What is Descartes' Legacy to Mathematics?

When discussing the history of mathematics, it is impossible to skip over Rene. A great scientist of the 17th century, Descartes left an influential mark on his era due to his innovations in the notation of geometry and his concept of analytic geometry. The creator of the infamous phrase “cogito ergo sum,” Descartes and his discoveries are taught in every school around the world. Aristotle, Spinoza, Kant, Pythagoras, philosophical thought, metaphysical foundations, reasoning, intuition, rational thinking – if calculation and philosophy is your passion, you’re sure to love Rene Descartes and his history!

The Life of the Mathematician

Born in 1596 in the village of the Haye, today re-baptized as simply Descartes, Rene Descartes is one of the most well-known French intellectuals. Raised in a bourgeois family, Rene Descartes was brought up by his father, an adviser to the Parliament of Brittany, and grandmother. His education started the Jesuit college of the town Fleche, a school that was notorious for its strict rules and heavy course load. Created by Henri IV, the school provided Descartes with the opportunity to develop his mathematical sense and scientific interests. Descartes' education sets him apart from some other famous mathematicians. In fact, much is not known about the early academic career for many ancient mathematicians, although we do know a lot about the schools they started.

He continued his higher education at the University of Poitiers, where he studied law and obtained a bachelor’s degree. Despite receiving his education in this domain, Descartes never actually worked in law or politics. Instead, the young man chose to enlist in the European army (the Bavarois army) and took advantage of this occasion to discover European countries. In 1682, Descartes decided to move to the Netherlands, where he prepared a work of science called “The World” or “Treatise on Light.” It was in this body of work that Descartes managed to described a number of physical phenomena that explained the functioning of the world. Most notably, Descartes explained, in following with the data of Copernicus and Galileo, that the world revolved around the sun. While Descartes goal was to publish the text, the scientist was condemned by the Church during in the context of the Inquisition in 1633. Because of this, he decided to postpone the publishing of the book for several more years. If you want to know more about the way some ruling bodies impeded the work of famous mathematicians, check out Pythagoras' story! After finishing this text, he wrote a work that to this day is still as famous and reputable as it was at its publishing: Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences, otherwise known as Discourse on the Method. Studied in schools the world over, this influential text was published in 1637. What was so astonishing, and what touched his contemporaries, was the fact that the book was written in French instead of Latin, as was tradition of scientific works of the time. The work was accompanied by three essays on geometric optics, the laws of refraction (also called dioptics), on meteors and meteorology, and the last on geometry. It was in this last essay that Descartes detailed the relationship between geometry and algebra. While this relationship is normal for us in the context of today’s current mathematics curriculum, he was actually the first person to describe it in what became known as analytic geometry. Descartes published many other works that marked his career, works that covered a broad range of topics and principles: from maths and philosophy to metaphysics and the history of philosophy. Some of these works include Principles of Philosophy in 1644 and The Passions of the Soul in 1649. Victim to the frigid Scandinavian temperatures, the scientist and philosopher died of pneumonia in 1650 while he was called upon by the queen of Sweden. Find maths tutors with a proven track record here.

Algebra in the Context of Descartes

While Descartes completed Discourse on the Method in the 17th century, the scientist made several choices that both challenged his contemporaries and marked the domain of algebra. Most notably, Descartes expressed unknown values and truths by letters. While today these notations seem normal to us, at the time these letters were not at all traditional. In fact, historians have struggled over the text of many ancient mathematical works for centuries! It was Francois Viète, a mathematical contemporary of Descartes, that was the first to introduce these letters to algebraic formulas. It was Descartes, however, who expanded upon this and applied it to the notation of the math performed the most celebrated essay in Discourse on the Method: the essay on geometry. It is in this last essay that we find the letters x,y,z designated to unknowns in the equations used, and the letters a,b,c to designate values already known. Descartes also used this method and applied it to the notation of exponents, where he changed the expression of powers from xxxx to x4. The equal sign was also not yet known in Descartes’ time. Subtraction, on the other hand, was actually expressed by two negative signs. The expression of the square went untouched by Descartes.

In the domain of algebra, Descartes also introduced the term “imaginary numbers” to talk about complex numbers, which are:

“An imaginary number is the product of ai, for any ai where a signifies a real number and i an imaginary one.”

Above all, Descartes is recognized in mathematics for having connected mathematical calculations with the geometry concerning planes. Descartes named this analytic geometry, and he used it to make the express the relationship between geometric shapes through equations, using both coordinates and graphical representations.

Math and Descartes: A System of Coordinates

While the names of many philosophers, scientists and mathematicians of the past remain obscure, Descartes is one name we have surely all heard in and outside of class before. There is a reason why maths, history and philosophy courses don’t skip over Descartes: he was the first to prove the relationship between lines and curves through mathematical equations. Analytic geometry began at the start of Descartes and it is defined as:

“The established correspondence between geometric shapes and algebraic equations, also known as coordinate geometry.”

The discovery of analytic geometry can be boiled down to a central principle, where Descartes reports that points of the same curve at two axes, but of the same origin, can be better explained thanks to a system of coordinates. Legend has it that Descartes first thought to use coordinates by observing a fly that was hovering over the squares in a window, and saw that the points where the fly landed on these squares could be used to establish the coordinates of the plane. While Descartes was influenced by many of antiquity's greatest mathematicians, it was his own knowledge that developed what is today known as Cartesian coordinates. While the system of coordinates were actually first invented by Leonardo de Vinci, Descartes was the first one to utilize then to translate curves and lines through calculating arithmetic. The parabolic curve actually owes its discovery to Descartes.

What differs from the system of coordinates we study today is that Descartes only took positive coordinates into account. These points represented the precise segments of the geometric shape where the values must be positive Descartes also gave his name to a system of equations, where the equation could be given as the shape a curve. The equation, called Cartesian equation, took the form:

ax + by + cz + d = 0 with (a,b,c) = / = (0,0,0)

For example: For one line passing through A (1,3), originating at -4, the Cartesian equation would be y= 7x – 4. For a plane passing through A(1,1,2), B(1,0,1) and C(0,2,1), the Cartesian equation would be: 2x +y – z = 1.

What is Descartes' Lasting Mark on Mathematics?

Trigonometry, algebraic reasoning, equation, fraction, logarithm - maths courses today are still marked by the scientific discoveries made by Rene Descartes. It is impossible to skip over this mathematical giant when describing the history of mathematics. All of our equations utilize letters to designate the known or unknown values. These modern notations are then the base of our mathematical learning which starts in primary school and follows us until high school. This notation becomes even more complex for those who choose to continue to study math in their higher studies. Without Descartes, many of the notations we use today would be completely unrecognizable; we would still mark “quadratus” and “cubus” to note the powers x² and x³. While notation is certainly important, Descartes also made the influential recognition that geometrical problems could be transformed into numerical ones. This analytical geometry now plays a major part in the what goes into the mathematical criteria of national education. Descartes is also associated with Cartesian thought, also known as Cartesianism, which is a form of philosophical and scientific school of thought that deals with metaphysicality and rationalism that went strictly against what was known as empiricism. Cartesians were encouraged to adopt a mindset wherein they were to view humans as dual, in that their mind and matter were two finite materials. While this might sound strange to wrap your head around, Cartesian thought inspired people like Sir Isaac Newton and Gottfried Wilhelm Leibniz, who both developed calculus.