See manuscript’s transcription

Overview of De curva Perraltii (1677–1693)

In this manuscript, whose catalogue title is De curva Perraltii (LH 35 XIII 2a Bl. 122), Leibniz discusses the nature of the tractrix, a curve traced by a clock dragged by a chain on a plane [Fig. 1]. Because of the friction of the clock on the plane, the chain in traction is always tangent to the traced curve. As the opening lines of the manuscript report, the curve was first described by the architect Claude Perrault, during a public demonstration which Leibniz also attended [Fig. 2].

[Fig. 1] Leibniz’s drawing of Perrault’s tractrix from the manuscript Zum Problem der Tractorie von Perrault.
[Fig. 2] Leibniz’s earliest representation of Perrault’s tractrix, from 1676 (A VII-6, n. 21, p. 259), perhaps soon after having attended Perrault’s demonstration.

Since large excerpts from the manuscript under examination were used by Leibniz as a draft for his Supplementum geometriae dimensoriae, appeared in Acta Eruditorum in 1693, our manuscript was composed before that date. By contrast, it is harder to find a terminus a quo. Certainly it was composed after 1676, when Leibniz saw Perrault’s demonstration for the first time. Another clue is perhaps given by Leibniz himself who, in a letter to Du Hamel dated 21/07/1684 (see A III-4, n. 63), mentions that he had studied the nature of Perrault’s curve using his newly discovered differential calculus. Was Leibniz referring to the manuscript we are describing?
The manuscript contains in fact a synthetic description of the tractrix and its construction, as well as an analytical study of the curve. In short, Leibniz shows that the curve can be described by the following differential equation:


that he tries to solve by reduction to the quadrature of an algebraic curve of equation:


In the figure below [Fig. 3], this is the curve LP in green. This quadrature cannot be solved algebraically, as it depends on logarithms: the tractrix is therefore a transcendental curve. Not content with Perrault’s tractrix, Leibniz also sketches possible generalizations of tractional motion, such as the generation of curves by dragging a string along a circular or, even more generally, curvilinear figure, or by changing the length of the string according to a given law.
If the manuscript was actually composed before or around 1684, Leibniz’s study of the tractrix would actually precede by almost 10 years the first article published on this curve by Christiaan Huygens, which appeared in 1693 (lettre à Basnage de Beauval, in Huygens, Oeuvres Complètes, vol. 10, p. 407-422). After seeing Huygens’ study, Leibniz promptly wrote to him, claiming his priority over the discovery of the tractrix and the study of its outstanding properties. However, Leibniz’s claims of priority, reiterated in his Supplementum geometriae dimensoriae, were met with some skepticism by Huygens. While it is implausible that Leibniz had studied the tractrix before 1676, as he actually states in Supplementum, the manuscript we present here may suggest that Leibniz’s claims of priority over Huygens were, after all, well-grounded.

[Fig. 3] Construction of Perrault’s tractrix by quadrature (above) and the same figure drawn by Leibniz’s hand (below). As we move point M on the circle, point A traces the quadratrix of the curve LP (in green in the figure above). Leibniz showed, using differential calculus, that the quadratrix curve traced by A (in grey in the figure above) is Perrault’s tractrix.

See manuscript’s transcription