EDITORIAL POLICY Mathematics Magazine aims to provide lively and appealing mathematical exposition. The Magazine is not a research journal, so the terse sty le appropriate for such a journal (lemma-theorem-proof-corollary) is not appropriate for the Magazine. Articles should include examples, applications, historical background, and illustrations, where appropriate. They should be attractive and accessible to undergraduates and would, ideally, be helpful in supplementing undergraduate courses or in stimulating student investigations. Manuscripts on history are especially welcome, as are those showing relationships among various branches of mathematics and between mathematics and other disciplines. A more detailed statement of .author guidelines appears in this Magazine, Vol. 74, pp. 75-76, and is available from the Editor or at www.maa.org/pubs/mathmag.html. Manuscripts to be submitted should not be concurrently submitted to, accepted for publication by, or published by another journal or publisher. Submit new manuscripts to Allen Schwenk, Editor-Elect, Mathematics Magazine, Department of Mathematics, Western Michigan University, Kalamazoo, Ml, 49008. Manuscripts should be laser printed, with wide line spacing, and prepared in a style consistent with the format of Mathematics Magazine. Authors should AUTHORS Jeff Suzuki dieted the known laws of physics and chemistry. In graduate school he combined his interests in mathematics, history, and physics in a dissertation on the history of the stability problem in celestial mechanics, and has subsequently focused on the mathematics of the 1 7th and 18th centuries because, in his words, it was "the last time it was possible to know everything." Current research interests include pre-Newtonian methods of solving calculus type problems (hence this article), the early history of analytic number theory, and Renaissance cookery. Matt Richey has been at St. Olaf College since 1986. He was an undergraduate at Kenyon College and received his Ph.D. in mathematical physics at Dartmouth College in 1985 under the direction of Craig Tracy. His primary interest is computational mathematics, especially randomized algorithms. When not attending to academic pursuits, he is working to convince the baseball strategists of the futility of the sacrifice bunt. His next paper is tentatively entitled "Baseball, Bunting, and Beane (Billy)". Paul Zorn was born and raised in southern India. He was an undergraduate at Washington University in St. Louis, and earned his Ph.D. in complex analysis at the University of Washington, Seattle, supervised by Professor E. L. Stout. He has been on the faculty at St. Olaf College since 1981. His professional interests include complex analysis, math­ ematical exposition, and textbook writing. In an earlier millennium he served a term as editor of Mathematics Magazine. Nathania! Burch is a senior at Grand Valley State University, where he is double majoring in mathematics and statistics. The past summers have found Nathania! working on two different funded under­ mail three copies and keep one copy. In addition, authors should supply the full five-symbol 2000 Mathematics Subject Classification number, as described in Mathematical Reviews. graduate research projects, one at Grand Valley State University and the other at the Worcester Polytechnic Institute REU program. He is a member of the Pi Mu Epsilon and Phi Kappa Phi honor societies. Upon completion of his senior year, Nathania! is planning on attending graduate school in applied mathematics. Interests beyond mathematics include his long-time passions for bowling, fishing, and billiards, along with a newly developed love for traveling. Cover image: Off on a Tangent, by Liam Boylan, based on a concept by Keith Thompson. Two articles in this issue address alternate ways to treat tangency. Featured on the cover are Descartes, Hudde, and Lanczos, along with representations of their contributions to this fruitful field of inquiry. Paul Fishback received his undergraduate degree from Hamilton College and his Ph.D. from the University of Wisconsin-Madison. Ever since beginning his college teaching career, he strongly desired to investigate the notion of least-squares differentiation, which his college classmate Dan Kopel had researched as an undergraduate. It is Liam Boylan graduated from Santa Clara University with a double major in Communications and Studio Art. Between careers at the moment, Liam seeks opportunities to bolster his portfolio. His parents have instructed him to marry rich. by sheer coincidence that he discovered that Dan's work was linked to the Lanczos derivative. In addition to having a broad array of analysis-related interests, he serves as a councillor for Pi Mu Epsilon and as associate editor of the MAA Notes series. His main nonmathematical interest is traditional, Greenland-style kayaking, and he has paddled extensively on Lake Michigan and Lake Superior. became a mathematician because his results in laboratory science frequently contra- Vol. 78, No. 5, December 2005 MATHEMATICS MAGAZINE E D I TOR Frank A. Farris Santa Clara University ASSOCIATE E D I TORS G l e n n D. App leby Santa Clara University Arth u r T. Benjam i n Harvey Mudd College Pau l J. Campbel l Beloit College Annal isa Cran nel l Franklin & Marshall College David M . James Howard University Elgi n H . Joh n ston Iowa State University Victor J. Katz University of District of Columbia Jen n ifer J. Qu i n n Occidental College David R. Scott University of Puget Sound Sanford L. Se g a l University of Rochester Harry Wa ldman MAA, Washington, DC E D ITORIAL ASSI STANT Martha L. G i a n n i n i STUD ENT E D I TORIAL ASSI STANT Keith Thompson MATHEMATICS MAGAZINE (ISSN 0025-570X) is published by the Mathematical Association of America at 1529 Eighteenth Street, N.W., Washington, D.C. 20036 and Montpelier, VT, bimonthly except July/August. The annual subscription price for MATHEMATICS MAGAZINE to an individual member of the Association is $131. Student and unemployed members receive a 66% dues discount; emeritus members receive a 50% discount; and new members receive a 20% dues discount for the first two years of membership.) Subscription correspondence and notice of change of address should be sent to the Membership/ Subscriptions Department, Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. Microfilmed issues may be obtained from University Microfilms International, Serials Bid Coordinator, 300 North Zeeb Road, Ann Arbor, Ml 48106. Advertising correspondence should be addressed to Frank Peterson ( ARTICLES The Lost Calculus (1637-1670): Tangency and Optimization without Limits J EFF S UZU K I Brooklyn College Brooklyn, NY 1121 0 jefLsuzuki®yahoo.com If we wished to find the tangent to a given curve or the extremum of a function, we would almost certainly rely on the techniques of a calculus based on the theory of limits, and might even conclude that this is the only way to solve these problems (barring a few special cases, such as the tangent to a circle or the extremum of a parabola). It may come as a surprise, then, to discover that in the years between 1637 and 1670, very general algorithms were developed that could solve virtually every "calculus type" problem concerning algebraic functions. These algorithms were based on the theory of equations and the geometric properties of curves and, given time, might have evolved into a calculus entirely free of the limit concept. However, the work of Newton and Leibniz in the 1670s relegated these techniques to the role of misunderstood historical curiosities. The foundations of this "lost calculus" were set down by Descartes, but the keys to unlocking its potential can be found in two algorithms developed by the Dutch mathematician Jan Hudde in the years 1657-1658. In modernized form, Hudde's results may be stated as follows : Given any polynomial j (X) = a n Xn + an -!Xn -I + · · · + azX2 + a!X + ao, 1 . if f (x) has a root of multiplicity 2 or more at x = a, then the polynomial that we know as f ' has f ' (a) = 0, and 2. if f(x) has an extreme value at x = a , then f ' (a) = 0. These results can, of course, be easily derived through differentiation, so it is tempting to view them as results that point "clearly toward algorithms of the calculus" [2, p. 375 ] . But Hudde obtained them from purely algebraic and geometric considerations that at no point rely on the limit concept. Rather than being heralds of the calculus that was to come, Hudde's results are instead the ultimate expressions of a purely algebraic and geometric approach to solving the tangent and optimization problems. We examine the evolution of the lost calculus from its beginnings in the work of Descartes and its subsequent development by Hudde, and end with the intriguing possibility that nearly every problem of calculus, including the problems of tangents, optimization, curvature, and quadrature, could have been solved using algorithms entirely free from the limit concept. Descartes's met hod of tangents The road to a limit-free calculus began with Descartes. In La Geometrie ( 1637), Descartes described a method for finding tangents to algebraic curves. Conceptually, 3 3 9 340 MATHEMATICS MAGAZI N E Descartes's method i s the following: Suppose we wish to find a circle that i s tangent to the curve OC at some point C (see FIGURE 1). Consider a circle with center P on some convenient reference axis (we may think of this as the x-axis, though in practice any clearly defined line will work), and suppose this circle passes through C. This circle may pass through another nearby point E on the curve; in this case, the circle is, of course, not tangent to the curve. On the other hand if C is the only point of contact between the circle and curve, then the circle will be tangent to the curve. Thus our goal is simple: Find P so that the circle with center P and radius CP will meet the curve OC only at the point C. F Figure 1 Descartes's method of tan g ents Algebraically, any points the circle and curve have in common correspond to a solution to the system of equations representing the curve and circle. If there are two distinct intersections, this system will have two distinct solutions; thus, in order for the circle and curve to be tangent and have just a single point in common, the system of equations must have two equal solutions. In short, the system of equations must have a double root corresponding to the common point C. Descartes, who wanted his readers to become proficient with his method through practice, never deigned to give simple examples. However, we will present a simple example of Descartes's method in modem form. Suppose OC is the curve y = JX, and let C be the point (a 2 , a). Imagine a circle passing through the point (a 2 , a), with radius r centered on the x-axis at the point (h , 0), with h and r to be determined. Then the circle has equation Expanding and setting the equation equal to zero gives The points of intersection of the circle and curve correspond to the solutions to the system of equations: l + x 2 - 2hx + h 2 - r 2 = 0 and y = ,JX. VOL. 78, NO. 5, D ECEM B E R 2 005 If we eliminate y using the substitution y = 3 4 1 JX, we obtain which is a quadratic and will in general have two solutions for x. By assumption, the circle and curve have the point (a 2 , a) in common; hence x = a 2 is a root of this equation. In order for the circle and curve to be tangent, we want x = a 2 to be the only root. Thus it is necessary that Expanding the right-hand side and comparing coefficients we find that 1 - 2h = -2a 2 and thus h = a 2 + 1 / 2. Therefore the circle with center (a 2 + 1 /2, 0) will be tangent to the graph of y = JX at the point (a 2 , a). This method of Descartes approached the problem of tangents by locating the center of the tangent circle. Today, we solve the problem by finding the slope of the tangent line. Fortunately there is a simple relationship between the two. From Euclidean geometry, we know that the radius through a point C on a circle will be perpendicular to the tangent line of the circle through C. In this case the radius PC will lie on a line with a slope -2a; hence the tangent line through C will have slope 1/(2a). This is, of course, what we would obtain using the derivative, but here we used only the algebraic properties of equations and the geometrical properties of curves. The method in La Geometrie is elegant, and works very well for all quadratic forms. Unfortunately, it rapidly becomes unwieldy for all but the simplest curves. For example, suppose we wish to find the tangent to the curve y = x 3 • As before, let the center of our circle be at (h, 0) ; we want the system x 2 - 2hx + h 2 + l - r 2 = 0 and y = x 3 to have a double root at the point of tangency (a, a 3 ). Substituting x 3 for y gives x6 + x 2- 2hx + h 2- r 2 = 0. (1) If w e wish to find the tangent at the point (a, a 3 ), this equation should have a double root at x = a; since the left-hand side is a 6th degree monic polynomial, it must factor as the product of (x - a) 2 and a fourth degree monic polynomial: 2 4 x6 + x 2 - 2hx + h 2 - r 2 = (x- a) 2 (x + Bx 3 + Cx + Dx + F). Expanding the right-hand side gives us x6 + x 2 - 2hx + h 2 - r 2 = x6 + (B - 2a)x5 + (a 2 - 2aB + C)x4 + (a 2 B - 2aC + D)x 3 + (a 2 C - 2aD + F)x 2 + (a 2D - 2aF)x + a 2 F. Comparing coefficients gives us the system B - 2a = 0 a 2 - 2aB + C = 0 a 2 B - 2aC + D = 0 (2) (3) (4) 3 42 MATH EMAT ICS MAGAZI N E a 2C - 2aD + F = 1 a 2 D - 2aF = -2h a 2 F = h 2 - r 2• (5) (6) (7) From Equation 2 we have B = 2a. Substituting this into (3) we have a 2 - 2a(2a) + C = 0; hence C = 3a 2• Substituting the values for B and C into (4) gives us hence D = 4a 3 . Substituting these values into (5) gives a 2 (3a 2 ) - 2a(4a 3 ) + F = 1 ; hence F = 1 + 5a4• Substituting into (6) gives hence h = a + 3a5 and the center of the tangent circle will be at (a + 3a5, 0) . As before, the perpendicular to the curve will have slope -a 3 /(3a5) = - 1 / (3a 2 ), and thus the slope of the line tangent to the curve y = x 3 at x = a will be 3a 2 . It is clear from this example that the real difficulty in applying Descartes's method is this: If y = f(x), where j(x) is an nth degree polynomial, then finding the tangent to the curve at the point where x = a requires us to find the coefficients of (x - a) 2 multiplied by an arbitrary polynomial of degree 2n - 2. The problem is not so much difficult as it is tedious, and any means of simplifying it would significantly improve its utility. Descartes discovered one simplification shortly after the publication of La Geometrie. He described his modified method in a 1638 letter to Claude Hardy [5, vol. VII, p. 6 lff1. Descartes's second method of tangents still relies on the system of equations having a double root corresponding to the point of tangency, but Descartes simplified the procedure by replacing the circle with a line and used the slope idea implicitly (as the ratio between the sides of similar triangles). In modern terms, we describe Descartes's second method as follows: The equation of a line that touches the curve f(x, y) = 0 at (a, b) is y = m (x - a) + b, where m denotes a parameter to be determined. In order for the line to be tangent, the system of equations f (x, y) = 0 and y = m (x - a) + b must have a double root at x = a (alternatively a double root at y = b). For example, if we wish to find the tangent to y = x 3 at (a, a 3 ), the equations y = x 3 can be reduced to and system of y = m(x - a) + a 3 by substituting the first expression for y into the second equation. In order for the line to be tangent at x = a, it is necessary that x = a be a double root, so (x - a) 2 is a VOL. 78, NO. 5, DECEMBER 2 005 3 4 3 factor of this polynomial; if we call the other factor (x - r), we can write x 3 - rnx + (rna - a 3 ) = (x - a) 2 (x - r) = x 3 - (r + 2a)x 2 + (a 2 + 2ar)x - a 2 r. Comparing coefficients gives us the system r + 2a = 0, a 2 + 2ar = -rn, and - a 2r = rna - a 3 . Solving this system gives us rn = 3a 2• This is, of course, the same answer we would obtain by differentiating y = x 3 , but obtained entirely without the use of limits. Either of the two methods of Descartes will serve to find the tangent to any algebraic curve, even curves defined implicitly (since, as Descartes pointed out, an expression for y can be found from the equation of the circle or the tangent line and substituted into the equation of the curve). For example, during a dispute with Fermat over their respective method of tangents, Descartes challenged Fermat and his followers to find the tangent to a curve now known as the folium of Descartes [5, vol. VII, p. 1 1 ] , a curve whose equation we would write as x 3 + l = p xy. The reader may be interested in applying Descartes's method to the folium. To find the line tangent to the folium at the point (x0, y0), we want the system x 3 + l = p xy and y = rn(x - xo) + Yo to have a double root x = x0• It should be pointed out that, contrary to Descartes's expectations, Fermat's method could be applied to the folium; Descartes subsequently challenged Fermat to find the point on the folium where the tangent makes a 45-degree angle with the axis (and again Fermat responded successfully). H udde's first letter: p olyn omial o perations The key to Descartes's methods is knowing when the system of equations that determine the intersection( s) of the two curves (whether a line and the curve, or a circle and the curve) has a double root, which corresponds to a point of tangency. An efficient algorithm for detecting double roots of polynomials would vastly enhance the usability of Descartes's method. Such a method was discovered by the Dutch mathematician Jan Hudde. Hudde studied law at the University of Leiden, but while there he joined a group of Dutch mathematicians gathered by Franciscus van Schooten. At the time van Schooten, who had already published one translation of Descartes's La Geornetrie from French into Latin, was preparing a second, more extensive edition. This edition, published in two volumes in 1 659 and 1 66 1 , included not only a translation of La Geornetrie, but explanations, elaborations, and extensions of Descartes's work by the members of the Leiden group, including van Schooten, Florimond de Beaune, Jan de Witt, Henrik van Heuraet, and Hudde. Hudde (along with Jan de Witt) would soon leave mathematics for politics, and eventually became a high official of the city of Amsterdam. When Louis XIV invaded The Netherlands in 1 672, Hudde helped direct Dutch defenses [2, 6] ; for this, Hudde became a national hero. De Witt was less fortunate: his actions were deemed partially responsible for the ineptitude and unpreparedness of the Dutch army in the early stage of the war, and he and his brother were killed by a mob on August 20, 1 672. 344 MATH EMATICS MAGAZI N E Hudde's return to politics may have saved The Netherlands, but mathematics lost one of its rising stars. Leibniz in particular was impressed with Hudde's work, and when Johann Bernoulli proposed the brachistochrone problem, Leibniz lamented: If Huygens lived and was healthy, the man would rest, except to solve your problem. Now there is no one to expect a quick solution from, except for the Marquis de l' Hopital, your brother [Jacob Bernoulli] , and Newton, and to this list we might add Hudde, the Mayor of Amsterdam, except that some time ago he put aside these pursuits [9, vol. II, p. 370] . As Leibniz's forecast was correct regarding the other three, one wonders what would have happened had Hudde not put aside mathematics for politics. Hudde's work in the 1 659 edition of Descartes consists of two letters. The first, "On the Reduction of Equations," was addressed to van Schooten and dated the "Ides of July, 1 657" (July 1 5 , 1 657). In the usage of the time, to "reduce" an equation meant to factor it, usually as the first step in finding all its roots. Thus the letter begins with a sequence of rules (what we would call algorithms) that can be used to find potential factors of a polynomial. These factors have one of two types: those corresponding to a root of multiplicity 1 , or those corresponding to a root of multiplicity greater than 1 . Since Descartes's method of tangents (and Hudde's method of finding extreme values) relies on finding multiple roots, this has particular importance. Key to Hudde's method of finding roots of multiplicity greater than 1 is the ability to find the greatest common divisor (GCD) of two polynomials. How can this be done? One way is to factor the two polynomials and see what factors they have in common. However this is impractical for any but the most trivial polynomials (and in any case requires knowing the roots we are attempting to find). A better way is to use the Euclidean algorithm for polynomials. For example, suppose we wish to find the GCD of f(x) = x 3 - 4x 2 + lOx - 7 and g(x) = x 2 - 2x + 1 . To apply the Euclidean algorithm we would divide j(x) by g(x) to obtain a quotient (in this case, x - 2) and a remainder (in this case, 5x - 5); we can express this division as x 3 - 4x 2 + lOx - 7 = (x 2 - 2x + l)(x - 2) + (5x - 5) . Next, we divide the old divisor, x 2 - 2x + quotient and remainder: x 2 - 2x + 1 1 , by the remainder 5x - 5 , to obtain a new = (5x - 5) (15 x - 51 ) + 0. The last nonzero remainder (in this case, 5x - 5) corresponds to the GCD; in general, it will be a constant multiple of it. While this is the way the Euclidean algorithm for polynomials is generally taught today, Hudde presented a clever variation worth examining. Since we are only interested in the remainder when the polynomials are divided, we can, instead of performing the division, find the remainder modulo the divisor. In particular, Hudde's steps treated the divisor as being "equal to nothing"; he set the two polynomials equal to zero and solved for the highest power term in each. In our example, we would have x 2 = 2x - 1 x 3 = 4x 2 - lOx + 7 . The first gives u s an expression for x 2 that can b e used t o eliminate the x 2 and higher degree terms of the other factor. Substituting and solving for the highest power remain