Descartes was born in La Haye en Touraine, in France, in 1596. However, he spent much of his life in Holland where he found the more liberal environment conducive to his philosophical views. Although best remembered as “the father of modern philosophy”, whose work is still the basis of much philosophical discourse, he was also very active as a scientist and as a mathematician.

His main focus in mathematical thought was to try to combine algebra and geometry. In this he succeeded spectacularly. His ground-breaking treatise “Discours de la méthode” had an appendix, “La Géométrie”, in which he described for the first time the representation of two-dimensional points on a plane using numbered coordinates, hence the term “Cartesian coordinates.” He discovered how points on a line could represent an algebraic function, such as *f(x)* = 2*x* + 1, and how functions such as quadratics, reciprocals, exponentials all led to families of curves – how exciting it must have been to see the results on graphs for the

very first time.

Being able to think of equations both algebraically and geometrically (“analytic geometry”) led to a massive new body of mathematics, from the straightforward solution of simultaneous equations to the development of calculus by both Leibniz and Newton. It also led to mathematical and philosophical considerations of higher dimensions – the rules which were developed in 2-D geometry can be applied to 3-D geometry and beyond. (For a wonderful account leading to a peep behind the curtain of the fourth dimension, read “Flatland” by the Victorian teacher and mathematician Edwin Abbott. In the first half of the book he describes the fictional Flatland, a two dimensional world; in the second half he describes a visit to Lineland, where all the beings live on a single line, and also a visit to Flatland by a three dimensional being. In this way, he helps us consider how a four dimensional being might appear to us in our 3-D world).

Although Cartesian geometry was undoubtedly Descartes’ greatest contribution to mathematics, he also left us modern algebraic notation whereby single letters in the first half of the alphabet represent constants, and those at the end of the alphabet represent variables (we should all understand this in an expressions such as *ax*^{2} + *bx* + *c*). He also invented the superscript notation for powers.

It is not surprising that he was able to develop whole new areas of mathematics, a discipline whose truths can be proved without reference to the outside world. His philosophical method, from an early age, was to doubt everything, and then build up again from ground level, using his own experience and observations. This also underpinned his scientific method, and led to such opposition from the church that his works were placed on the Catholic church’s list of prohibited books in 1653, 13 years after his death in Sweden.

### Descartes: interesting facts

1. His birthplace has been renamed Descartes in his honour.

2. “I think, therefore I am.”

3. During one night in 1619 he had three visions which led to all his future work.

4. He never got out of bed before 11am.

5. He made all his money early in life from a good investment (hence fact number 4)!