- THE EARLIEST GREEK GEOMETRY 281
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Β Ã ÆFIGURE 9. Pappus' construction of a neusis using a rectangular hyperbola.catch-all category consisted of problems requiring all manner of more elaborate and
less regular curves, which were harder to visualize than the first two and presumably
required some mechanical device to draw them. Pappus says that some of these
curves come from locus problems, and lists the inventors of some others, among
them a curve that Menelaus called the paradox. Other spirals of the same type, he
says, are the quadratrices, the conchoids, and the cissoids. He goes on to say that
geometers regard it as a major defect when a planar problem is solved using conies
and other curves.
Based on this classification of problems, the first geometers were
unable to solve the abovementioned problem of [trisecting] the an-
gle, which is by nature a solid problem, through planar methods.
For they were not yet familiar with the conic sections; and for
that reason they were at a loss. But later they trisected the angle
through conies, using the convergence described below.
The word convergence (neusis) comes from the verb neuein, one of whose mean-
ings is to incline toward. In this particular case, it refers to the following construc-
tion. We are given a rectangle ΑΒÃΑ and a prescribed length m. It is required to
find a point Ε on ÃÄ such that when AE is drawn and extended to meet the exten-
sion of ΒÃ at a point Z, the line EZ will have length m. The construction is shown
in Fig. 9, where the circle drawn has radius m and the hyperbola is rectangular, so
that AA ÃÄ = \H • ZH.
Given the neusis, it becomes a simple matter to trisect an angle, as Pappus
pointed out. Given any acute angle, label its vertex A, choose an arbitrary point Ã
on one of its sides, and let Ä be the foot of the perpendicular from à to the other
side of the angle. Complete the rectangle ΑΒÃΑ, and carry out the neusis with
m = 2ΑÃ. Then let Ç be the midpoint of ZE, and join TH, as shown in Fig. 10.