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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. 
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. Find an online Math tutor for your kids here. 
“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. 
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.
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