## Redefining geometrical exactness in pre-axiomatic contexts: on the distinction between geometrical and mechanical curves

Abstract:

In this paper, I would like to tackle the following broad question: “by which means, available within an historically given pre-axiomatic mathematical practice, mathematicians could succeed in establishing the subject matter of their theories?” by exploring a well-defined problem in the history of mathematics, namely the constitution of the fundamental boundary between geometrical and mechanical curves brought to the fore in Descartes' Géométrie (1637).
Descartes' program intended to frame an accomplished solution to this meta-theoretical question by illustrating the necessary and sufficient properties that ought to be ascribed to an object in order to recognize it as (or as a part of) an acceptable geometrical argument, but the content of his proposals remains problematic.
According to the exactness standards in force within La Géométrie, acceptable curves must be traced by articulated devices, which we may call “geometrical linkages”, basically working as systems of joints allowing one degree of freedom movements between the two links that they connect.
On the other hand, the simple fact that a curve has not been described via a cartesian linkage does not suffice to exclude that a device cannot be conceived and constructed for this purpose (Mancosu, 2007). In order to give a full proof of the non-geometrical, or “mechanical” nature of a curve one must show that the curve is not constructible by one of the canonical devices: an achievement which is far from easy to obtain in the cartesian setting.
In the light of this asymmetry, the only mathematical resources available to Descartes in order to argue against the acceptance of special curves in geometry seem to be reducible to local criteria.
In order to circumvent this dilemma, I suggest that an important (although non exclusive) motivation behind distinction between geometrical and mechanical curves can be retrieved in an aspect of the ancient mathematical practice Descartes was acquainted with.
A main point of discrimination between geometrical and mechanical curves concerns the motions involved in their constructions. More precisely, while both geometrical and mechanical curves are generated by motions in the plane, the fundamental properties, or symptomata of the first class of curves can be expressed independently from their generating motions, via an algebraic equations. On the contrary, the symptomata of the latter class of curves cannot be coded into algebraic equations.
My point is that this distinction can be envisaged as a reformulation, in the light of fundamental technical and conceptual innovations, of a dichotomy, already apparent in Pappus' Mathematical Collection, between two ways of studying the symptomata of curves: on one hand, by determining a quantifiable relation, like an equality or a proportion, that characterizes curves without appeal to the mode of their generation (it is the case of the circle and the conic sections); on the other, by reading the symptomata off directly from the genesis of the curve (as in the case of “linear” curves: the spiral and the quadratrix). Hence, Descartes, who was certainly acquainted with Pappus' text, might have perceived his overall distinction between geometrical and mechanical curves as a plausible and robust one, especially because it was grounded in the tradition of classical mathematics.