# Mathematical Achievements of Rene Descartes

## Introduction

Rene Descartes was a celebrated mathematician, philosopher, and a scientist. He was born in France in 1596 in the province of La Haye, Tourane (Clarke, 2006). His father was called Joachim Descartes. His mother died when he was hardly two years old. Upon celebrating his eighth birthday, he was enrolled in Society of Jesus run institution called La Fleche where he was basically taught Mathematics and philosophy. After graduating from La Fleche he proceeded to study law at the University of Poitiers. With an intention of pursuing a military career, he joined the service of Prince Maurice Nassau of the provinces of the Netherlands. He later served in other armies but he saw his future more in Mathematics and Philosophy. This research paper endeavors to give an outline, albeit detailed, the achievements that Rene Descartes made in the Mathematical realms.

## Rene Descartes contribution to Mathematics

A close look at the works of the 17^{th} century Mathematicians no doubt leaves an impression on the interested party that there was indeed some glaring rift in mathematical thoughts advanced by these great thinkers. These thoughts were generally embodied in the traditional and the revolutionary modes. This is evident in divergent views which were held by among others Pierre de Ferment who maintained that pioneer mathematicians were powerless and could not even afford to pose. However, Pierre’s contemporary and foe Descartes, embraced a kind of mathematics that was a hybrid of both traditional and revolutionary modes (Mahoney, 1968). Descartes is credited for having come up with mathematical method that no other Greek had ever contemplated coming up with.

Rene Descartes was able to demonstrate to Isaac Beeckman, a thirty year old medical student, how algebra and Mathematics could be applied in real life situations like how Mathematics could be used in tuning lute stings, how algebraic formula could be used in explaining a rise in water level especially when an object is placed in such waters, and drawing geometric graph that could be used in predicting the speed at which a pencil dropped in a vacuum accelerates. With the help of Mathematics, he was also able to explain how a man can remain airborne borrowing from the ability of the spinning top to stay upright. Rene Descartes believed that Mathematics and all other sciences were interrelated and should better be treated wholly other than being studied individually as characterized in his fourth rule that postulates that Mathematics can be applied in all measurable entities (Mahoney, 1968). The universal mathematics that he was alluding to has since been used in optics and other disciplines that he indeed foresaw and others that he never stated.

In mathematical cycles, especially geometry, Descartes is credited for having opined that straight edge and compasses were indeed mechanical means of construction. Despite the fact that the straight edge and compass integrated aspects of counted units, an inheritance from the ancient Greek ontology, Descartes introduced the idea of number as the thing counted. Descartes further developed algebraic symbolism in the sense that instead of using Viete’s capital letters he instead resorted to using x, y , z to denote the unknowns. He however used a, b, c for the parameters. A further improvement on Viete’s work was witnessed when he used particular revealing symbolism instead of Viete’s last vestiges of verbal algebra (Balam, 2000). For example instead of using Viete’s 2A Cubus, he instead devised 2x^{3}. In his endeavor to construct algebra of line segments, Descartes attempted to show the six basic operations of algebra by integrating aspects of rising power of the roots , addition, subtraction, multiplication, and division. These, in the field of numerical algebra, posses a geometrical interpretation (Klein, 1968). This therefore implies that any number of line segments can be added. Apparently, smaller segments can be subtracted from greater segments. In the formative stages, multiplication of line segments posed some considerable challenge to Descartes because the classical procedure where one had to construct a rectangle proved ridiculous because a rectangle in itself was not a line segment.

Descartes asserts that not all roots of a given equation can be obtained using geometrical or numerical means like in the equation x^{3}-1=0 can be factored into the form (x-a)(x-b)(x-c)=0, however, when a=1, the values for b and c cannot be at first attained. He came up with imaginary roots of which he never said much about. He freed the concept of number from its natural enclave (Mahoney, 1968). By analyzing the structure of algebraic expressions, he asserts that its applicability to numerical problems if the concept of number is married to aspects of integers, fractions, and irrationals.

Descartes midwifed the theory of equations that dealt much with the nature of the extremes, the maximum and the minimum. In geometric writing sphere, Descartes distinguished order and the method of demonstration. He alluded that demonstration can be done by way of first analyzing the situation and then subsequently synthesizing the outcome of the analysis. By analyzing, one gets to know how something was found methodically. This enables a reader to develop a clear understanding of the concept. Some sense of ownership is developed. However, synthesis requires that a reader to come up with a string of definitions. The reader is then required at the end of the day to assent to certain underlying facts. The assent can be either hostile or stubborn. Moreover, he asserts that ancient Mathematicians made use synthesis in most of their writing not because they were not aware of analysis but because analysis was more of a secret that should only be known by them. He attests to the fact that analysis is the only best way of teaching mathematics.

While setting up mathematical symbolism, Descartes points out that mathematical symbolism is indeed a applicable in algebra hence its universality. According to him, the symbolism is intended to spare the memory and consequently free the mind to enable it to attend to other compelling mathematical problems. Mathematical symbolism is primarily intended to unravel the core of the problem rather than preoccupation with non essential aspects. Through symbolism, the structure of the problem can thus be illuminated irrespective of its geometric or arithmetic nature.

Rene Descartes has been referred to as a revolutionary mathematician by friends and foes alike because of his ability to marry the ideas of the ancient/traditional mathematicians with those of revolutionary mathematicians. He merged all the ancient Greek sciences with a view to coming up with universal mathematics that has played pivotal role in solving day-to-day life challenges in the modern world. It is therefore not in dispute that the privileges that man enjoy today because of revolution that has taken place in Mathematical spheres is owed to Rene Descartes.

## Reference List

Balam, R. (2000). *Algebra or Doctrine of Equations*. London: Routledge

Clarke, D. (2006). *Descartes: A Biography*. Cambridge: Cambridge University

Klein, J. (1968). *Greek Mathematical Thought and the Origin of Algebra*. Cambridge, Massachusetts: Cambridge.

Mahoney, M.S. (1968). Another Look at Greek Geometrical Analysis. *Archive for **History of Exact, *5, 331-338.