Monday, April 21, 2008 4/21/08-Descartes

In today's excerpt--Rene Descartes (1596-1650), one of the key figures of the Scientific Revolution; highly influential philosopher, mathematician, and writer; and most famously author of the philosophical proclamation--"I think, therefore I am":

"Towards the end of 1619, the most important event of Descartes's life occurred, and we know exactly when and where it happened because he tells us in his book The Method. ... It was on 10 November 1619 that Descartes first saw the road to his own philosophy and also had one of the greatest mathematical insights of all time.

"Idly watching a fly buzzing around in the corner of the room, Descartes suddenly realized that the position of the fly at any moment in time could be represented by three numbers, giving its distance from each of the three walls that met in the corner. Although he instantly saw this in three-dimensional terms, the nature of his insight is now known to every schoolchild who has ever drawn a graph. Any point on the graph is represented by two numbers, corresponding to distances along the x axis and up the y axis. In three dimensions, you just have a z axis as well. The numbers used in the system of representing points in space (or on a piece of paper) in this way are now known as Cartesian co-ordinates, after Descartes. ...

"Descartes's discovery meant that any geometric shape could be represented simply by a set of numbers--in the simple case of a triangle drawn on graph paper, there are just three pairs of numbers, each specifying one of the corners of the triangle. And any curved line drawn on paper (or, for example, the orbit of a planet around the Sun) could be represented, in principle, by a series of numbers related to one another by a mathematical equation. When this discovery was fully worked out and eventually published, it transformed mathematics by making geometry susceptible to analysis using algebra, with repercussions that echo right down to the development of the theory of relativity and quantum theory in the twentieth century.

"Along the way, it was Descartes who introduced the convention of using letters at the beginning of the alphabet (a, b, c ...) to represent known or specified quantities, and letters at the end of the alphabet (especially x, y, z) to represent unknown quantities. And it was he who introduced the now familiar exponential notation, with x2 meaning x times x, x3 meaning x times x times x and so on. If he had done nothing else, laying all these foundations of analytical mathematics would have made Descartes a key figure in seventeenth-century science. But that was far from all he did."

John Gribbin, The Scientists, Random House, Copyright 2002 by John and Mary Gribbin., pp. 110-112.


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