Descartes, Pascal, and the Epistemology of Mathematics: The Case of the Cycloid Douglas M. Jesseph North Carolina State University This paper deals with the very different attitudes that Descartes and Pascal had to the cycloid—the curve traced by the motion of a point on the periphery of a circle as the circle rolls across a right line. Descartes insisted that such a curve was merely mechanical and not truly geometric, and so was of no real mathematical interest. He nevertheless responded to enquiries from Mersenne, who posed the problems of determining its area and constructing its tangent. Pascal, in contrast, saw the cycloid as a paradigm of geometric intelligibil- ity, and he made it the focus of a series of challenge problems he posed to the mathematical world in 1658. After dealing with some of the history of the cycloid (including the work of Galileo, Mersenne, and Torricelli), I trace this difference in attitude to an underlying difference in the mathematical epistemologies of Descartes and Pascal. For Descartes, the truly geometric is that which can be expressed in terms of ªnite ratios between right lines, which in turn are expressible as closed polynomial equations. As Descartes pointed out, this means that ratios between straight and curved lines are not geometrically admissible, and curves (such as the cycloid) that require them must be banished from geometry. Pascal, in contrast, thought that the scope of geometry included curves such as the cycloid, which are to be studied by em- ploying inªnitesimal methods and ratios between curved and straight lines. Introduction Mathematics has long held a prominent place in philosophical theorizing, either as a source of philosophical problems or as a paradigm of human in- tellectual achievement. Mathematical knowledge is generally regarded as more certain and secure than other claims to knowledge, and it is no won- der that mathematical examples have been central to epistemological dis- cussions at least since Pythagoras. Yet mathematics also raises some im- Perspectives on Science 2007, vol. 15, no. 4 ©2007 by The Massachusetts Institute of Technology 410 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.4.410 by guest on 26 September 2021 Perspectives on Science 411 portant puzzles. At a minimum, the ontological status of mathematical objects is hardly obvious: triangles and prime numbers are not the sort of thing one expects to stumble into on the way home from work, and it can be difªcult to articulate the sense in which these things exist and to ac- count for their relation to the more mundane things. Beyond that, the epistemological status of mathematics is far from straightforward. The certainty characteristic of mathematical knowledge stems from the fact that such knowledge is grounded in demonstrative proofs, yet it is dif- ªcult to specify just which methods of proof are truly rigorous, and the historical development of mathematics is ªlled with instances where the status of various proof techniques was unclear and contested. My purpose here is to consider an episode in the history and philosophy of mathemat- ics that can highlight some of these epistemological and conceptual issues. The case I have in mind is that of an interesting curve known as the cycloid, which was the focus of intense mathematical research from the early seventeenth century until well into the eighteenth century. The curve proved to have some astonishing properties, but it also provoked a remarkable number of disputes. The eighteenth-century French historian of mathematics Jean-Étienne Montucla compared it to “la pomme de discorde,” ([1799–1802] 1960, 2: 52) and in a curious mixing of his clas- sical metaphors then termed it “cette Hélène des géomètres” ([1799– 1802] 1960, 2: 55). The focal point of these disputes was the question of priority for the discovery of its many remarkable properties.1 Although I am not principally interested in the many squabbles that arose in the course of investigations into the cycloid, it should become clear that part of what made the curve so problematic was the fact that it raised issues about what kinds of curves count as genuinely geometrical and what methods of proof are truly rigorous. My primary concern is with the differing reactions of Descartes and Pascal to the study of the cycloid, but the case inevitably demands a treat- ment of the contributions of others. This investigation is divided into three sections. The ªrst gives a brief introduction to the cycloid and its place in seventeenth-century mathematics, emphasizing the mathematical and conceptual difªculties posed by the curve. The second considers Des- cartes’ oddly dismissive attitude toward a curve that others had deemed to be of tremendous mathematical signiªcance. The ªnal section gives a brief account of Pascal’s approach to the cycloid, and particularly his rather un- 1. Among the most intriguing of its properties are the fact that the cycloid is the brachistochrone (the curve of fastest descent under gravity) and the tautochrone (the curve yielding equal time of descent under gravity from any point along the curve). These two features were ªrst demonstrated by Jean Bernoulli in the eighteenth century. Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.4.410 by guest on 26 September 2021 412 Descartes, Pascal, and the Cycloid Figure 1 A simple cycloid generated by the motion of the circle AEC across the line AB characteristic behavior in using the cycloid as the subject of a series of challenge problems he posed to the mathematical world in 1658. In the end, I hope to explain how the very different epistemologies of Descartes and Pascal could lead the one to see the cycloid as a triºe of no intrinsic mathematical signiªcance, while the other could take it as a paradigmatic mathematical object, the study of which would introduce powerful new methods.2 1. The Cycloid and its History The cycloid is the curve traced by the point on the periphery of a circle as the circle rolls without slipping across a right line. In Figure 1, the circle with diameter AC moves to the right with a rotational velocity equal to its rectilinear velocity, rotating as it moves until point A arrives at B, making the line AB equal to the circumference of the circle. The cycloid is the curve ADB traced by the composition of rectilinear motion parallel to the line AB and rotational motion in the circumference AEC.3 The curve is simple enough to deªne and is the sort of thing that seems to arise quite naturally from the experience of a wheel rolling across a ºat surface. Nev- ertheless, the curve itself was not studied by Greek geometers and seems ªrst to have attracted the attention of mathematicians in the sixteenth century. As Pascal noted in the opening of his 1658 Histoire de la Roulette, 2. Mahoney (1990) considers similar issues regarding the cycloid in the late seven- teenth century. 3. Using parametric equations in the Cartesian plane, the curve is deªned as x ϭ rt Ϫ r sin(t) and y ϭ r Ϫ r cos(t), where r is the radius of the circle and t the angle of rotation. More complex cycloidal curves can be deªned by placing the generating point on the ra- dius inside the rolling circle (the curtate cycloid) or outside the circle (the prolate cycloid). I will only be concerned with the simple case, however. Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.4.410 by guest on 26 September 2021 Perspectives on Science 413 the curve “is described so often before the eyes of everyone that there is reason to be astonished that it had never been considered by the ancients, among whom we ªnd nothing about it” (Pascal 1963, 117). The reasons for this long neglect of something that appears quite evi- dent and natural are no doubt complex, but an essential part of the story is the restrictive framework of classical Greek geometry. Classical geometers distinguished among classes of problems on the basis of the constructions required to solve them. The simplest problems require only the compass and rule constructions of Euclidean geometry. These “planar” problems are paradigms of mathematical intelligibility because they demand noth- ing for their solution beyond straight lines and circles—the two simplest geometric objects, and ones that can be constructed in a two dimensional plane. In contrast, “solid” problems require more complex constructions. The conic sections, to take the standard example, are “solid” curves de- ªned in terms of the intersection of a plane with a cone. Problems that can be solved only by recourse to the construction of such curves are in some important sense more recondite and less accessible to the untutored intel- lect. Still more recherché problems are the so-called “linear” problems that require curves of a more complex nature.4 A classiªcation of curves arises naturally from this classiªcation of problems—the simplest curves being “plane loci” in contrast to solid and linear loci. As the fourth century (C.E.) geometer Pappus of Alexandria put it: [t]he ancients maintained that there are three types of problems, one of which is called plane, another solid, and the third linear. Therefore those problems which can be solved by right lines and the circumferences of circles are justly called plane, since the lines by which they are solved have their origin in the plane. But those problems whose resolution is found by one or more sections of the cone are called solid problems, for it is necessary in the construction to use the surfaces of solids, namely cones.