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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:

$ax+bx^2+cx^3=d$

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.

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