Giovanni Poleni, Epistolarum mathematicarum fasciculus, Ioannis Poleni ad virum celeberrimum Hermannum, epistola.

In this section, we present a number of manuscripts that Leibniz devoted to the construction of curves, both algebraic and transcendental. This was an important area of research for Leibniz, as proven by numerous manuscripts in which he sought new methods to trace curves by a single continuous motion. Our research aims to reconstruct, in particular, Leibniz’s work on curves described by tractional motion, namely the motion of a string or a chain dragging a heavy body. This motion played a significant role in the geometrical foundations of calculus.

Overviews

Constructor ([LH 35, III, A 20, Bl. 1-5])

Overview of Constructor (1674)

The construction of curves represented a major interest in Leibniz's mathematical research, as attested by several documents of his Nachlass devoted to the study of curves and the invention of new machines for this purpose. Leibniz devoted several pages to mechanisms for the tracing of transcendental curves, namely curves that could not be described by algebraic equations, which constituted at the time the least known domain of geometry, but he also studied machines for the construction of algebraic curves.

Algebraic curves were a central topic of Cartesian geometry. In fact, in the Géométrie (1637) Descartes provided a canonical method for their constructions by means of apparata constituted by pivoting ruler and moving curves. The construction of curves had not only a foundational role, insofar as it allowed to ground higher geometry on the same foundational basis as Euclidean geometry. In fact, the construction of curves was also studied for theoretical reasons related to the solution of geometric problems. Descartes showed that algebra could be used in the analysis of those geometrical problems that... view

De Curva Perraltii ([LH 35, XIII, 2a, Bl. 122])

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.... view
De Tractrice ([LH 35, XIII, 2a, Bl. 98])

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... view

Nova ratio construendi lineas differentialiter datas, per motum ([LH 35, VI, 1, Bl. 11])

Overview of Nova ratio construendi lineas differentialiter datas, per motum (1676 – 1716)

This manuscript and De Tractrice 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... view