Descartes's La Geometrie example essay topic

Rene Descartes was born on March 31, 1596, in a small town in Touraine called La Haye, now called La Haye-Descartes or simply Descartes (3: 1). Rene was the second of two sons and one daughter (4: 1). When he was about the age of nine, his father sent him to a new Jesuit College, Henri IV at La Fleche (3: 1). Due to his delicate health, Descartes was allowed to sleep late in the mornings, a custom to which he always followed (4: 1). Descartes left La Fleche in 1614 to study law at Poitiers.

Two years latter he received his Baccalaureate and Licentiate degrees in law. In 1618 he joined the army of Prince Maurice of Nassau. Descartes used this military experience as a means of exploring the world (4: 1). During a stay in Breda, Descartes was walking through the streets where he saw a poster written in Dutch, which aroused his curiosity. He asked the first pedestrian he met to translate it into French or Latin. The pedestrian, who happened to be Isaac Beeckman, the head of Dutch College of Dort, said he would if Descartes would answer it; the poster being a challenge to all the world to solve a certain geometrical problem.

Descartes worked it out within a couple of hours, and a warm friendship between him and Beeckman was the result (1: 15-16). While stationed in Germany, on November tenth, 1619, Descartes envisioned a new philosophy of analytic geometry. He regarded this day as the most critical day in his life, and one that determined his whole future. In the spring of 1621, he resigned from military service (4: 1-2). Through much of the 1620's, Descartes threw himself into the pursuit of the good life.

Traveling through Paris, gambling, and dueling seemed to catch his attention while also studying mathematics. In 1628, Cardinal de Berulle encouraged Descartes to give up this way of life and devote it to the examination of the truth. He agreed and departed for Holland, where he would spend the next 20 years (3: 1) During the first four years, 1629-1633, of his stay in Holland he wrote Le Monde. Even though this work was never published, it was an attempt to explain the universe. He then devoted himself to explain metaphysics and universal science in the work entitled Discourse on the method for conducting one's Reason Rightly an for searching for the truth in the sciences (4: 2).

In this book he made famous the phrase I think therefore I am. However in saying that Descartes argued that there is a God. Before he could write this with a clear conscious, he had to prove to himself that God did exist (1: 16). Along with this work came these appendices: La Dioptrique, Les Meteors, and La Geometrie. Descartes's La Geometrie is divided into 3 books. The first book explains the principles of analytical geometry and discusses a problem that had stumped Pappus, Euclid, and Apollonius.

It was in the attempt to solve this problem that Descartes invented analytical geometry (3: 2). In the second book, Descartes divided curves into two classes, geometrical and mechanical. He defined geometrical curves as those which can be generated by the intersection of two lines each moving parallel to one co-ordinate axis with commensurable velocities. By commensurable he means that the derivative of y with respect to x (dy / dx ) is an algebraical function. Descartes goes on to define a mechanical curve as: when the ratio of velocities of the two lines is incommensurable velocity.

By this he means the derivative (dy / dx ) is a transcendental function. He also paid a lot of attention to the theory of tangents. The, then current, definition of a tangent at a point was a straight line through a point so that no other straight line could be drawn between it and the point. Descartes proposed that the tangent was the limiting position of the secant (4: 2-3).

The method Descartes used to find the tangent or normal at any point of a given curve is as follows: He determined the center and radius of a circle, which should cut the curve in two consecutive points there. The tangent to the circle at that point will be the required tangent to the curve. He then selected a circle as the simplest curve and one to which he knew how to draw a tangent; he so fixed his circle as to make it touch the given curve at the point in question, and thus reduced the problem to drawing a tangent to a circle. (1: 17) The third book of La Geometrie contains an analysis of the then current algebra. In this book Descartes was the first to use last letter of the alphabet to symbolize unknown quantities and the first letters to symbolize known quantities.

He also introduced a system of indices (such as x 2) to express the powers of numbers. In addition he introduced the rule for finding the number of positive and negative roots for any algebraic equation (4: 3). Of the other two appendices to Discourse on Method one was devoted to optics. In doing so, he was the first to publish the fundamental law of reflection: that the angle of incidence is equal to the angle of reflection.

Descartes's treatment of light as a type of pressure in a solid medium paved the way for the undulatory theory of light (4: 3). The final appendix, on meteors, explains numerous atmospheric phenomena such as the rainbow (2: 1). Descartes's physical theory of the universe, containing most of what was in his unpublished work Le Monde, was published in his work Principia in 1644. In it he starts with a discussion on motion and then lays down ten laws of nature. Of which the first two are almost identical with Newton's first two laws; the remaining eight are false. He then proceeds to discuss the nature of matter and many other metaphysical questions of the day (4: 4).

Not only was Descartes a great mathematician, he was also a great philosopher. According to Descartes, God created two classes of substances that make up reality. One class was thinking substances, or minds, and the other was extended substances, or bodies (1: 18). With this in mind, he began to publish philosophical works. Among them were Meditations on First Philosophy in 1641 and The Principles of Philosophy in 1644. The latter volume was dedicated to Princess Elizabeth Stuart of Bohemia, whom Descartes had formed a deep friendship with.

He also wrote Passions of the Soul. In it Descartes explored such topics as the relationship of the soul to the body, the nature of emotion, and the role of the will in controlling the emotions (4: 4-5). In 1649 Queen Christina of Sweden convinced Descartes to come to Stockholm to teach her about his philosophy. He came to Stockholm during the brutal winter of 1650 and caught pneumonia in February. After more than a week of suffering, he died on February 11, 1650 (2: 2).

Bibliography

1. Barrett, William. Death of the Soul from Descartes to the Computer. Garden City, New York: Anchor Press / Doubleday 1986.2. web 3. web hawk / descaresh. html 4. web 322.