History 1596 A.D

1596 A.D.
Rene Descartes-The Father of Analytical Geometry
or Ludolph Van Ceulen and the history of pi

Rene Descartes was born on March 31, 1596 in Touraine, France. His family lived in the estate of Les Cartes at La Haye, France and his father was a Councilor in the city of Rennes, France. The second child of a family of two sons and one daughter, Descartes was sent to the Jesuit School at La Fleche at the age of 8. There he studied classics, logic and traditional Aristotelian philosophy and math. While in school, Descartes suffered from poor health and was granted permission by his school to remain in bed until 11 o'clock in the morning. He continued this practice throughout his life. When he visited Pascal in 1647, he said "the only way to do good work in mathematics and preserve my health is to never allow anyone to make me get up in the morning before I feel inclined to do so"(Ball, p. 1)

When he left school in 1612, he spent 2 years in Paris studying mathematics with his schoolboy friend Mersenre. During this time period, young men were encouraged to go into the military or the church. Descartes chose the military and joined the army in 1617 under Prince Maurice of Orange. There was no mention of his sleeping schedule during this time, but it probably changed to accommodate army life. After a couple of years in the army he grew disinterested, but not before he was introduced to a problem that would change the course of his life. One day, while walking along the street, he saw a Dutch sign telling of a geometric problem. He asked Isaac Beeckman, Head of the Dutch College at Dort who was unknown to him at the time, to translate it. It took Descartes just a few hours to solve the problem and a friendship grew between them. (MacTutor)

With the advent of the Thirty Years' War, Descartes was persuaded to volunteer under Count de Bucquoy in the army of Bavaria. But he continued studying mathematics in his spare time. The turning point in his life came during his tour with the army. He "had a dream" on the night of Nov 10, 1619 while on a campaign at Danube. From this dream, he developed his first ideas on a new philosophy of analytical geometry. Whether the dream occurred or not is left to speculation, but it was around this time that Descartes began formulating his geometry. In 1621, after a few years in the army, his passion for math got the better of him and he resigned from the army. He spent the next five years traveling and studying pure mathematics. (MacTutor)

Rene Descartes settled in Paris in 1626 where he continued to formulate his mathematical philosophy. While in Paris, he met Cardinal de Berulle, founder of the Oratorians. Taken by Rene, Cardinal de Berulle persuaded Descartes to examine the "truth" in life. Descartes moved to Holland at the "height of his power" (Eves, p. 347) to devote his life to the study of philosophy and mathematics. Descartes spent the next 20 years studying philosophy and mathematics and decided that:

"Science may be compared to a tree; metaphysics is the root, physics is the trunk, and the three chief branches are mechanics, medicine, and morals, these forming the three applications of our knowledge, namely, to the external world, to the human body, and to the conduct of life" (Ball, p. 1)

During his time in Holland, he published many books. He wrote Le Monde in which he attempted to give a physical theory of the universe. However, nearing the completion of his book he learned that the Church placed Galileo on house arrest for publishing his ideas on the universe. Instead of becoming a martyr for his ideals, Descartes decided against having the book published. The incomplete manuscript was later published in 1664, 14 years after his death.

After Le Monde, he started working on Discours de la methode pour bien conduire sa raison et chercher la verite dans les sciences, a philosophical treatise on universal science. This manuscript was accompanied with three appendices entitled La Dioptrique, Les Meteores, and La Geometrie. Descartes' contribution to analytical geometry appears in the last appendix of Discours. The birth of analytical geometry can be dated to the publication of La Geometrie. J F Scott, the author of The Scientific Work of Rene Descartes summarizes the importance of Descartes' work in four points:

1. He makes the first step towards a theory of invariant, which at later stages derelativises the system of reference and removes 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. (Ball, p. 2)

Descartes' major contribution to analytical geometry was his realization that a point in a plane could be completely determined if its distances, x and y coordinates, are given from two fixed lines that make right angles in the plane given (the Cartesian plane). Also, a given equation f(x, y) can satisfy an infinite number of x values with their corresponding y values on a given curve. (Ball, p. 2)
The complete work of La Geometrie is divided into three parts. The first part contains an explanation of some of the ancient Greek principles of algebraic geometry. The second part to La Geometrie deals with a now-obsolete classification of curves and a method of constructing tangents to curves. The third part of La Geometrie deals with the language of the subject by fixing the custom of employing letters to geometric variable so that letters at the beginning of the alphabet denote known quantities and those at the end of the alphabet to denote unknown quantities. (Ball, p. 2) He was one of the earliest mathematicians to move all the terms of an equation to one side in order to make the problem easier to solve.

La Geometrie was "written with intentional obscurity" and, as a result, was too difficult to be widely read. (Eves, p. 352) Perhaps Descartes wrote it with obscurity because he wanted only pure mathematicians to understand what he was doing. However, in 1649 a new version of La Geometrie became widely read because F. de Beaune accompanied it with explanatory notes and a commentary by Frans Van Schooten the Younger. But it still took one hundred years for the subject to achieve its present-day form. The words "coordinate, abscissa, and ordinate" were not used as terms in analytic geometry until 1692 when Leibniz contributed these words to the subject.

In 1641, Descartes published a work called Meditations on First Philosophy, which explained, in length, his views on philosophy as sketched out in the Discours. Descartes issued his Principia Philosophiae in 1644, which contains some inaccurate laws of nature and an inconsistent cosmological theory of vortices. A greater portion of this work was devoted to physical science, especially laws of motion and the theory of vortices. Descartes "attempted to put the whole universe on a mathematical foundation" by reducing it to a computational study. (MacTutor) For his twenty years of discoveries, he received a pension from the French Court in 1647. In 1649 Descartes received an invitation from Queen Christina of Sweden. Queen Christina liked to draw tangents at 5 a. m. and as a result, disrupted Descartes' not-getting-up-before-11-in-the-morning rule. However walking in the palace in the cold mornings of the northern climate made Descartes sick and he eventually died of pneumonia on February 11, 1650 in Stockholm, Sweden. He died just a few months after he had accepted the invitation from the Queen. Seventeen years after his death, Descartes' remains were returned to France. All of his bones were reinterred in Paris; except for his right hand, which was kept as a souvenir by the French Treasurer-General who had arranged for the bones to be transported. (Eves, p. 347)

Author: Sergio Gonzalez

References:
1. Ball, W. W Rouse. History of Math--A Short Account of the History of Mathematics, 4th Edition. 1908. 1998. http://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html

2. Eves, Howard. An Introduction to the History of Mathematics: second edition. Published by Saunders College Publishing, Florida, 1992.

4. Jui-ling Chao. Glass Bead Game. 1996. 1998. http://userwww.sfsu.edu/~rsauzier/Descartes.html.

5. O'Connor, John J and Robertson, Edmund F Mathematical MacTutor. 1996. 1998. http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Descartes.html

Ludolph Van Ceulen was born on January 28, 1540 in Germany and died on December 31, 1610 in the Netherlands. He was both a mathematician and a fencer and he taught both fencing and mathematics in Delft. He is most famous for calculating pi to 35 places. He used the same method that Archimedes used in solving for pi, but Van Ceulen used polygons with 2⁶² sides. He spent his entire life accomplishing his feat. He was so proud of his accomplishment that when he died, his wife had all 35 places of pi engraved on his tombstone.

Author: Sergio Gonzalez

References:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Van_Ceulen.html

1. Beckmann, Petr. A History of pi. Published bby the Golem Press, Colorado, 1970.

2. Borwin, Jonathan M, and Borwein, Peter B. Pi and the AGM. Published by John Wiley and Sons, New York, 1987.

Last updated December 1998