- ANALYTIC AND ALGEBRAIC GEOMETRY 355
FIGURE 3. Descartes' analysis of the ç-line locus problem.general: "the most brilliant victory ever achieved by a man whose genius was
applied to reducing the need for genius" (quoted by Davis and Hersh, 1986, p. 7).
This point was ignored by Newton in a rather ungenerous exhibition of his own
remarkable mathematical talent (Whiteside, 1967, Vol. IV, pp. 275-283). Newton
said that Descartes "makes a great show" about his solution of the three- and four-
line locus problems, "as if he had achieved something so earnestly sought after by
the ancients." He also expressed a distaste for Descartes' use of symbolic algebra
to solve this problem (a distaste that would be echoed by other mathematicians),
saying that if this algebra were written out in words, it "would prove to be so
tedious and entangled as to provoke nausea." One is inclined to say, on Descartes'
behalf, "Precisely! That's why it's better to use algebraic symbolism and avoid the
tedium, confusion, and nausea."
1.3. Newton's classification of curves. Like Descartes, Newton made a clas-
sification of curves according to the degree of the equations that represent them,
or rather, according to the maximal number of points in which they could inter-
sect a straight line. As Descartes had argued for the use of any curves that could
be generated by one parameter, excluding spirals and the quadratrix because they
required two independent motions to be coordinated, Newton likewise argued that
geometers should either confine themselves to conic sections or else allow any curve
having a clear description. In his Universal Arithmetick, he mentioned in particular
the trochoid,^4 which makes it possible to divide an angle into any number of equal
parts, as a useful curve that is simple to describe.
1.4. Algebraic geometry. As we have just seen, Descartes began the subject of
algebraic geometry with his classification of algebraic curves into genera, and New-
ton gave an alternative classification of curves also, based on algebra, although he
included some curves that we would call transcendental, curves that could intersect
a line in infinitely many points. The general study of algebraic curves p(x, y) = 0,
where p(x, y) is a polynomial in two variables, began with Colin Maclaurin (1698-
1746), who in his Geometria organica of 1720 remarked that a cubic curve was not
uniquely determined by nine points, even though nine points apparently suffice to
determine the coefficients of any polynomial p(x, y) of degree 3, up to proportional-
ity and hence determine a unique curve p(x, y) = 0. Actually, however, two distinct
(^4) A trochoid is the locus of a point rigidly attached to a rolling wheel. If the point, lies between
the rim and the center, the trochoid is called a curtate cycloid. If the point lies outside the rim,
the trochoid is a prolate cycloid. If the point is right on the rim, the trochoid is called a cycloid.
The names come from the Greek words trokhos (wheel) and kyklos (circle).