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Rene Descartes

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The philosophical and mathematical advancements sparked by Rene Descartes, a French philosopher of the 17th century and Scientific Revolution, contributed to many of the main ideas seen currently in geometry. He is attributed with the development of influential ideas, theorems, and formulas in modern geometry, despite his ideas being from centuries ago. Arguably some of his most groundbreaking ideas of his are quantified in the Discussion of the Method, where his method of scientific and mathematical thinking that helped him to create some of his most well-known concepts, such as analytical geometry, are revealed.

His step by step, analytical mindset and format influenced the geometry seen in the modern day classroom.

Descartes was born in La Haye en Touraine in the Kingdom of France on March 31, 1596. He was the youngest of three children and born into a family of relatively high political status. While his mom and sibling both died during childbirth when he was only one year old, his father, Joachim worked as a member of the Parliament of Brittany, the court of justice at the time.

Because of this, Descartes was able to obtain a small amount of nobility.

As a result of his parents’ absence, he lived most of his life in the home of his grandmother and great uncle, but kept in contact with his father throughout his life. Descartes’ family life was far from perfect, his father was so busy and had little time to spend with his children, thus why Rene lived with his grandma and great uncle.

His relationship with his sibling was shaky as well. His brother, Pierre did not even notify him about the loss of his father in 1640. The home life of Rene Descartes was in no way ideal, but that imperfection shaped his life. As a result of his insufficient upbringing as a child, he learned independence and self-thinking. This would serve as beneficial during his educational years.

Descartes’ father enrolled him into the Jesuit college of La Fleche in 1606 at approximately ten years of age. He left in 1614 and in 1615 he began studying at the University of Poitiers where he obtained a license in canon and civil law. Rene always had dreams of being in the military, but his father forced him to give up that dream and pursue a career in law. Nonetheless, in 1618 Descartes enlisted into the Dutch States army of Prince Maurice of Nassau where he spent the next three years as a soldier. It is believed that he spent the majority of his time in the army working on mathematical codes to aid in the building of military vehicles and weapons.

As for his military education, Descartes was stationed at an academy for noblemen as well. The academy was somewhat “structured around the educational model of Lipsius” (Gaukroger 65-66). Lipsius being a Dutch political theorist who studied at the Jesuit school, Cologne. Descartes met a man during his time stationed in Breda called Isaac Beeckman. That relationship is responsible for instilling Descartes’ interest in sciences again. Beekman's questions also inspired Descartes to create the Compendium Musicae, one of the most popular books about musical mathematics during the late seventeenth century.

Rene Descartes’ “La Géométrie” contained some of the most influential mathematical theorems and ideas to date, with many of his concepts still being used today in their original forms. “La Géométrie” was an appendix in Descartes’ famous philosophical book Discourse on Method. While it was in a book about philosophy, “La Géométrie” was completely independent of the philosophical aspects of the rest of the books. He had many goals and concepts in this appendix, but his main desire was to find a way to solve geometric problems in a way that could be represented with compasses and rulers. He was able to accomplish this by using algebra in geometry, which had never been done before. Descartes’ use of algebra in geometry, which is found in his book “La Géométrie”, was one of the most important discoveries in all of geometry, with it still being used today in many ways.

This algebra in geometry, which is known today as analytical or Cartesian geometry, came about through the use of real-life devices which he used to help turn the previously theoretical concepts of geometry into ones that can be used in the real world. Descartes created what he called “compasses” out of dirt and sticks on the ground. He would slide the sticks around in sliding grooved members in order to replicate curves, cubes, and trisecting angles on the earth. Descartes considered these compasses to be as useful and geometrical as any other tool, and this did turn out to be true. Using these compasses he was able to create the mathematical concept of curves, which were essential in the creation of analytical geometry.

These curves he created showed that algebra could be incorporated into geometry, which allowed many previously unsolvable problems to be solved. He was able to represent curves as having an x value, which represented its horizontal location, and a y value, which represented its vertical location. Once he had those values, he was able to convert the curves into algebraic problems and vice-versa, which was a game-changing discovery. Now, many “unsolvable” problems could easily be solved through geometry. By involving curves and algebra into geometry, Descartes was able to change the field of geometry into what is considered geometry today.

Descartes’ creation of the x and y-axes not only created a new way to solve problems, but also created a whole new coordinate plane that is the exact same one used in classrooms around the world today. Descartes did not actually label the axes x and y; they were just implied to be those letters. Even though he did not label these axes, his labeling of their values ending up creating the Cartesian coordinate system. Using his ideas that the x value was the horizontal value and the y value was the vertical value, he was able to label numbers along those axes that corresponded to the values. When two or more points on this chart were connected, they formed a line that could then easily be turned into an algebraic expression. Through his simple and easy to understand coordinate system, he was able to allow geometrical shapes to be easily converted in algebraic equations.

Aside from his creation of modern-day analytical geometry, he was also able to come up with many other important ideas and theorems that, while not as impactful as analytical geometry, are still useful. Descartes was actually the first person to ever use letters at the beginning of the alphabet (a, b, c, etc.) to represent known variables and to use letters at the end of the alphabet (x, y, z) to represent unknown values. The use of these particular variables can be seen in all types of equations from the simplest to the most complex, which helped unify all of algebra. He also was the one who started using exponential notation to signify the power of a number ( x2, x3, etc.) This simplified what had once been a tedious task of having to write a value times itself up to hundreds of times into an uncomplicated notation. Through his simplification and unification of concepts in algebra, he was able to make a small, but still very impactful, imprints on other parts of mathematics that were not analytical geometry.

The revolutionary influence that Descartes imposed on the geometric form of mathematics created more possible ways to reach solutions and an organized system of proving them. Having this framework of skepticism at hand, he doubted everything until it was proven which is arguably one of the main ideas of the proof system used in geometry. In Discussion of the Method, we are enlightened on the true beginning of the geometric exploration done by Descartes.

Part of the Cartesian method, put well by the “Internet Encyclopedia of Philosophy” as the “Method of Doubt,” was what drove the algebraic format to get involved in geometry, seeing as it was a way to conclude so many problems that were previously thought to be unsolvable. Descartes’ method when it came to scientific or mathematical discoveries based itself on four criteria: lack of assumption, simplicity, order, and inclusion. As a system used for both scientific and mathematical advancements, Descartes’ method stuck and provided the problem-solving aspect we see in class. Although the concept of analytical geometry is not directly noted and the coordinates are not explained to be Cartesian, the systematic approach of the mathematical process is still seen.

Descartes’ influence was not limited to the geometric field, integrating the algebraic concept actually created the form of calculus used today. His idea of analytical geometry established the foundation for Isaac Newton and Gottfried Leibniz to construct the form of calculus, and developing the rule of signs, Descartes allowed the highest possibility of solutions to be found in a polynomial equation. Additionally, he is credited with devising the standard superscript notation to show exponents. Furthermore, Descartes contributed to the majority of mathematical equations displayed in education. Speaking on behalf of his solving method Descartes explains that “each problem will be solved according to its own nature.” This sparked not only his infatuation for innovation but also his mindset of thinking.

Throughout the field of mathematics, Descartes’ influence can be found through his ways of thinking and revolutionary concepts. His ability to think through situations logically has helped him to impact the field of geometry in ways as massive as analytical and as small as exponents. He truly was one of the most influential people of all time, with him creating ideas in fields as vastly different, but equally important, as mathematics and philosophy. His influence in mathematics even spread past geometry into calculus, since his work was one of the major causes of Isaac Newton’s creation of calculus. Descartes’ innovative way of thinking led to the creation of many important ideas, theorems, and beliefs, which proves that his method of thinking through a situation logically and methodically is one a superior way of thinking that should be applied to all fields thinking in the modern-day.

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