René Descartes (1596 - 1650)

Descartes had a habit of getting out of bed at 11am but became a critical character in the intellectual development of the world and is regarded as the modern founder of both mathematics and philosophy.

He wrote the first important book in the French language (rather than Latin) and enabled mathematicians to understand each other - even if no one else could. Mathematics is a language, and like any language it has different dialects and alphabets.

Choosing the right notation is one way of making it universal, and indeed it is possible, for example, for a Chinese mathematician with no English to write mathematics that an English speaking mathematician can read and understand.

If you read pre-Descartes mathematics book you would have difficulty understanding the mathematics because the notation would be totally unfamiliar. For example x^3 would be written as xxx.

Descartes was the first to start using modern style notation. He didn’t use our equals sign, but he did use plus and minus and represented known entities with a, b, c and unknowns with x, y, z.

He was born in Toulouse, France and after completing a law degree, he lived the life of a gentleman travelling to Paris and through Europe before settling in Holland. This was a haven in a continent ripped apart by religious intolerance and Descartes settled down to write a book on physics.

Just as he was about to publish, the religious troubles caught up with him. He heard of Galileo’s arrest and as his book was also based on the Copernican view he stopped publication.

Instead he turned to write Discours de la methode, a treatise on science. This dismissed the Aristolian logic on which most European thought was based. Mathematics, he felt, was the only certain thing and all thought should be based on this.

An appendix to this work was on geometry. It led to what we now know as Cartesian geometry and brought all the algebraic tools to the geometric arena - this developed into a subject which was rich in results and techniques. Oughtred and al-Khwarizmi had also attempted to do this, but Descartes was more thorough than his predecessors.

It is said that the idea for coordinates came to Descartes in his bed. Lying in bed he saw a spider crawling on his ceiling, and realised that its position could always be determined by its distances from the edges.

But mathematical discovery cannot always be done in bed. In 1649 Descartes broke the habit of a lifetime. Queen Christine of Sweden persuaded him to come to Stockholm and be her tutor. The winter was bitter that year but the young Queen insisted that her lessons commence at 5am. Poor Descartes, used to his leisurely, meditative mornings succumbed to pneumonia and did not last the winter.

Descartes’ mathematics

Descartes basic idea was to use axes to define all the points in a plane. Today, we denote the vertical axis as the y axis and the horizontal one as the x axis. Each point then has coordinates (x,y) where x and y denote the distance from these two axes.

Using this it is easy to find the algebraic form of a curve or locus. The circle is the locus of points that are at a fixed distance from its centre. Consider any point P on the circle with coordinates (x,y). If the radius of the circle is r, then by Pythagoras’ Theorem we have

r^2=x^2+y^2.

We call this the equation of the circle. Using this we can discover algebraically the many geometric properties that it has. Let us deal with the unit circle whose equation is,

x^2+y^2=1.

For example, let us introduce a line whose slope is 1. It’s equation will be of the form

y=x+c

where c is the intercept on the y axis. Does the line intersect the circle - and if so where? Simple, we need to solve the equations simultaneously - when they meet they share the same values for x and y. We’ll concentrate on the y coordinate. The equation of the circle can be written,

y^2=1-x^2

and substituting the equation of the line into this gives,

(x+c)^2=1-x^2

this equation tells us the x coordinate of the points where the line and the circle meet - and it is points, because the equation is a quadratic with two roots. Big deal you say - you’ve done all that algebra just to prove that a line intersects a circle in two points! Not so fast. It gives a great deal more. Let’s first write the equation in the usual form,

2x^2+2cx+c^2-1=0.

The roots of this equation are determined by the value of a quantity called the discriminant.

For the equation ax^2+bx+c=0 this is b^2-4ac. In our case it is,

4c^2-8(c^2-1)

which may be simplified to,

4(2-c^2).

Can the line be a tangent? In this case, the two roots of the equation must be equal and the discriminant zero. So the quantity inside the first bracket must be zero, that is,

2-c^2=0.

This equation for c has the two roots c=\pm\sqrt{2}; the line is a tangent to the circle for these values of c:

The method works just as well in three dimensions - though now there will be 3 coordinates. It may be extended to any number of dimensions, and the properties of curves and surfaces may be explored through their equations. Geometry was never the same again following Descartes breakthrough.