The Key to Geometry:
A Pair of Perpendicular Lines

Deepak Kandaswamy

René Descartes (1596-1650) is primarily associated with Philosophy: his Discourse on Method and Meditations have even led him to be called the "Father of Modern Philosophy." In his most celebrated argument, Descartes attempted to prove his own existence via the now hackneyed argument, "I think therefore I am." However, it should not be forgotten that René Descartes applied his system to investigations in physics and mathematics, with real success, playing a crucial role in the development of a link betw een algebra and geometry - now known as analytic geometry, a subject defined by Webster's New World Dictionary as "the analysis of geometric structures and properties principally by algebraic operations on variables defined in terms of position coordinates." Simply put, analytic geometry translates problems of geometry into ones of algebra. Prior to the Cartesian plane and analytic geometry, most mathematicians considered (synthetic) geometry and (diophantine) algebra to be two quite different fields of study. To anyone that has taken a high school course in analytic geometry, that notion s eems ridiculous, or even incomprehensible, but to mathematicians of 500 years ago or more, solving geometric problems using the methods of algebra probably seemed equally absurd.

In fact, as will be evident later in the paper, much of our tenth grade "vocabulary" (using x2 to represent the equation of a parabola, using terms a, b, c to denote indeterminate parameters, etc...) can trace their roots directly back to the work o f René Descartes, building on the algebra of the late 16th century.

How did it transpire that someone who had more interest in determining whether or not we live in a dream world than in, for example, determining the mean and extreme ratio mathematically, come to fundamentally change not only the way we do geometry, but also the way we think about geometry? To understand the answer, it will be useful to examine the life of René Descartes and the period in which he flourished.

Descartes' father was a lawyer and judge, and his parents belonged to the noblesse de robe, the social class of lawyers, between the bourgeoisie and the nobility. As such he received and excellent education, and had the financial resources to continue hi s studies at the Jesuit College of the town of La Flhche in Anjou [9, pp. 1-2]. Men are a product of their times, and René Descartes was no exception. After hearing that Galileo Galilei, among others, both pronounced, and persuasively argued, that the sun did not revolve around the Earth, but rather vice versa, and that, in addition , the earth made a complete revolution daily, Descartes began to question whether any of the senses could be trusted as a source of information. After all, his sense of motion clearly demonstrated that the Earth is stationary, while it was "truly" rotating and moving at a great speed through space. If his senses could be wrong in regard to something so basic, was not it possible to be equally mis taken in other fundamental areas as well? Nonetheless, according to Descartes "I concluded that I might take as a gen-eral rule the principle that all things which we very clearly and obviously conceive are true: only observing, however, that there is some difficulty in rightly determining the o bjects which we distinctly conceive." Descartes held knowledge up to a very severe standard. According to Descartes, the four rules of logic were:
1.) To accept as true only those conclusions which were clearly and distinctly known to be true.
2.) To divide difficulties under examination into as many parts as possible for their better solution.
3.) To conduct thoughts in order, and to proceed step by step from the simplest and easiest to know, to more complex knowledge.
4.) In every case to take a general view so as to be sure of having omitted nothing.
[9, p.16] Because of his severe standard, Descartes' quest for underlying truths blossomed into a distinct penchant for mathematics, where proofs were just that - undeniable knowledge. Descartes' fourth standard conveys more than just a hint of the mathematician a s well as the philosopher. Often in mathematics, solving a simple problem can be trivial. However, the formulation of a general rule to solve the problem can be infinitely more useful. Descartes seems to say in his fourth rule that the general case is the one of great importance, not the specific problem. Eventually Descartes published his ideas in a little book, or appendix, titled La Géomitrie, in 1637. Descartes major contribution in this book is considered to lie in the idea of a coordinate system, allowed problems that were considered to be strictly geometric to pass over into algebra. Although the association of algebra and geometry was proposed even by the Greeks [8, p.84], and taken up anew as a program by Vihte, no satisfying procedure had been found to merge the two disciplines into one ( unti l the development of the Cartesian plane. Thus, Descartes was not the first to attempt to develop a coordinate plane, but his method has been the one that achieved the desired goal. Both the Greeks and Egyptians had developed a numerical coordinate system (driven by its relevance to astronomy and cartography), but with little mathematical development. "Hipparchus (B.C. 150) and Ptolemy (150 A.D.), to name but two, both employed a system of latitude and longitude to locate stars on the celestial sphere." [9, p.85] The Greeks even employed a system that made use of two axes at a right angle. However, nothing systematic or permanent came out of the study of specific problems using two axes as part of the solution. Heath says that "the essential difference between t he Greek and modern method is that the Greeks did not direct their efforts to making the fixed lines of a figure as few as possible, but rather to expressing their equations between areas in as short and simple a form as possible." [10, p.26 bottom footno te] The first real development of a geometrical coordinate system comes in the work of Apollonios of Perga (ca. 240 - ca. 174 B.C.). Apollonios of Perga, or the "Great Geometer" as he was known, wrote a book called Conics, which, among other things, introduc ed the world to the terms parabola, ellipse, and hyperbola. In his Conics, Apollonius used a system of coordinates to solve problems regarding second-order curves (conic sections). [5, p. 211] The next person to significantly advance the creation of the coordinate system was Frangois Vihte (1540-1603). In his In artem analyticem isagoge (Introduction to the Analytical Art) published in 1591, Vihte announced a program to "[bring] together the a ncient geometrical methods of Euclid, Archimedes, Apollonius, and Pappus" [1, p.268], with ancient algebraic methods to produce his logistica speciosa, a way to formulate and solve algebraic problems. Among other things, this text uses consonants to repr esent given quantities and vowels to denote unknown quantities. This led to Vihte's nickname, "The father of modern algebra." The degree of Descartes' originality remains a subject of controversy, as will be addressed at greater length below, a controversy that has persisted in the three and a half centuries since his death. In Descartes' La Géomitrie, he uses the letters a, b , c, etc., to express already known magnitudes and x, y, z, etc., for unknown ones. Later on, Descartes unveils what appears to be the birth of a fixed set of coordinate systems in a passage begining, "Let AB, AD, EF, GH, any number of straight lines given in position..." [10, p.26] Smith points out here "it should be noted that t hese lines are given in position but not in length. They thus become lines of reference or coordinate axes, and accordingly they play a very important part in the development of analytic geometry. In this connection we may quote as follows: 'Among the p redecessors of Descartes we reckon, besides Apollonius, especially Vihte, Oresme, Cavalieri, Roberval, and Fermat, the last the most distinguished in the field; but nowhere, even by Fermat, had any attempt been made to refer several curves of different or ders simultaneously to one system of coordinates, which at most possessed special significance for one of the curves. It is exactly this thing which Descartes systematically accomplished.' [3, p.229-230] However, Scott does not agree with this assessmen t, as will be seen below. Another person who played a key role in the creation of analytic geometry was Pierre Fermat (1601 - 1665), although it is unclear whether or not Descartes knew of Fermat's work (a subject to which we shall return), Ad Locos Planos et Solidos Isagoge. In an effort to recover some of the lost proofs of Apollonius, Fermat used a system of coordinates to refer to various curves. There was a large advance in the use of the coordinate system between Apollonios and Fermat. "In [Fermat's] published works, too, there is incontrovertible evidence that he had hit upon the idea of expressing the nature of curves by means of algebraic eq uations. How clearly in fact, he had grasped the fundamental principles of analytic geometry becomes evident after a study of the opening pages of the Isagoge, the substance of which is as follows: 'Whenever two unknown quantities are found in a final equation we have a locus and the extremity of one of them describes a right angle line or a curve. The straight line is simple and unique; the curves are infinite in number and embrace the circle, par abola, ellipse, etc...'[9, p.86] Fermat goes on to list various equations of geometric interest, such as the equation of a straight line through the origin (x/y = b/d), the equation of any straight line (b/s = (a-x)/y), the equation of certain types of circle (a2-x2=y2), the equation of certain types of ellipse (a2-x2=ky2), and the equations of certain types of hyperbola (a2+x2=ky2). These formulas should leave no doubt that Fermat understood the underlying principles of analytical geometry, and helped lay the foundation for its develop ment. The ideas with which La Géomitrie had to deal, at least potentially, were of three types according to the formulation of J.F. Scott [9, pp.88-89]: 1. The employment of coordinates as a mere instrument of description 2. Algebra and geometry collaborate on single problems 3. Transference of system and structure By analyzing these individually we can see how influential they were in the development of analytic geometry, and consider more carefully which of them are actually attributable to Descartes, according to Scott. The first item, according to Scott, consti tutes the most visible connection between Descartes' work and the Cartesian plane. In La Géomitrie, Descartes uses a system of coordinates adapted to each problem. When studying multiple curves, he uses a system of lines to unify all the separate coordi nate systems into one giant system. This account clashes with the opinion of Fink and Smith, according to whom Descartes' coordinate system was set up in advance for a general set of curves, not a particular one. As far as the second point, it is the most important in Descartes' work. Using algebra to solve geometric problems greatly enhanced the flexibility of geometry. This became a legitimate way to solve a problem, and as is often found in mathematics, the m ore ways there are to approach a class of problems, the better. An example of this given at the outset in La Géomitrie was the solution of a problem of Pappus (ca. 300 A.D.), which Descartes claimed had not been completely solved by anyone [9, p.97]. In a letter to his friend Mersenne, Descartes wrote, "J'y risous un e question qui par le timoignage de Pappus n'a p{ estre trouvie par qucun des Anciens, et l'on peut dire qu'elle ne l'a p{ estre non plus par aucun des Modernes." ("I solve a problem which defeated the ancients and the moderns alike.") Pappus' problem reads, "There being three, or four, or a greater number of right lines given in position in a plane, it is first required to find the position of a point from which we can draw as many other right lines, one to each of the given lines, mak ing a known angle with it, such that the rectangle contained by two of these drawn from this point has a given proportion either to the square on the third, if there are only three, or to the rectangle contained by the other two, if there are four. Or if there are five, the product of the remaining two lines so drawn has a given proportion to the product of the remaining two and another line, and so on." [9, p. 97] Descartes originally attempted to solve this problem using pure geometry, and was unable to. This aided Descartes in his pursuit to find another method to solve the problem. Using his newly developed analytic methods, Descartes wrote in a letter to his friend that he was able to solve the problem in just five or six weeks. Unsurprisingly, Sir Isaac Newton was the first one to solve this problem using methods of pure geometry [9, p. 97]. As to the third point that Scott raises in regard to the major achievements in La Géomitrie, it appears to be rather similar to the second, and possibly not necessary. As Scott puts it, "The structure of a whole region of geometrical theory is transferre d to a region of algebraical theory, where it brings about an instructive rearrangement of the matter and raises algebraical problems which otherwise might not have imposed themselves" [9, 89]. Among the achievements of La Géomitrie, there are many methods that are still used today. Descartes proposes a method of simultaneously handling several unknown quantities at once. Also introduced is a clearer distinction between real and imaginary root s, that helped lead to modern mathematics. Scott also says, "It is a momentous liberation when Descartes throws aside the dimensional restrictions of [Vihte] and lets the arithmetical second power a2 measure a length as well as an actual square, and the arithmetical first power a measure a square as well as an actual length." [9, p.89] In La Géomitrie, Descartes views curves of degree 2n and 2n-1 as having the same complexity, and thus as closely related. Scott even claims, "Descartes notes that this number is independent to the choice of organic coordinates. In modern language it is an invariant under change of axes. Here is a first case of invariance (A celebrated later case is Relativity). When employing coordinates we are forced to make an arbitrary choice of axes and even of the type of coordinates, and in this way we impart an arbitrary element into our methods" [9, p. 90]. Scott summarized the work of Descartes under four headings: 1.) He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes its arbitrariness. 2.) Algebra makes it possible to recognize the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all. 3.) Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method. 4.) Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all. [9, pp.92-93] Much of the work that is thus accredited to René Descartes is the subject of controversy. His reputation came under attack while he was alive, attacks which have been renewed in the 350 years since his death. Even at the time of his publication of La Gi omitrie, Descartes was forced to defend himself against claims that the work was in large part derived from the work of Pierre de Fermat and Frangois Vihte. There is no doubt that Fermat compiled his work in 1629, eight years before Descartes published La Géomitrie. However, this work of Fermat did not appear in print until 1679 (posthumously, in Opera Varia), approximately thirty years after Descartes' deat h. The question then is whether or not Descartes had access to his fellow countryman's compilation prior to it being published. Fermat gave his papers to M. Despagnet around 1629, but it is unclear whether or not Despagnet circulated these works farther . Descartes did not remain silent about such allegations. He vehemently defended himself, saying even that he had nothing to learn from his contemporary mathematicians, because they were unable to solve the ancient problems. "...and in particular he [Desc artes] leaves his readers in no doubt that he did not rate the achievements of Fermat very highly." [9, p.87]

One may wonder whether maybe the opposite was true: could Fermat have "borrowed" from Descartes? This possibility can be excluded. According to Scott, who appears to be a partisan of Descartes, Fermat's letters revealed his character to be of the highest moral caliber. One may also argue that had Fermat been familiar with Descartes' work, he would likely have adopted Descartes' notation, far superior to his own. There is in any case no evidence that Fermat ever saw Descartes' work prior to its public ation, much less prior to his own work in 1629, nor were any such allegations ever made. Scott comes to the conclusion that "It seems not impossible, therefore, that Descartes and Fermat had each made considerable progress in the new methods unconscious of what had been achieved by the other." [9, p.88] He asserts that history has numerous examples of discoveries of great importance that were made simultaneously and independently. Frangois Vihte was another mathematician whom Descartes has been ac-cused of robbing. In Vihte's In Artem Analyticam Isagoge (1591), he uses a notational system to represent algebraic equations similar to the one employed by Descartes in La Géomitrie. T his has led to speculation that much of Descartes' accomplishments were merely restatements of work Vihte had done 45 years earlier. "But Descartes' clumsy cossic notation, derived in all probability from Clavius' (a 16th and 17th century teacher at the Jesuit Collegio Romano in Rome) Algebra, which he had studied while in college, indicates that he was not familiar with Vihte's work a t this point, for Vihte's notation is clearly superior, and had he been familiar with it he could not have favored that of Clavius. Descartes was obliged to rediscover these relations, to formulate the problems in his own terms, and to develop his own me ans to solving the problem, something he was to do in a way that went far beyond Vihte's pioneering work" [5, pp. 98-99]. On the other hand, had Descartes wanted to take credit for another's ideas, it is doubtful that he would have been so overt as blatan tly to copy Vihte's notation. In this regard, Descartes wrote, "As to the suggestion that what I have written could easily have been gotten from Vihte, the very fact that my treatise is hard to understand is due to my attempt to put nothing in it that I believed to be known by either him or by anyone else...I begin the rules of my algebra with what Vihte wrote at the very end of his book, De emendatione aequationum... Thus, I begin where he left off" [10, p. 10, first paragraph of footnote]. This does of course openly acknowledge fami liarity with Vihte.

One final person declared Descartes in no uncertain terms to be a plagiarist - John Wallis (1616-1703). Wallis repeatedly and very publicly said that the main principles of coordinate geometry had already been published in Artis Analyticf Praxis by Thom as Harriot (1560-1621). Wallis wrote in Algebra (1685), a treatise designed to promote the ideas of Harriot, which were first published in 1631, that "Harriot hath laid the foundation on which Des Cartes hath built the greatest part of his Algebra or Ge ometry" [9, pp. 138-139]

"Whilst there appears little doubt that Descartes did not hesitate to avail himself of the knowledge of Harriot in his treatment of equations, it is difficult to find anything in Harriot's published works to suggest that he had devoted any attention to the subject of coordinate geometry." [7, p. 117]

How René Descartes came up with the ideas presented in his La Géomitrie is unclear. What is clear is that regardless of the source of these ideas, La Géomitrie is a work of great importance that fueled the adoption of the Cartesian plane and the develop ment of analytic geometry, allowing problems of geometry to be solved by algebraic methods.

It seems only fitting to end this paper the way Descartes ended his La Géomitrie - with a little humor and more than a little arrogance. "Et i'espere que nos neueux me sgauront gri, non seulement des choses que iay icy expliquies; mais aussy de celes que iay omises volontairemen [sic], affin de leur laisser le plaisir de les inuenter." Or as David Eugene Smith and Marcia L. Latham have it: "I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery."


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