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Overview of De Tractrice (1690 – 1716)

This manuscript and Nova ratio construendi lineas differentialiter datas, per motum contain Leibniz’s exploration in the geometry of curves and their constructions. The dating of these manuscripts is uncertain, although both were written during the Hannover period, i.e. after 1676. The Nova ratio was certainly written after 1676 while the other, De Tractrice can be dated after 1690.

The construction of curves was a central topic in Leibniz’s mathematics since the beginning of his studies in Paris, where he was influenced by two major works: on one hand, Descartes’ Géométrie and, on the other, Huygens’ Horologium Oscillatorium. In his Géométrie, Descartes managed to provide a canonical way for constructing algebraic curves by machines composed by a finite number of movable, interconnected rods (linkages), which imparts a one-degree-of-freedom motion. For him all and only the curves constructed in this way were geometrical, while mechanical, or transcendental curves such as spirals and quadratrices did not qualify for geometry.

However, the latter acquired more and more importance in the second half of the 17th century because they often appeared as solutions to differential equations and as descriptions of physical phenomena. One of the major issues related to the construction of transcendental curves was foundational. Unlike the case of Cartesian geometry, where a hierarchy of constructive means can be defined with enough precision, mechanical or transcendental curves were generated via countless types of motions, both purely mathematical and more concrete, or physical ones. But for Leibniz, unlike Descartes,  these curves should be treated as legitimate geometrical objects, hence the question whether the various kinds of transcendental motions could be ordered in a hierarchy and extend Cartesian ones became of some theoretical importance. However, the manuscripts displayed here suggest that Leibniz’s research was also motivated by the simple desire to experiment with increasingly new types of motions and machines and with their combinations, without necessarily having theoretical concerns in mind.

[Fig. 1]
[Fig. 2]

The Nova ratio construendi Lineas differentialiter datas, per motu contains the description of a new ideal machine [Fig. 1] consisting of a solid moving body and a connected device endowed with a tracing pin. The goal of this machine, as indirectly suggested in the title, is to trace a curve H(H) orthogonal to another, given one. Leibniz’s interest seems to lie here in the possibility of playing around with the components of this machine and making hypothesis on its physical realizability and the kinds of motions, and thus curves, that may result.

In the other manuscript considered here, Leibniz studies the curve resulting from the combination of tractional motion, i.e. the motion resulting from dragging a heavy object over a board or table, and evolutional motion, which occurs when a thread is unwrapped along a curve. In the case studied by Leibniz, the weight D in [Fig. 2] unwraps along the given curve AB, to which is attached by a thread, and at the same time is dragged along the plane.

In both manuscripts the curves are described only qualitatively, which confirms the experimental, exploratory character of Leibniz’s research.

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