Lecture 24. Descartes and AnalyticGeometry

In Calculus, rectangular coordinate is called Descartes coordinate. In philosophy, hisstatment I think, therefore I am became a fundamental element of Western philosophy.

Figure 24.1 Rene Descartes and a house in which he was born.

Biography of Descartes Rene Descartes (1596-1650) was a French mathematician andphilosopher. He is famous for, together with Fermats work, having made an importantconnection between geometry and algebra, which allowed for the solving of geometricalproblems by way of algebraic equations.

Rene Descartes was born in La Haye, which is now called La Haye-Descartes, in theFrench province of Touraine. He was born into a family of the old French nobility.

Descartes father was a lawyer and judicial officer, and his mother died when Descarteswas one year old. He, as well as his brother and sister Pierre and Jeanne, was raised by

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his material grandmother and a nurse in La Haye. Although his father Joachim Descarteswas often away from home, he observed Descartes exceptional curiosity, calling him littlephilosopher.

In 1606, at around ten years of age, he was enrolled in the Jesuit College of La Fleche.The young Descartes was given a special privileges at school, because of his intellectualpromise and because he was sickly throughout his youth: he was permitted to have his ownroom and to stay in bed until late in morning. Such special treatment lead him to developa lifelong habit of spending several hours in bed in morning thinking and writing. By theway, such habit eventually may cause his life.

The morning thinking led him to conclude that little he had learned in school was certain,and then he became so full of doubts. As reported in his Discours on Method, he wrote

I used the rest of my youth to travel, to see courts and armies, to frequentpeople of differing dispositions and conditions, to store up various experiences, toprove myself in the encounters with which fortune confronted me, and everywhereto reflect upon the things that occurred, so that I could derive some profit fromthem.

Figure 24.2 King Henry IV. He was assassinated by Francois Ravaillac

On 14 May 1610, King Henry IV was assassinated in Paris and was stabbed to deathwhile he rode in his coach. Henry was buried at the Saint Denis Basilica. Since HenryIV was also the founder of the school where Descartes attended, his death was a profound

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shock. In the funeral ceremony, the climax was the burial of the kings heart. Descartes wasone of the 24 students chosen to participate the ceremony.

He left Jesuit College and entered the University of Poitiers in 1615, where a year laterhe received his Baccalaureate and License.

When in Breda, a city in the southern part of the Netherlands, on November 10, 1618,Descartes saw a mathematical problem posted on a well. Since his Dutch was not yet fluent,he asked a bystander to translate it for him. This was how Descartes met Isaac Beeckman1 , who became his first instructor in mathematics and a lifelong friend.

In 1628, Descartes moved to Holland where he spent most of the rest of his life. He liveda simple but leisurely life, and finally settled down to working out the idea conceived nineyears ago: only accept truths that would cause no doubt, and follow mathematical methodsto achieve perfect certainty in human knowledge by mathematical reasoning through simple,logical steps.

In fact, Descartes is often regarded as the first thinker to provide a philosophical frame-work for the natural sciences as these began to develop. Most famously, his statment Ithink, therefore I am became a fundamental element of Western philosophy. 2

He soon wrote a major treatise on physics. However just a moment before publishingthe treatise, Descartes heard of the news about Galileos condemnation by the Church, andthen he decided not to publish it for fear that a small error might lead to the banning of hisentire philosophy.

1Isaac Beeckman, (1588 - 1637) was a Dutch philosopher and scientist who may have virtually givenbirth to modern atomism.

2It means: If someone is wondering whether or not his exist, that is the proof that he does exist (because,at the very least, there is an I who is doing the thinking).

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Later he changed his mind and decided that he should share his new ideals with theworld. As a result, Descartes published his Discourse on Methods in 1637, which includesthree essays on optics, meteorology, and geometry designed to show the efficacy of the method. During a twenty-year period of secluded life in Holland, he produced the body ofwork that enhanced and secured his international philosophical reputation.

In 1649 Descartes moved to Sweden as invited by Queen Christina of Sweden to cometo tutor her. He reluctantly accepted, but it turns out that his health could not withstandthe severity of the northern climate, especially since Christina required him to rise at anearly hour, which is contrary to his long-established habits. Descartes soon contracted alung disease so that he did not survive his first winter at Stockholm.

Figure 24.3 Descartes and Queen Christina

Analytic geometry The basic idea of analytic geometry is to establish a link betweengeometry and analysis such that one uses analysis to study geometry. For example, eachcurve in plane or in the space can be represented as the zero set of some equation(s). Inother words, studying of curves in the plane is reduced to study the equations

(, ) = 0,

which defining the curve. Thus a great new field was opened, and geometry and algebrabecame inseparable.

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It has to be pointed out that using equations to study curve is not the whole idea. Infact, the Greek mathematician Apollonius used equation to study conics as by product ofgeometric argument but we do not regard that to be analytic geometry. The idea of coordi-nates emerged appeared in the work of Nicole Oresme (1323-1382) 3. He used longitudeand latitude, (i.e., coordinates) in astronomy and geography, which is beyond the Greekmathematics. However without sufficient algebraic techniques, he simply was unable to gofurther.

Analytic geometry needs more: beyond the equations that represent curves, one needs thetechniques to manipulate equations to obtain information about curves. Such manipulationhas not been done by Greek mathematics.

The crucial step that finally made analytic geometry feasible was the techniques of solvingequations and the improvement of algebraic notation in the sixteenth century. This was doneby Fermat (see the next lecture) and Descartes.

Figure 24.4 Descartes

Geometric curves Descartes studied curves defined by a polynomial equation, andcalled them geometric curves. The terminology geometric curves could come from theGreek mathematics in which curves are the product of geometric constructions. In modernmathematical language, such curves are called algebraic curves because they are definedby algebraic equations.

Recall that Greeks considered some transcendental (i.e., non-algebraic) curves construc-tions, such as rolling one circle on another, which are capable of producing transcendental

3A French mathematician who invented coordinate geometry long before Descartes.

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curves. Descartes called such curves mechanical and exclude them by his definition ofgeometric curves. He wrote:

I could give here several other ways of tracing and conceiving a series ofcurved lines, each curve more complex than any preceding one, but I think thebest way to group together all such curves and then classify them in order is byrecognizing the fact that all points of those curves which we may call geometric,that is, those which admit of precise and exact measurement, must bear a definiterelation to all points of a straight line, and that this relation must be expressedby means of a single equation.

Descartes rejection of transcendental curves was short-signed, as the calculus soon pro-vided techniques to handle them, nevertheless it was fruitful to concentrate on algebraiccurves. The notion of degree was an important measure of complexity. First-degree curvesare lines; second-degree are conic sections. With third-degree curves have new phenomenaof inflections, double points, and cusps (cusp, a singular point of a curve).

Figure 24.5 His most famous curve is the folium of Descartes .

Descartes rule Descartes rule of signs is also a commonly used method to determinethe number of positive and negative roots of a polynomial. More precisely, the number ofpositive roots (counting multiplicity) of an equation

() = 0 + 1

1 + ...+ 1+ , 0 > 0

with real coefficients is equal either to the number of variations in the signs of its coefficientsor to this number decreased by a positive even integer. Newton formulated it more preciselyin his Arithmetica Universalis (1707). 4

Find normal directions Until the early 1600s, constructing tangents to curves had notbeen seriously studied. Motivated from his optical studies, Descartes studied the problem.

4David Burton, The History of Mathematics, p.374.

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He devised a purely algebraic method for finding the normal at any given point on a curvewhose equation is known; the tangent line could be taken as the perpendicular through thepoint to the normal.

Coordinate geometry paved a way to applications to physics. An example was Newtonsderivation of the Kepler laws of gravitation. Keplers first law says that the planetary orbitsare ellipses with the sun as their common focus. The proof was possible only after an analytictheory of conics has been established. 5

5S.S. Chern, What is geometry? A. M. Monthly 97(1990), 548-555.

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