in the understanding of quantity. In the first, the condition for a thing to be considered as a quantity is that it can be said lesser or greater because it is contained in or contains the other. This is the sense in which the angle of contact is said smaller than any rectilinear angle. But Leibniz pointed out that this approach is lax because it would be tantamount to saying that the point, since it is contained in the line, is a quantity. In the second approach, stricter [“arctiore”], he considered that for things to be quantities, they must have parts that are homogeneous to the whole. Consequently, to be in (be contained in) a thing is not sufficient to be a part of it. This second approach would not be particularly original if Leibniz did not propose a new definition of homogeneity. He considered that homogeneous things are those which are similar or which can become so by a transformation preserving the magnitude. Leibniz recalled the Euclidean anthyphairesis: to compare two magnitudes one removes the smaller from the greater as many times as possible, then the rest from the smaller, and the second rest from the first, and so on ... the result is either an exact common measure, or a magnitude that can be rendered as small as one wishes so that the error becomes less than any given. In Euclid’s Elements, the existence of a ratio between two magnitudes implies that these magnitudes are yet homogeneous (def. 3 of Book V). Thus, as many of his contemporaries, Leibniz interpreted Euclid's algorithm for determining a common measure as Euclid's definition of homogeneity between magnitudes, even if Euclid never defined the term "homogeneity". Moreover, Leibniz pointed out that his way of defining homogeneity is “simpler” and more general.
Leibniz tested these definitions on rectilinear and circular angles. He introduced a way of measuring circular angles by assuming that when tangent circles have their diameters in geometrical progression, the successive circular angles are equal. This measure enabled him to formulate a “Theorema mirabile” which states a comparison between circular angles, angles of contact and rectilinear angles: a circular angle is to an angle of contact as the logarithm of a finite number is to the logarithm of the infinite number, and an angle of contact is to a rectilinear angle as the logarithm of the infinite number is to the infinite number. Leibniz reproduced this development in De Angulus Linearum plane nova. This theorem seems to contradict the assumption asserted by Leibniz in other texts that circular and rectilinear angles belong to the same degree of infinity, and therefore it conduces to question the validity of the measure of circular angles proposed above.