280 10. EUCLIDEAN GEOMETRY
FIGURE 8. The quadratrix of Hippias.the regular heptagon. Surprisingly, however, the regular 17-sided polygon can be
constructed using only compass and straightedge. Since 9 = 3 · 3, it would seem
natural to begin by trying to construct this figure, that is, to construct an angle of
40°. That would be equivalent to constructing an angle of 20°, hence trisecting the
angles of an equilateral triangle.
Despite the seeming importance of this problem, less has been written about
the ancient attempts to solve it than about the other two problems. For most of the
history we are indebted to two authors. In his commentary on Euclid's Elements,
Proclus mentions the problem and says that it was solved by Nicomedes using
his conchoid and by others using the quadratrices of Hippias and Nicomedes. In
Book 4 of his Synagogi (Collection), Pappus says that the circle was squared using
the curve of Dinostratus and Nicomedes. He then proceeds to describe that curve,
which is the one now referred to as the quadratrix of Hippias.^8
The quadratrix is described physically as follows. The radius of a circle rotates
at a uniform rate from the vertical position AB in Fig. 8 to the horizontal position
AA, while in exactly the same time a horizontal line moves downward at a constant
speed from the position ΒÃ to the position A A. The point of intersection Æ traces
the curve BZH, which is the quadratrix. The diameter of the circle is the mean
proportional between its circumference and the line AH. Unfortunately, Ç is the
one point on the quadratrix that is not determined, since the two intersecting lines
coincide when they both reach AA. This point was noted by Pappus, citing an
earlier author named Sporos. In order to draw the curve, which is mechanical, you
first have to know the ratio of the circumference of a circle to its diameter. But if
you knew that, you would already be able to square the circle. One can easily see,
however, that since the angle ZAA is proportional to the height of Z, this curve
makes it possible to divide an angle into any number of equal parts.
Pappus also attributed a trisection to Menelaus of Alexandria. Pappus gave a
classification of geometric construction problems in terms of three categories: pla-
nar, solid, and [curvi]linear. The first category consisted of constructions that used
only straight lines and circles, the second those that used conic sections. The last,(^8) Hippias should be thankful for Proclus, without whom he would apparently be completely
forgotten, as none of the other standard commentators discuss him, except for a mention in
passing by Diogenes Laertius in his discussion of Thales. Allman (1889, pp. 94-95) argues that
the Hippias mentioned in connection with the quadratrix is not the Hippias of Elis mentioned in
the Eudemian summary, and other historians have agreed with him, but most do not.