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.
De Curva Perraltii
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].
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:
[latex]dx=\frac{dy}{y}\sqrt{a^2-y^2}[/latex]
that he tries to solve by reduction to the quadrature of an algebraic curve of equation:
[latex]z=a\frac{\sqrt{a^2-y^2}}{y}[/latex]
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.
De Tractrice
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.
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.
Nova ratio construendi lineas differentialiter datas, per motum
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 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.
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.
Constructor
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 required to find a certain locus, namely a curve obeying given conditions, or construct an unknown segment, as in the classical problems of doubling the cube or trisecting an angle. A polynomial equation in one or two unknown was the standard outcome of the Cartesian analysis of a problem, but it was not acceptable as such as a solution: the unknown or unknowns had to be constructed geometrically by the intersection of curves in the plane. To grant the existence of these intersections, one has to describe them through a continuous tracing. This tracing was often interpreted as a purely mental operation just like, in Euclid's Elements, the construction of circles and straight lines through a continuous motion does not imply the use of a physical compass or a physical ruler.
In these documents, which together with the Ms. CC827 form Leibniz's long study dedicated to an analogical computation device, Leibniz generalized some of Descartes' discoveries in the domain of the construction of equations by presenting a "geometrico-mechanical" machine, called constructor, to solve finite algebraic equations, such as the following:
[latex]ax+bx^2+cx^3=d[/latex]
At the basis of Leibniz's idea there is the Cartesian correspondence between equations and proportions. Thus, this equation can be transformed into a chain of proportions and, by assigning to each coefficient a certain length, the machine can be suitably moved to represent the proportion graphically, through an appropriate configuration of segments and similar triangles. In this way, provided a segment with length 1 is fixed, the unknown x can be determined as a segment within the resulting configuration. We note that, unlike Descartes' linkages, Leibniz's machine was not meant to trace curves. On the basis of the indications contained in the manuscript, Leibniz possibly intended to have this machine constructed.
Moreover, as Leibniz specifies in the end, this machine generalizes Descartes' theory of equations because it also allows to solve numerical equations by assigning given numerical values to the lengths.