Customer Review

November 24, 2005
This is the theme of the book: Euclid built up the whole Elements with just ruler and compass, but, as the Greeks knew perfectly well, for many problems it is necessary to allow higher curves or curve-tracing devices, or other suspicious tricks. This is very bothersome; we don't want things to get out of hand. What methods should be allowed in true geometry? An appreciation of these issues it is absolutely essential for a proper understanding of Descartes' Geometrie, which is in a sense a culmination of this whole tradition (and which of course many of us have tried to read for very different reasons).

The pre-Cartesian picture can be summed up as follows. "In 1588 the publication of Pappus' Collection had provided a new impetus to the interest in geometrical problems solving. Soon afterward the writings of Viete set in motion the principal dynamics in the field: the creation and elaboration of algebraic methods of analysis." But by the 1630's impetus was lost among "a proliferation of special studies on special problems, several, but unstructured methods of analysis, and little interest in elaborating the new algebraic techniques beyond their application to solid problems [i.e., conics]." (p. 221)

In other words, algebraic methods were subordinated to the classical geometrical framework of constructions. In retrospect we can say that Descartes brought synthesis to the chaotic proliferation of geometrical methods by inverting this relationship. But this was not a step taken lightly; rather it grew gradually out of Descartes's extensive work on constructions in a more old-fashioned style.

In 1619 Descartes proposed to add to the Euclidean construction tools "new compasses, which I consider to be no less certain and geometrical than the usual compasses by which circles are traced." He justified his "new compasses" on the grounds that they traced curves "from one single motion," contrasting it with the "imaginary" curves traced by "separate motions not subordinate to one another" (p. 232), such as the quadratrix and exponential curves. At this point "Descartes had not yet developed the idea of a correspondence between a curve and an equation" (p. 247).

Around 1625 Descartes discovered a general method for constructing the roots of third and fourth degree equations by the intersection of a parabola and a circle. According to a contemporary source, "Descartes values this invention so much that he avows never to have found anything more outstanding, indeed that nothing more outstanding has been found by anybody" (p. 255). This discovery meant that a problem could be considered solved according to the classical construction standards whenever it had been reduced to an equation of degree three or four, especially since Descartes's method gave the simplest possible construction (an important requirement classically). This discovery therefore legitimised a shift of attention to algebraic equations as primary means of dealing with curves. Furthermore, since equations of degree one and two correspond to ruler and compasses, it also suggested that the classical criterion of always using the simplest mode of construction corresponded precisely to the degree of the equation.

In 1631, prompted by a challenge, Descartes tackled Pappus's line-locus problem. His solution of this problem was based on yet another kind of "new compasses" (the "turning rules and moving curve" of section 19.4, which fits the criterion that the curves it generates "arise from one single motion"). Descartes believed that the curves that could be traced in this way were exactly the set of all algebraic curves and also exactly all curves that were solutions to Pappus problems. In particular, the "imaginary curves" that Descarte was so keen to exclude in 1619 fall outside of this class.

Only when algebraic curves were legitimised in this way did Descartes consider his reconception of the foundations of geometry in algebraic terms, and the justification for it in terms of tracing motions remained crucial in his Geometrie. We thus see that Descartes's revolutionary algebraisation of geometry was in no way conceived as a radical break with the past; rather it grew out of and aligned perfectly with Descartes's pre-algebraic vision of the foundations of geometry.
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