282 10. EUCLIDEAN GEOMETRYFIGURE 10. Trisection of an arbitrary angle by neusis construction.A mechanical (curvilinear) solution of the neusis problem. Finding the point Ε in
the neusis problem is equivalent to finding the point Z. Either point allows the
line AEZ to be drawn. Now one line that each of these points lies on is known. If
some other curve that Æ must lie on could be drawn, the intersection of that curve
with the line ΒÃ would determine Æ and hence solve the neusis problem. If we use
the condition that the line ZE must be of constant length, we have a locus-type
condition for Z, and it is easy to build a device that will actually draw this locus.
What is needed is the T-shaped frame shown in Fig. 11, consisting of two pieces
of wood or other material meeting at right angles. The horizontal part of the Ô
has a groove along which a peg (shown as a hollow circle in the figure) can slide.
The vertical piece has a fixed peg (shown as a solid circle) at distance ΑÄ from its
top. Onto this frame a third piece is fitted with a fixed peg (the hollow circle) at
distance m from its end and a groove between the peg and the other end that fits
over the peg on the vertical bar. The frame is then laid down with its horizontal
groove over the line ÃÄ and its fixed peg over A. When the moving piece is fitted
over the frame so that its peg slides along the horizontal groove over ÃÄ and its
groove slides over the peg at A, its endpoint (where a stylus is located to draw the
curve) traces the locus on which Æ must lie. The point Æ lies where that locus
meets the extension of ΒÃ. In practical terms, such a device can be built, but the
rigid pegs must be located at exactly the distance from the ends determined by the
rectangle and the fixed distance given in the neusis problem. Thus the device must
be modified by moving the pegs to the correct locations for each particular problem.
If oxymoron is permitted, we might say that the practical value of this device is
mostly theoretical. The locus it draws is the conchoid of Nicomedes, mentioned by
Pappus and Proclus.
Because of the objections reported by Pappus to the use of methods that were
more elaborate than the problems they were intended to solve, the search for planar
(ruler-and-compass) solutions to these problems continued for many centuries. It
was not until the 1830s that it was proved that no ruler-and-compass solution
exists. The proof had no effect on the cranks of the world, of course. The problems
continue to be of interest since that time, and not only to cranks who imagine they
have solved them. Felix Klein, a leading German mathematician and educator in