1. Introduction

Thanks to the work of D. T. Whiteside, there is no doubt that close study of Descartes's three-book Géométrie (Reference Descartes1637) helped shape Newton's early researches into mathematics. Whether and to what extent Descartes's work influenced Newton's understanding of the relationship between geometry and mechanics is my present concern. Some recent commentators have interpreted Newton's famous dictum that “geometry is founded on mechanical practice” against the backdrop of the “constructivist” philosophy of geometry proposed by Descartes in Reference Descartes1637. However, there is evidence in Newton's lectures on algebra (1673–1683) as well as the unpublished treatise, Geometria (ca. 1692), that strongly suggests Newton rejected Cartesian constructivism and specifically, the interpretation of ancient geometry on which this approach to geometrical intelligibility is grounded.Footnote ¹ By examination of these pre- and post-Principia manuscripts, I hope to make clear the divide separating Newton from Descartes in regard to the association between the intelligibility and the constructibility of geometrical objects. I will also make suggestions for how these texts shed light on the claims presented at the outset of the Principia regarding geometry and mechanics. I begin my discussion with a very brief overview of Descartes's interpretation of ancient geometry and his well-known development of a new interpretation of geometrical curves.

2. Cartesian Constructivism in the Géométrie

In the opening section of book 2 of his Géométrie, Descartes famously introduces his criteria for accepting curves into geometry. He claims to borrow a classification proposed by the ancients, whereby so-called “geometrical” curves used in the solution to ‘plane’ and ‘solid’ problems were accepted into geometry, while the more complex curves invoked for the solution to ‘linear’ problems were not considered treatable by the geometer and thus called “mechanical” (Descartes Reference Descartes1925, 40). After a brief discussion of the ambiguities surrounding the grounds for this ancient division, Descartes shifts attention to his own classification. He accepts the fundamental construction postulates of EuclidFootnote ² and adds, “Now to treat all the curves which I mean to introduce here, only one additional assumption is necessary, namely, two or more lines can be moved, one upon the other, determining by their intersection other curves” (Descartes Reference Descartes1925, 43). With this assumption now proposed and motion assuming a central role in geometrical postulates, Descartes offers the beginnings of his criteria for those curves that can be accepted into geometry. He writes:

The requirement of “continuous and successive motions” does not, however, exhaust the possible methods of construction for “geometrical” curves. In general, it is the clear and distinct idea of the motion necessary to construct a “geometrical” curve that differentiates it from a “mechanical” one. As Descartes explains in his later discussion, “geometrical” curves are “those which admit of precise and exact measurement” (Descartes Reference Descartes1925, 48), whereas mechanical curves “must be conceived of as described by two separate movements whose [ratio] does not admit of exact determination” (Descartes Reference Descartes1925, 44).Footnote ⁴ And although Descartes asserts that “an exactness of reasoning” rather than an exactness of instruments is sought by the geometer, as we see from several cases presented in the Géométrie, the “exact determination” of motions that Descartes requires for the construction of “geometrical” curves is intertwined with construction by means of straightedge and compass, or more precisely, by means of idealized straightedges and compasses—the straight lines and circles invoked in reasoning. As Bos (Reference Bos1981) explains, the instruments employed by Descartes when describing acceptable constructions, whether they be rulers or strings or some other physical mechanism, “served as acceptable generalizations of ruler and compass” (Bos Reference Bos1981, 298). Or alternatively, as described by Friedman (Reference Friedman, Sher and Teiszen2000), the constructions presented by Descartes are instances of construction by means of idealized rulers and compasses along with translation and rotation. That is, we begin with “the continuous or several successive motions” of our idealized instruments and can then translate and rotate the constructions generated at this initial stage. Translations and rotations can themselves be continuously and successively applied to the fundamental straightedge and compass constructions, and proceeding in this fashion, we can iterate all “geometrical” curves (Friedman Reference Friedman, Sher and Teiszen2000, 196).

As is well known, however, Descartes's notion of geometrical intelligibility does not rest on methods of construction alone. The very same curves describable by straight lines and circles, along with translations and rotations, are those curves that have a corresponding closed polynomial that describes the curve, or more specifically, that describes the behavior of any point included on the curve. Combining this “analytic” requirement with the allowable methods of construction, Descartes groups “geometrical” curves into distinct classes. For instance, curves of “the first and simplest class” (viz., the circle, parabola, hyperbola, and ellipse) are described by a first- or second-degree polynomial equation and can be constructed by straight lines and circles (cf. Bos Reference Bos, Goldstein, Gray and Ritter1996, Mahoney Reference Mahoney, Lindberg and Westman1990). Following the methods delimited by Descartes, only the so-called “algebraic” curves, those with a corresponding closed polynomial, are to be included in the sphere of geometry. In contrast, so-called “transcendental” curves are rendered geometrically unintelligible.

As Descartes has it, his new classification improves upon the more restrictive ancient classification, for his new scheme allows for the admission of some ‘linear’ curves, such as the conchoid and cissoid, into geometry. Descartes's characterization of the ancient distinction between “geometrical” and “mechanical” curves may underscore what he takes to be the novelty of his approach, but he leads us astray in claiming that the ancients considered a particular class of curves not treatable in geometry. Ambiguities notwithstanding, the boundary of geometrical intelligibility in Greek geometry does not appear to lie at the divide between “geometrical” and “mechanical” curves, as Descartes suggests.

We must turn, in particular, to Pappus's fourth-century a.d. Mathematical Collections, the treatise consulted by Descartes, Newton, and a host of other seventeenth-century mathematicians for an account of the ancient classification of geometrical problems.Footnote ⁵ It is in book 4 of the Collections that Pappus distinguishes ‘plane,’ ‘solid,’ and ‘linear’ problems (Collections 270.1–272.3 [in Cuomo Reference Cuomo2000, 151]). What is important to note here is that Pappus invokes this classification to prescribe the proper methods for solving a given problem rather than to delimit the scope of geometry. After presenting the classification of problems, Pappus writes, “It seems to the geometers that it is no small mistake [or, as translated by Bos (Reference Bos2001, 49), “a considerable sin”] when [the construction of] a planar problem is discovered by someone by means of conics or of linear [curves], and in general when it is solved by means of a kind not its own” (Collections 270.1–272.3 [in Cuomo Reference Cuomo2000, 151; my emphasis]). Further evidence of an overriding methodological concern is found in Pappus's criticism of a geometer who attempted to solve a ‘solid’ problem by means of ‘planar’ methods (book 3 of the Collections). The geometer in question provides a solution to the problem of finding the two mean proportionals of two given lines, a problem equivalent to that of duplicating a cube, by using constructions in the plane.Footnote ⁶ As a means of illustrating the deficiency of such an approach, Pappus offers four “proper” solutions to the problem, one which is ‘linear.’ In light of the criticism charged at the geometer attempting to provide a ‘planar’ solution to a ‘solid’ problem and given the inclusion of a ‘linear’ solution to a ‘solid’ problem, one might conclude that the “superior” classes of curves could come to bear on “lower” classes, but not vice versa. So for instance, it is admissible to solve ‘plane’ or ‘solid’ problems by use of ‘linear’ curves, but only ‘linear’ curves can be invoked for the solution to a ‘linear’ problem. Even if that is the case, we are still left with a tension between Pappus's solutions and his methodological pronouncement.Footnote ⁷

Perhaps a greater misfortune is that Pappus is also not entirely clear on the ancient distinction between “geometrical” and “mechanical” curves. At several instances, the basis for the distinction appears to rest entirely on the origin of a curve, so that “geometrical” curves are simply those that do not require mechanical instruments for their construction; they are constructed via simple motions in the imagination (cf. Molland Reference Molland1976). Thus, a circle drawn on paper by means of a compass is in fact “mechanical,” just as any line segment produced by a straightedge would be. Whether, on Pappus's account, such instrumentally constructed curves can be situated in the deductive system of geometry is a matter of interpretative debate (cf. Cuomo Reference Cuomo2000, 154; Molland Reference Molland1976, 30). But what does seem clear is that, in the Collections, “linear” and “mechanical” were by no means two designations for the same class of curves as Descartes presents. The construction of ‘linear’ curves may only be describable in reference to mechanical procedures involving rulers and others instruments, but the status of the ‘linear’ curves as “geometrical” was not in this way forfeited. For curves that take their origin in mechanical practice can be reproduced by simple motions in the imagination, rendering these reproductions “geometrical.” Thus, the ancient designation of ‘plane’ and ‘solid’ problems as “geometrical” and ‘linear’ problems as “mechanical” appears to be entirely of Descartes's own devising. Molland (Reference Molland1976) points out that this interpretative move allowed Descartes to establish his distinction between “algebraic” and “transcendental” curves as a distinction between those curves admissible into geometry and those not. But, as we will see, the mixing of geometry and algebra that underwrites this distinction and the interweaving of constructibility and intelligibility that Descartes invokes to redefine the proper domain of geometry is a move that Newton rejected.

3. Newton's Anti-Cartesian Geometry

Newton's early lectures on algebra, composed during the pre-Principia period of 1673–1683, seemingly stand as a testament to Descartes's analytic-algebraic approach to treating geometrical problems. Newton's own mixture of equations and curves is quite evident throughout his discussion, and he explicitly states that when an equation can be obtained from the geometrical description, “all difficulties in problems propounded concerning [the curve are] reduced to that equation” (Mathematical Papers (Reference Newton1967–1981) volume 5, page 183, hereafter Reference NewtonMP, 5:183). However, whatever methods he may have appropriated from Descartes, Newton is also explicit that his application of Cartesian analysis does not commit him to a redefinition of geometry itself. In particular, though the reduction of curves to equations may service the geometer in solving problems, any criteria centered on the structure of these equations has no bearing whatsoever on the nature of geometry. For instance, Newton claims that we should not use the dimensions (or degrees) of equations corresponding to curves as a means for classifying curves as simple or complex, for by doing so we are led to the false conclusion that geometrical constructions by means of parabolas are simpler than those by circlesFootnote ⁸ (Reference NewtonMP, 5:425). As he has it, a curve's construction, rather than its equation, alone determines its simplicity (Reference NewtonMP, 5:425).

In general, Newton asserts that algebraic criteria centered on the structure of equations used to describe curves must be kept separated from geometric criteria centered solely on the construction of curves in the plane. We can say that “algebraic” curves have a corresponding closed polynomial equation, but to use this algebraic criterion to determine the scope of geometry as Descartes does is to confuse the two sciences. Newton writes:

This same sentiment is echoed in the preface to Newton's post-Principia treatise, the Geometria (ca. 1692), where, after presenting the ancient distinction between analysis and synthesis (or resolution and composition), Newton relegates algebra to the realm of analysis. While admitting that the reduction of a problem to an equation may in fact assist the geometer in formulating a solution, the solution itself, for both Newton and the ancients, rests entirely on the synthesis, or composition, of a problem—a synthesis founded entirely on the construction of curves in the plane (Reference NewtonMP, 7:249–255).Footnote ¹⁰

Whatever insight we may gain into Newton's interpretation of the relationship between algebra and geometry from this treatise, the value of the Geometria I believe ultimately lies in its lengthy discussion of the relationship between mechanics and geometry, the same relationship mentioned in the author's preface to the reader included in the Principia (1687). Recall that at the outset of the Principia, Newton states that

Similar and, in some cases, identical statements are offered in book 1 of Newton's Geometria. As in the Principia, he tells us that “the genesis of the subject-matter” and the “fabrication” of geometry's postulates pertain to mechanics and that geometry “postulates a technician who knows how to form straight lines and circles, and teaches him how through their formation appointed magnitudes are to be measured” (Reference NewtonMP, 7:287–291).

In their respective attempts to make sense of these claims, both Peter Dear (Reference Dear1995) and James Garrison (Reference Garrison1987) draw considerable attention to the relationship Newton erects between geometry and mechanics and specifically, Newton's claim in the Geometria that geometry was devised for “workaday use.” On Garrison's reading, Newton supplies a “constructivist approach to mathematical foundations,” whereby geometry depends on the practice of mechanics for its postulates (Garrison Reference Garrison1987, 613). As such, the accuracy of construction is the only feature that separates the two fields. Dear does not accept this accuracy criterion but does follow Garrison in reading Newton's claim that “geometry is founded on mechanical practice” as an indication that proper geometrical objects are constructed (or constructible) by the mechanic. As Dear has it, geometry's postulates are for Newton “assumptions of possibility rooted in experience”—assumptions pertaining specifically to the possible mechanical construction of objects (Dear Reference Dear1995, 226). Admittedly, neither Dear nor Garrison makes explicit reference to Descartes's brand of constructivism in their respective assessments of Newton's philosophy of geometry. Yet following their reading of Newton, mechanical constructibility determines the scope of geometrical intelligibility, inching Newton closer to the Cartesian camp.

On the other hand, Molland's (Reference Molland and Rashed1991) interpretation of the above Principia remarks relies on Newton's assumed acceptance of the Cartesian understanding of geometry. Pointing to the passage in which Newton distinguishes geometrical and mechanical problems and famously claims that geometry is founded on mechanical practice, Molland asserts that this passage “can symbolise the move from the ancient intellectualist imagination of perfect simple movements to the more ‘practical’ orientation of the seventeenth-century vision, and can allow us to speak of the mechanisation of the geometrical picture, as a companion to that of the world picture” (Molland Reference Molland and Rashed1991, 196).

With Newton's emphasis on the exactness of geometry and the geometer's consideration of perfectly constructed straight lines and circles, there appears to be legitimate grounds for interpreting Newton's remarks against the seventeenth-century backdrop of Cartesian constructivism. Yet by closer examination of the remainder of Newton's discussion in the Geometria we find that Descartes's distinction between “geometrical” and “mechanical” curves is in fact wholly rejected. In particular, although the origins of geometry rest on mechanical practice, for Newton, the domain of geometry is not restricted to those curves actually constructible by straightedges and compasses. The geometer does not simply start from those curves constructible by rulers and compasses and then idealize such constructions by substituting instruments with straight lines and circles. To do so would be to distinguish geometry and mechanics solely on the basis of the exactness of instruments employed in these fields, and for Newton, “this common belief is a stupid one” (Reference NewtonMP, 7:289).

In the later discussion of the Geometria, Newton is quite explicit that the scope of geometry is not determined by what the mechanic is actually capable of drawing by means of instruments. For instance, Newton addresses whether, as some of his contemporaries have argued, an additional postulate should be added to geometry so that ‘solid’ and ‘linear’ problems could be constructed by the geometer. Newton does not hesitate to point out that this proposed postulate “of moving any lines whatever according to appointed laws and by their intersections describing new lines is to be rejected for many reasons” (Reference NewtonMP, 7:295).Footnote ¹¹ On Newton's account, this postulate would render Euclid's three construction postulates superfluous, and more importantly, it is not necessary because the geometer can, in fact, treat ‘solid’ and ‘linear’ problems in the plane using straight lines and circles. There is a reduction, as it were, of these more complex problems to the plane. For instance, when dealing with figures not constructible by straightedge and compass, the geometer refers to the construction by “any manual operation which shall seem simplest” (Reference NewtonMP, 7:302–303). As Newton further explains in reference to Archimedes's methods,

The “speculative” power that geometry has over mechanical constructions grounds the wide-ranging scope of those objects treatable by the geometer. For in allowing the geometer to hint at the possible modes of construction or to simply assume a construction as given, neither straightedge and compass nor their idealized counterparts determine the objects that can be accepted into geometry. As Newton declares, “Geometrical constructions…are ones which are achieved by means of straight lines and circles and any figures whatever in the plane” (Reference NewtonMP, 7:303, my emphasis).

In contrast, then, to Descartes's treatment of “geometrical” curves, constructibility, by means of idealized instruments, does not determine which objects may be accepted into geometry. In the absence of well-defined rules for construction, the geometer is free to speculate upon the possible rules for the construction of a complex curve, and whether one can actually generate the curve by means of the rules proposed remains a matter only for the practical mechanic. The geometer may employ instruments and mechanical procedures for constructions, but geometry does not thereby assume the limitations of mechanical methods. Constructibility and intelligibility thus remain divorced in Newton's understanding of geometry, precisely because constructibility is an issue for the mechanic alone.

4. Conclusion

In light of the above characterization of Newton's philosophy of geometry, and in particular, given that the scope of geometry remains insulated from the actual constructions of practical mechanics, we can now return to the prefatory remarks included in the Principia. In the first instance, I want to suggest that we take Newton's claim that “geometry is founded on mechanical practice” as a historical claim. Geometry was first developed to solve the problems of earth-measure plaguing practical mechanics, but with the wide-ranging scope Newton grants geometry, it appears that, as a branch of pure mathematics, geometry severs its ties from the empirical arts.Footnote ¹²

And what now of Newton's notion of rational mechanics, the discipline in which geometry is united with mechanics? This union is not one between geometry and the constructions of practical mechanics, but rather, as Newton states, between geometry and the study of motions in general. As such, no considerations of constructibility enter into rational mechanics, and it thus appears that the same freedom imparted to the geometer to treat an unbounded variety of curves in the plane is directly carried over to the natural philosopher, where for him, these curves represent motions in nature. Had Newton made any restriction on the curves treatable by the geometer on the basis of constructibilility as Descartes had done, he would at the same time have restricted the types of motion, and hence the types of forces, that could be considered in rational mechanics. By keeping intelligibility and constructibility divorced, in both geometry and natural philosophy, we can, I believe, better understand Newton's declaration in the Preface to the Principia that the rational mechanic has the power not only to investigate “the motions that result from any forces whatever,” but also “the forces that are required for any motions whatever” (Newton Reference Newton1999, 382, my emphasis).