150 A History ofMathematics
considered Descartestherevolutionary who had freed them from bondage to the tedious methods
of the ancient Greeks, by reducing hard geometric problems to simple algebraic ones. This is a view
which is often now regarded with some suspicion, although Descartes himself promoted it:I have given these very simple [methods] to show that it is possible to construct all the problems of ordinary geometry
by doing no more than the little covered in the four figures that I have explained. [That is, the figures which construct
a sum, a product, a quotient, and a square root.] This is one thing which I believe the ancient mathematicians did
not observe, for otherwise they would not have put so much labour into writing so many books in which the very
sequence of propositions shows that they did not have a sure method of finding at all, but rather gathered together
those propositions on which they had happened by accident. (Descartes 1954, p. 17)It is noteworthy that Descartes here isnotclaiming to be rediscovering an ancient technique. In
fact, the simplicity of his methods, he claims, is a proof that the ancients did not have them—or
they would have found his results. It is sometimes claimed that he was unoriginal—the graphical
representation came from Oresme, and the algebra from Viète. Descartes did acknowledge his debt
to Viète, specifically defending himself against charges of difficulty by claiming (which he nowhere
states in theGéométrie) that he supposed his readers to be familiar with theAnalytic Art.^14 In
any case, his project was different and specific: the relation of geometry and algebra. A standard
modern textbook criticizes Descartes for not being more practical:Our account of Descartes’ geometry should make clear how far removed the author’s thought was from the practical
considerations that are now so often associated with the use of coordinates. He did not lay out a coordinate frame
to locate points as a surveyor or a geographer might do, nor were his coordinates thought of as number pairs...
La géométriewas in its day just as much a triumph of impractical theory as was theConicsof Apollonius in antiquity.
(Boyer and Merzbach 1989, pp. 385–6)This criticism is interesting, but, I think, misplaced. Coordinate geometry even today is not ‘intrins-
ically’ practical—even the statistician who studies whether points in a scatter graph lie near a
straight liney=ax+b, let alone the geometer who wishes to picture the curvey^2 =x^3 +x^2
(Fig. 2) are not thinking as surveyors or geographers. On the other hand, forsomepractical tasks,
the new ideas were very well adapted, as Newton and Leibniz were to understand. Galileo takes a
great deal of trouble to establish using Apollonius’Conicsthat a projectile describes a parabola, a
fact which follows very easily by finding its equation; and while Descartes does not deal with results
of this kind (his physics was too different from Galileo’s, and mostly confused), they are simplified
and clarified by using the methods which are to be found in his book. To see this, and to see how,
unlike Viète, he avoided the Euclidean heritage of formal definitions, propositions, and proofs,
I have given the basic construction in which coordinates first appear as Appendix B. The idea is to
draw a curve by using a simple-minded machine (a ruler which pivots, subject to constraints), and
to find the equation of the curve. The description of the machine seems more complicated than it
is in practice, and the derivation of the equation is not hard. At the end, the curve is said to be ‘of
the first kind’, by which Descartes means a conic section; the reason being that its equation is of
the second degree (quadratic) inxandy. Note that the use of machines for drawing curves could
be seen as a typically practical Renaissance innovation; but like much else, it has a long heritage,
both Greek (Eratosthenes) and Islamic, though neither is acknowledged by Descartes.
However, besides inventing a new method and a new notation, Descartes was introducing a new
style of writing mathematics, which was also to have considerable influence. All previous books in
- Letter to Mersenne, 1637 (Descartes 1939 t. II, p. 66.)