Rene Descarates Contd

One might ask how we know that these curves are in fact hyperbolas. Descartes said that this is implied by Equation 1. In his commentaries on Descartes, van Schooten gives us more detail [11, p. 55, note 86]. Once again these mathematicians assumed that their readers were familiar with a variety of ratio properties from Book 2 of the Conics of Apollonius [2, 15] that are equivalent to Equation 1. I will not give a full set of proofs, but instead suggest means for exploring these relations.

Several of the theorems of Apollonius concerning the relations between tangents and asymptotes are beautiful, and easily explored in this setting. Using the asymptotes of the curve in Figure 5 as edges to define rectangles, one sees that the points on the curve define a family of rectangles which all have the same area (see Figure 6). Letting M and N be any two points on the curve, Equation 3 implies that the areas of OPMS and OQNR are both equal to a.b, the product of the constants used in the drawing of the curve. Another geometric property of interest is that the triangles TSM and NQU are always congruent. This congruence provides one way to dissect and compare these rectangles in a geometric way [17].

Approaching these questions analytically, assume that the curve in Figure 6 has the equation x. y = k (using O as the origin). Let M = (m, k/m) and N = (n, k/n), i.e. OP = m and OQ = n. Writing the equation of the line through M and N, one obtains

TS = = NQ. Since triangles TSM and NQU are clearly similar, TS=NQ implies that they are congruent and that TM = NU. Now let the points M and N get close to each other. The line MN then gets close to a tangent line, and one can perceive a theorem of Apollonius:

Given any tangent line to a hyperbola, the segment of the tangent contained between the two asymptotes is always bisected by the point of tangency to the curve [2, 15].

This property is a defining characteristic of hyperbolas. This simple and beautiful theorem immediately implies, for example, that the derivative of is , by simply looking at the congruent triangles and computing the rise over run for the tangent. This gives a student an independent geometrical check on the validity of the calculus derivation.

This bisection property of hyperbolic tangents is not restricted to the right hyperbola. Looking back at Figure 3, and Equation 2, one sees that any hyperbola coordinatized along both its asymptotes will always have an equation of the form x. y = k for some constant k. To see this, subtract off the linear and constant terms from the y-coordinate, and then rescale the x-coordinates by a constant factor that projects them in the asymptotic direction (i.e., in Figure 7 the new x-coordinate in this skew system is OQ). In the general case the curve can be seen as the set of corners of a family of equi-angular parallelograms which all have the same area. In Figure 7, for any two points on the curve, M and N, the parallelograms (OQNR and OPMS) have equal areas. Since the triangles TSM and NQU are congruent, by letting M and N get close together one sees that any tangent segment TU is bisected by the point of tangency (M or N).

An alternative view of the situations just described is to imagine any line parallel to TU meeting the asymptotes and the curve in corresponding points T', M', and U'. Then the product T'M' x M' U' = TM x MU. That is to say, parallel chords between the asymptotes of a hyperbola are divided by the curve into pieces with a constant product. This follows from our discussion, because the pieces are constant projections of the sides of the parallelograms just discussed. It is this form of the statement that was most often used by van Schooten, Newton, Euler and others in the seventeenth century. This statement (from Book 2 of Apollonius [2, 15]) was traditionally used as an identifying property of hyperbolas. This constant product was given as a proof by van Schooten that the curve drawn by Descartes' device was indeed a hyperbola [11, p. 55]. Apollonius derived these properties directly from sections of a general cone.

In this way it is possible to make an investigation of hyperbolas, using both geometric and algebraic representations that creates a complete cognitive feedback loop. Neither representation is being used as a foundation for proof, instead one is led to a belief in a relative consistency between certain aspects of geometry and algebra through a checking back and forth between multiple representations. A calculus derivation of the derivative of y = 1/x becomes, in this setting, a limited special case of the bisection property of hyperbolic tangents. It can be very satisfying to see symbolic algebra arrange itself into answers that are consistent with physical and geometric experience. Students of calculus can then experience the elation of Leibniz, as they build up a vocabulary of notation that becomes viable, because it can be checked against independently verifiable physical and geometric experience. Mathematical language is then seen as a powerful and viable code for aspects of experience, rather than as the sole dictator of truth.

The hyperbolic device is only the beginning of what appears in Descartes' Geometry. He discussed several cases where curve drawing constructions can be progressively iterated to produce curves of higher and higher algebraic degree [11, 10]. It is usually mentioned in histories of mathematics that Descartes was the first to classify curves according to the algebraic degree of their equations. This is not quite accurate. Descartes classified curves according to pairs of algebraic degrees, i.e. lines and conics form his first class (he used the term genre), curves with third or fourth degree equations form his second class, etc. [11, p. 48]. This classification is quite natural if one is working with mechanical linkages and loci. With most examples of linkage iteration, each iteration raises the degree of the curve's equation by two, with some special cases that collapse back to an odd algebraic degree [7].[5] What follows is an example of such an iteration based on the hyperbolic device.

Descartes generalized the previous hyperbola construction method by replacing the triangle KLN with any previously constructed curve. For example, let a circle with center L be moved along one axis and let the points C and C' be the intersections of the circle with the line LG, where G is any fixed point in the plane and LG is a ruler hinged at point L just as in the hyperbolic device (see Figure 8). Then C traces out a curve of degree four, known in ancient times as a conchoid [11, p. 55]. The two geometric parameters involved in the device are the radius of the circle (r), and the distance (a) between the point G and the axis along which L moves.

Figure 8 shows three examples of conchoids for a > r, a = r, and a < r. If the curve is coordinatized along the path of L, and a perpendicular line through G (OG), then its equation can be found by looking at the similar triangles GOL and CXL (top of Figure 8). Since GO=a, LC=r, CX=y, OX=x, XL= , one obtains the ratios of the legs in the triangles as: = , which is equivalent to x2y2 = (r2–y2) (a–y)2, which is of fourth degree, or of Descartes' second class. The squared form of the equation has both branches of the curve, above and below the axis, as solutions.

This example demonstrates Descartes' claim that, as one uses previously constructed curves to draw new curves, one gets chains of constructed curves that go up by pairs of algebraic degrees. Descartes called the conchoid a curve of the second class (i.e. of degree three or four). Dragging any rigid conic-sectioned shape along the axis, and drawing a curve in this manner will produce curves in the second class. Dragging curves of the second class will produce curves of the third class (i.e. degree five or six), etc. Descartes demonstrated this general principle through many examples [11, 7, 10], but he did not offer anything like a formal proof, either geometric or algebraic. His definition of curve classes was justified by his geometric experience.

Notice that when a ≤ r, the point G becomes a cusp or a crossover point. When singularities like cusps or crossover points occur, these tend to occur at important parts of the apparatus, like pivot points (e.g. point G), or at a point on an axis of motion. Other important examples of this phenomena can be found in Newton's notebooks [22, 23]. I am not stating any particular or explicit mathematical theorem here. This general observation is based upon my own historical research and empirical experience with curve drawing devices. There are probably several ways to make this observation into an explicit mathematical statement, subject to proof (Newton attempted several, [23]). There are many important open questions concerning these forms of curve iteration and the relations between parts of the devices and singularites of the curves [7]. Students might benefit enormously from such empirical experience regardless of the extent to which they eventually formalize that experience in strictly algebraic or logical language. An instinctual sense of where curve singularites might occur is fundamentally useful in many sciences [1]. Modern computer software makes such investigations routinely possible with a minimum of technical expertise.

Descartes wrote his Rules for the Direction of the Mind in 1625, twelve years before he would publish his famous Geometry . In this earlier work he emphasized the importance of making strong connections between physical actions and their possible representations in diagrams and language. Here are a few quotes from [12].

Rule 13: If we understand a problem perfectly, it should be considered apart from all superfluous concepts, reduced to its simplest form, and divided by enumeration into the smallest possible parts.

Rule 14: The same problem should be understood as relating to the actual extension of bodies and at the same time should be completely represented by diagrams to the imagination, for thus will it be much more distinctly perceived by the intellect.

Rule 15: It is usually helpful, also, to draw these diagrams and observe them through the external senses, so that by this means our thought can more easily remain attentive.

These lines from Descartes sound almost like parts of the hands-on, problem-solving educational philosophy of mathematics put forth by the National Council of Teachers of Mathematics [21]. Descartes' entire approach to mathematics had problem solving as its foundation [14], but we must not allow ourselves to read into him too modern a perspective. He was constructing a new method of mathematical representation that responded to both the new symbolic language of his time (algebra) and to the new technology of his time (mechanical engineering). He was not seeking the broad educational goals of N.C.T.M., and in fact his Geometry was not widely read in the seventeenth century until it was republished in 1657 with extensive commentaries by Franz van Schooten.

Nonetheless, Descartes' approach to geometry through curve-drawing devices and locus problems has important implications for education. His work connects important classical and Arabic traditions with modern algebraic formalisms [7]. It provides the missing linkages (pun intended). These linkage and loci problems combined with the new dynamic geometry software allow for a new kind of exploration of curves leading to a type of investigation which could go far towards ending the isolation of geometry in our mathematics curriculum. One can use geometrical curve generation to recreate calculus concepts such as tangents and areas in a much more elementary and physical setting [7, 8, 10], as well as to explore complicated questions about algebraic curves left open since the seventeenth century [7, 23]. Computer graphic techniques have already led to new branches of mathematical investigation (e.g., fractals). Perhaps a new phase of computer-assisted empirical geometrical investigation of curves and surfaces has already begun. If this new beginning is as revolutionary as the century that began with Descartes' Geometry then we are in for some very exciting times ahead.

1. V. I. Arnol'd, Hygens & Barrow, Newton & Hooke , Birkhäuser Verlag, Boston, 1990.

2. Apollonius of Perga, On Conic Sections. In Vol. 11 of The Great Books of the Western World, Encyclopedia Britannica, Chicago, 1952.

3. I. I. Artobolevskii, Mechanisms for the Generation of Plane Curves , Macmillan, New York, 1964.

4. F. Cajori, Controversies on mathematics between Wallis, Hobbes, and Barrow, The Mathematics Teacher, Vol. XXII Num. 3 (1929) 146 - 151.

5. J. M. Child, The Early Mathematical Manuscripts of Leibniz., Open Court, Chicago, 1920.

6. J. Confrey, A Theory of Intellectual Development. For the Learning of Mathematics. (Published in three consecutive issue) 14, 3, (1994) 2-8; 15, 1, (1994) 38-48; 15, 2 (1995).

7. D. Dennis, Historical Perspectives for the Reform of Mathematics Curriculum: Geometric Curve Drawing Devices and their Role in the Transition to an Algebraic Description of Functions. Unpublished Doctoral Dissertation, Cornell University, Ithaca, New York, 1995.

8. D. Dennis and J. Confrey, Functions of a curve: Leibniz's original notion of functions and its meaning for the parabola, The College Mathematics Journal, 26, (1995), 124-130.

9. D. Dennis and J. Confrey, The Creation of Continuous Exponents: A Study of the Methods and Epistemology of Alhazen and Wallis. In J. Kaput & E. Dubinsky (Eds.) Research in Collegiate Mathematics II. CBMS Vol. 6, pp. 33-60. Providence, RI: American Mathematical Society, 1996.

10. D. Dennis and J. Confrey, Drawing Logarithmic Curves with Geometer's Sketchpad: A Method Inspired by Historical Sources. In J. King & D. Schatschneider (Eds.), Geometry Turned On: Dynamic Software in Learning, Teaching and Research. Washington D.C.: Mathematical Association of America, 1997.

12. R. Descartes, Rules For the Direction of the Mind, Bobbs-Merrill, New York, 1961.

13. M. Foucault, The Order Of Things: An Archeology Of The Human Sciences, Pantheon Books, New York, 1970.

14. J. Grabiner, Descartes and Problem Solving, Mathematics Magazine, 86, (1995), 83-97.

15. T. L. Heath, Apollonius of Perga: Treatise on Conic Sections, Barnes & Noble, New York, 1961.

16. T. L. Heath, The Thirteen Books of Euclid's Elements, Dover, New York, 1956.

17. D. Henderson, Experiencing Geometry on Plane and Sphere, Prentice Hall, Engelwood Cliffs, NJ, 1996.

18. N. Jackiw, Geometer's Sketchpad™ (version 3.05N) [Computer Program], Key Curriculum Press, Berkeley, CA, 1996.

19. J. Klein, Greek Mathematical Thought and the Origin of Algebra, M.I.T. Press, Cambridge, MA, 1968.

20. T. Lenoir, Descartes and the geometrization of thought: The methodological background of Descartes' geometry, Historia Mathematica, 6, (1979), 355-379.

21. National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics, N.C.T.M., Reston VA, 1991.

22. I. Newton, The Mathematical Papers of Isaac Newton, Vol. 1 (1664-1666), Cambridge University Press, 1967.

23. I. Newton, The Mathematical Papers of Isaac Newton, Vol. 2 (1667-1670), Cambridge University Press, 1968.

24. B. Rotman, Signifying Nothing: The Semiotics of Zero, Stanford University Press, 1987.

25. S. Shapin and S. Schaffer, Leviathan And The Air Pump, Hobbes, Boyle And The Experimental Life, Princeton University Press, 1985.

26. D. Schatschneider and J. King (Eds.), Geometry Turned On: Dynamic Software in Learning, Teaching and Research. M.A.A., Washington DC, 1997.

[1] This research was funded by a grant from the National Science Foundation.

[2] See, for example, Book 10, Lemma 1 before Prop. 29, where Euclid generates all Pythagorean triples geometrically even though he violates the dimensional integrity of his argument. Areas, in the form of "similar plane numbers," are multiplied by areas to yield areas. There seems to be no way to reconcile dimension and still obtain the result.

[3] Descartes' contemporary, Fermat, did begin graphing equations but his work did not have nearly the philosophical or scientific impact of Descartes'. Fermat's original problematic contexts came from financial work rather than engineering and mechanics.

[4] Animated figures made in Geometer's Sketchpad are available from the author by e-mail.

[5] Descartes' linkages led directly to Newton's universal method for drawing conics which is essentially a projective method [7, 23]. This same classification by pairs of degrees is used in modern topology in the definition of "genus." The "genus" of a non-singular algebraic plane curve can be thought of topologically as the number of "handles" on the curve when defined in complex projective space. In complex projective space, linear and quadratic non-singular curves have genus 0, and are topologically sphere-like. Similarly, curves of degrees 3 and 4 are topologically torus-like, and have genus 1. Curves of degrees 5 and 6 are topologically double-holed and have genus 2, etc. In the real model, (i.e. when considering only real solutions of one real equation in 2 variables) the genus 0 curves consist of at most one oval when you join up the asymptotes. The genus 1 curves will have 2 ovals, which is what you'd expect when cutting through a toric by a plane, etc. (This comment was made to me by Paul Pedersen.)