## 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.

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."

### Editor's Remarks

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