stoutfellow

There is, no doubt, much more to be said about Descartes' impact on mathematics, but I'll confine myself to one more subject. Euclid had restricted the attention of mathematics to the straight line and the circle, and though Apollonius had studied conic sections and Archimedes spirals, they sat uneasily within the Euclidean framework. Descartes, questioning these strictures as he questioned everything else, chose to break the bounds Euclid had imposed.

In Book II of La Géométrie, Descartes questions the ancient restrictions.

The ancients were familiar with the fact that the problems of geometry may be divided into three classes, namely, plane, solid, and linear problems. This is equivalent to saying that some problems require only circles and straight lines for their construction, while others require a conic section and still others require more complex curves. I am surprised, however, that they did not go further, and distinguish between different degrees of these more complex curves, nor do I see why they called the latter mechanical, rather than geometrical.

He goes on to point out that, if it comes to the depiction of curves, the straightedge and compass are as much machines as any more complex devices, and that, so long as the development of a curve can be precisely described, there is no justification for ignoring it. He describes, in general terms, a class of machine, composed of sliding rods, ligatures, and pen-holding points, and declares that any curve which can be drawn by such a machine deserves mathematical consideration.

Here he enters slightly murky water, for it is not altogether clear what the limits on the machines are, nor what distinguishes curves which can be so drawn from curves which cannot. He does, a bit later, propose that curves which can be described by polynomial relations (not that he used the phrase) are the curves in question: conic sections, describable by quadratic equations; curves "of the second class", represented by third or fourth degree equations; and so on up the ladder, each class allowing the degree to rise by two. (Note that this classification is only possible in the light of Descartes' invention of exponents!) Why this set of curves should be the same as that given by his machines is not clear. Furthermore, such interesting curves as the cycloid - studied by Galileo, among others of Descartes' predecessors - are not constructible by Descartes' machines, though they are easy to draw by mechanical means, and those means are every bit as precise as those Descartes described. (Let a circle roll along a straight line; the path followed by a point on the circumference of the circle is a cycloid. Any child who's played with a SpiroGraph has encountered this curve and its relatives.)

Descartes wanted to escape from the narrow bounds of straightedge and compass, but he was unwilling to admit complete freedom; indeed, he wanted to justify his new curves as being as precisely definable as the circle and the line. However, the cork once off the bottle, it was impossible to confine the new realm of geometry even within Descartes' looser bounds. The curves that mathematics was henceforward to study grew stranger and stranger, until at last, in the nineteenth century, it was realized that some sort of restrictions did need to be imposed. That, though, is a story for another day.

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