René
Descartes' Curve-Drawing Devices:
Experiments in the
Relations Between Mechanical Motion and Symbolic Language
David Dennis
University of Texas at El Paso
Published in Mathematics Magazine, Vol. 70, No. 3, June 1997,
pp. 163-174. Washington D. C.: Mathematical Association of America.
Introduction
By the beginning of the seventeenth century it had become possible to represent a wide variety of arithmetic concepts and relationships in the newly evolved language of symbolic algebra [19]. Geometry, however, held a preeminent position as an older and far more trusted form of mathematics. Throughout the scientific revolution geometry continued to be thought of as the primary and most reliable form of mathematics, but a continuing series of investigations took place which examined the extent to which algebra and geometry might be compatible. These experiments in compatibility were quite opposite from most of the ancient classics. Euclid, for example, describes in Book 8-10 of the Elements a number of important theorems of numbers theory cloaked awkwardly in a geometrical representation [16]. The experiments of the seventeenth century, conversely, probed the possibilities of representing geometrical concepts and constructions in the language of symbolic algebra. To what extent could it be done? Would contradictions emerge if one moved freely back and forth between geometric and algebraic representations?
Questions of appropriate forms of representation dominated the intellectual activities of seventeenth century Europe not just in science and mathematics but perhaps even more pervasively in religious, political, legal, and philosophical discussions [13, 24, 25]. Seen in the context of this social history it is not surprising that mathematicians like René Descartes and G. W. von Leibniz would have seen their new symbolic mathematical representations in the context of their extensive philosophical works. Descartes' Geometry was originally published as an appendix to his larger philosophical work, the Discourse on Method. Conversely, political thinkers like Thomas Hobbes commented extensively and on the latest developments in physics and mathematics [25, 4]. Questions of the appropriate forms of scientific symbolism and discourse were seen as closely connected to questions about the construction of the new apparatuses of the modern state. This is particularly evident, for example, in the work of the physicist, Robert Boyle [25].
This paper will investigate in detail two of the curve drawing constructions from the Geometry of Descartes [11] in such a way as to highlight the issue of the coordination of multiple representations [6]. The profound impact of Descartes' mathematics was rooted in the bold and fluid ways in which he shifted between geometrical and algebraic forms of representation, demonstrating the compatibility of these seemingly separate forms of expression. Descartes is touted to students today as the originator of analytic geometry but nowhere in the Geometry did he ever graph an equation. Curves were constructed from geometrical actions, many of which were pictured as mechanical apparatuses. After curves had been drawn Descartes introduced coordinates and then analyzed the curve-drawing actions in order to arrive at an equation that represented the curve. Equations did not create curves; curves gave rise to equations. Descartes used equations to create a taxonomy of curves [20].
It can be very difficult for a person well schooled in modern mathematics to enter into and appreciate the philosophical and linguistic issues involved in seventeenth century mathematics and science. We have all been thoroughly trained in algebra and calculus and have come to rely on this language and grammar as a dominant form of mathematical representation. We inherently trust that these symbolic manipulations will always give results which are compatible with geometry; a trust that did not fully emerge in mathematics until the early works of Euler more than a century after Descartes. Such trust became possible because of an extensive set of representational experiments conducted throughout the seventeenth century which tested the ability of symbolic algebraic language to faithfully represent geometry [5, 7]. Descartes' Geometry is one of the earliest and most notable of these linguistic experiments. Because of our cultural trust in the reliability of symbolic languages applied to geometry, many of those schooled in mathematics today have learned comparatively little about geometry in its own right.
Descartes wrote for an audience with opposite predispositions. He assumed that his readers were thoroughly acquainted with geometry, in particular the works of Apollonius on conic sections from ~200 BC [2, 15]. In order to appreciate the accomplishments of Descartes one must be able to check back and forth between representations and see that the results of symbolic algebraic manipulations are consistent with independently established geometrical results. The seventeenth century witnessed an increasingly subtle and persuasive series of such critical linguistic experiments in the work of Roberval, Cavalieri, Pascal, Wallis, and Newton [8, 9]. These led eventually to Leibniz's creation of a general symbolic language capable of fully representing all known geometry of his day, that being his "calculus" [5, 7].
Since many of the most simple and beautiful results of Apollonius are scarcely known to modern mathematicians, it can be difficult to recreate one essential element of the linguistic achievements of Descartes, that being the checking of algebraic manipulations against independently established geometrical results. In this paper I will ask the reader to become a kind of intellectual Merlin and live history backwards. After we explore one of Descartes curve-drawing devices, the resultant bridge between geometry, and algebra will be used to regain a compelling result from Apollonius concerning hyperbolic tangents. The reader can than choose to regard the contents of this investigation as a philosophical demonstration of the consistency between algebra and geometry, or as a simple analytical demonstration of a powerful ancient result of Apollonius. By adopting both views one gains a fully flexible cognitive feedback loop of the sort that I and my students have found most enlightening [6].
I was recently discussing my work on the history of curve drawing devices and their possible educational implications with a friend. His initial reaction was, "Surely you don't advocate the revival of geometrical methods; progress in mathematics has been made only to the extent to which geometry has been eliminated." This claim has historical validity especially since the eighteenth century, but my response was that such progress was only possible once mathematicians had achieved a basic faith in the ability of algebraic language to represent and model geometry accurately. I argued that one can not appreciate the profundity of calculus unless one is aware of the issue of coordination of multiple independent representations. Many students seem to learn and even master the manipulations of calculus without ever having questioned or tested the language's ability to model geometry precisely. Even Leibniz, no lover of geometry, would feel that such a student has bypassed the main point of his symbolic achievement [5]. My friend and I reached agreement on this issue.
Descartes' curve-drawing devices poignantly raise the issue of technology and its relation to mathematical investigation. During the seventeenth century there was a distinct turning away from the classical Greek orientation that had been popular during the Renaissance in favor of pragmatic and stoic Roman philosophy. During much of the seventeenth century a class in "Geometry" would concern itself mainly with the design of fortifications, siege engines, canals, water systems, and hoisting devices. i.e., what we would call civil and mechanical engineering. Descartes' Geometry was not about static constructions and axiomatic proofs, but instead concerned itself with mechanical motions and their possible representation in algebraic equations. Classical problems were addressed but they are all transformed into locus problems, through the use of a wide variety of motions and devices that went far beyond the classical restriction to straight-edge and compass. Descartes sought to build a geometry that included all curves whose construction he considered to be "clear and distinct" [11, 20]. An examination of his work shows that what he meant by this was any curve that could be drawn with a linkage (i.e. a device made of hinged rigid rods). Descartes' work indicates that he was well aware that this class of curves is exactly the class of all algebraic curves although he gave no formal proof of this. This theorem is scarcely known among modern mathematicians although it can be proved straight-forwardly by looking at linkages that add, subtract, multiply, divide and generate integer powers [3]. Descartes' linkage for generating any integer power was used repeatedly in the Geometry and has many interesting possibilities [10].
This transformation of geometry from classical static constructions to problems involving motions and their resultant loci has once again raised itself in light of modern computer technology, specifically the advent of dynamic geometry software like Cabri, and Geometer's Sketchpad. An abundance of new educational and research possibilities have emerged recently in response to these technological developments [26]. It seems that seventeenth century mechanical geometry may yet rise from the ashes of history and regain a new electronic life in our mathematics classrooms. It has always had a life in our schools of engineering where the finding of equations that model motion has always been a fundamental concern. My own explorations of seventeenth century dynamic geometry have been conducted with a combination of physical models and devices combined with computer animations made in the software Geometer's Sketchpad [18]. The first figure in this paper is taken directly from Descartes while all of the others were made using Geometer's Sketchpad. This software allows for a more authentic historical exploration since curves are being generated from geometrical actions rather than as the graphs of equations. Static figures in print can not vividly convey the sense of motion that is necessary for a complete understanding of these devices. In the generation of the figures in this paper no equations were typed into the computer.
Descartes' Hyperbolic Device
Figure 1:
Descartes' Hyperbolic Device
Figure 1 is reproduced from the original 1637 edition of Descartes' Geometry [11, p. 50]. Descartes described this curve drawing device as follows:
Suppose the curve EC to be described by the intersection of the ruler GL and the rectilinear plane figure NKL, whose side KN is produced indefinitely in the direction of C, and which, being moved in the same plane in such a way that its diameter KL always coincides with some part of the line BA (produced in both directions), imparts to the ruler GL a rotary motion about G (the ruler being hinged to the figure NKL at L). If I wish to find out to what class this curve belongs, I choose a straight line, as AB, to which to refer all its points, and on AB I choose a point A at which to begin the investigation. I say "choose this and that," because we are free to choose what we will, for, while it is necessary to use care in the choice, in order to make the equation as short and simple as possible, yet no matter what line I should take instead of AB the curve would always prove to be of the same class, a fact easily demonstrated. [11, p. 51]
Descartes addressed here several of his main points concerning the relations between geometrical actions and their symbolic representations. His "classes of curves" refer to the use of algebraic degrees to create a taxonomy of curves. He is asserting that the algebraic degree of an equation representing a curve is independent of how one chooses to impose a coordinate system. Scale, starting point and even the angle between axes will not change the degree of the equation, although this "fact easily demonstrated" is never given anything like a formal proof in the Geometry. Descartes also mentioned here the issue of a judicious choice of coordinates, an important scientific issue that goes largely unaddressed in modern mathematics curriculum until an advanced level, at which point geometry is scarcely mentioned.
Descartes went on to find the equation of the curve in Figure 1 as follows. Introduce the variables (Descartes used the term "unknown and indeterminate quantities") AB = y, BC = x, (i.e. in modern notation C = (x,y)), and then the constants ("known quantities") GA = a, KL = b, and NL = c. Descartes routinely used the lower case letters x, y, and z as variables, and a, b, and c as constants, and our modern convention stems from his usage. Descartes, however, had no convention about which variable was used horizontally, or in which direction (right or left) a variable was being measured (x is measured to the left here). There was, in general, no demand that x and y be measured at right angles to each other. The variables were tailored to the geometric situation. There was a very hesitant use of negative values (often called "false roots"), and in most geometric situations they were avoided.
Continuing with the derivation, since the triangles KLN and KBC are similar, we have = , hence BK = x , hence BL = x – b . From this it follows that
AL = y + BL = y + x – b . Since triangles LBC, and LAG are similar, we have = . This implies the following chain of equations.
=
x ( y + x – b ) = a ( x – b )
xy + x2 – bx = x – ab
x2 = cx – xy + ax – ac (1)
Descartes left the equation in this form because he wished to emphasize its second degree. He concluded that the curve is a hyperbola. How does this follow? As we said before Descartes assumed that his readers were well acquainted with Apollonius. We will return to this issue shortly.
If one continues to let the triangle NLK rise along the vertical line, and keeps tracing the locus of the intersection of GL with NK, the lines will eventually become parallel (see Figure 2), and after that the other branch of the hyperbola will appear (see Figure 3).
Figure 2:
Descartes' Device in the Asymptotic Position
These figures were made with Geometer's Sketchpad, although I have altered slightly the values of the constants a, b, and c from those in Figure 1. In Figure 2, the line KN is in the asymptotic position (i.e., parallel to GL). I will hereafter refer to this particular position of the point K, as point O. In this position triangles NLK, and GAL are similar, hence the length of AK = AO = + b (the y-intercept of the asymptote). The slope of the asymptote is the same as the fixed slope of KN, i.e., b/c (recall that KL = b,
NL = c, and GA = a).
To rewrite Equation 1 using A as the origin in the conventional modern sense, with x measured positively to the right, one must substitute –x for x . Making this substitution and solving Equation 1 for y one obtains the following equation.
y = ab + x + ( + b) (2)
Figure 3:
Geometrical Display of the Terms in the Hyperbolic Equation
One sees in Equation 2 that the linear equation of the asymptote appears as the last two terms of the equation. In Figure 3, I have shown, to the right, the lengths that represent respectively the values of the three terms in Equation 2, for the point P (1-inverse term, 2-linear term, 3-constant term). Term 3 accounts for the rise from the x-axis to the level of point O (the intercept of the asymptote). Adding term 2, raises one to the level of the asymptote, and term 1 completes the ordinate to the curve.
As a geometric construction, the hyperbola is drawn from parameters which specify the angle between the asymptotes (angle NKL), and a point on the curve (G). If one changes the position of the point N without changing the angle NKL, the curve is unaffected as in Figure 4. The derivation of the equation depends only on similarity, and not on having perpendicular coordinates. As long as GA (which determines the coordinate system) is parallel to NL, the derivation of the equation is the same except for the values of the constants NL = c, and GA = a (which have both become larger in Figure 4). Of course this equation is in the oblique coordinate system of the lines GA (x-axis) and AK (y-axis). It is the same curve geometrically, with the same form of equation, but with new constant values that refer to an oblique coordinate system. As long as angle NKL remains the same, and G is taken at the same distance from the line KL, the device will draw the same curve. This form of a hyperbolic equation, as an inverse term plus linear terms, depends only on using at least one of the asymptotes as an axis.
Figure 4: Hyperbola in Skewed Coordinates
I have encountered many students over the years who are well acquainted with the function y = , and yet are entirely unaware that its graph is an hyperbola. Descartes' construction can be adjusted to draw right hyperbolas. Consider the special case when the line KN is parallel to the x-axis (see Figure 5). The point G is on the negative x-axis. Let KC = x, and AK = y (i.e. C = (x,y)) , AG = a, and KL = b. Now AL = y–b, and since triangles LKC, and LAG are similar, we have = , or, equivalently, = . Hence the curve has the following equation.
y = ab + b (3)
A vertical translation by b would move the origin to the point O, and letting a = b = 1, would put G at the vertex (-1,-1), yielding a curve with an equation of y = 1/x.
Figure 5:
Device Adjusted to Draw Right Hyperbolas
Equation 3 can be seen as a special case of Equation 2, obtained by substituting ∞ for c, where c is thought of as the horizontal distance from L to the line KN. In this case the linear term disappears. All translations and re-scalings of the multiplicative inverse function can be directly seen as special members of the family of hyperbolas, via this construction.