- ANALYTIC AND ALGEBRAIC GEOMETRY 353
AC Ε GFIGURE 1. Descartes' linkage for generating curves. The curve
x4n = a^2 (x^2 + y^2 )2n~l is shown for ç = 0,1,2,3.the point on the oblique line directly above C is D. The line perpendicular to
the oblique line at D intersects the horizontal line at E, from which a vertical
line intersects the oblique line at F, and so forth in a zigzag pattern. Descartes
imagined a mechanical linkage that could actually draw these curves.
Descartes regarded determinate curves of this sort, depending on one parame-
ter, as we would say, as legitimate to use in geometry. He offered the opinion that
the opposition to "mechanical" curves by ancient Greek mathematicians arose be-
cause the curves they knew about—he mentioned the spiral of Archimedes and the
quadratix—were indeterminate. In the case of the spiral of Archimedes, which is
generated by a point moving at constant linear velocity along a line that is rotating
with constant angular velocity, the indeterminacy arises because the two velocities
need to be coordinated with infinite precision. For the quadratrix, the same prob-
lem arises, as the ratio of the velocity of a rotating line and that of a translating
line needs to be known with infinite precision.
Descartes' Geometrie resembles a modern textbook of analytic geometry less
than does Fermat's Introduction to Plane and Solid Loci. He does not routinely use
a system of "Cartesian" coordinates, as one might expect from the name. But he
does remove the dimensional difficulties that had complicated geometric arguments
since Euclid's cumbersome definition of a composite ratio.
[U]nity can always be understood, even when there are too many
or too few dimensions; thus, if it be required to extract the cube
root of a^262 — b, we must consider the quantity a^2 b^2 divided once
by unity, and the quantity b multiplied twice by unity. [Smith and
Latham, 1954, P- 6]Here Descartes is explaining that all four arithmetic operations can be per-
formed on lines and yield lines as a result. He illustrated the product and square
root by the diagrams in Fig. 2, where AB = 1 on the left and FG = 1 on the right.
Descartes went a step further than Oresme in eliminating dimensional con-
siderations, and he went a step further than Pappus in his classification of locus
problems. Having translated these problems into the language of algebra, he real-
ized that the three- and four-line locus problems always led to polynomial equations
of degree at most 2 in a; and y, and conversely, any equation of degree 2 or less rep-
resented a three- or four-line locus. He asserted with confidence that he had solved