as others.”^7 It seems that even in ancient Greece, they felt that another
name for the runner-up is “loser.”
Other problems in geometry, although not appearing to have conse-
quences as significant as plague prevention, perplexed the Greek mathe-
maticians. Two of these problems, squaring the circle and trisecting the
angle, were solved by the Greeks by reaching outside the realm of classi-
cal straightedge-and-compass constructions. The third problem, the con-
struction of regular polygons (a polygon is regular if all its sides are the
same length and if all the angles formed by adjacent sides are equal—the
square and the equilateral triangle are regular) with an arbitrary number
of sides, eluded them.
The term squaring the circle—constructing a square whose area is the
same as that of a given circle—is often used as a shorthand for an impossi-
ble task. As with doubling the cube, the task was not impossible. Archimedes
described a neat construction that began by “unrolling” the given circle to
produce a line segment whose length was the circumference of the given
circle.^8 However, unrolling is not a straightedge-and-compass construction.
Similarly, the task of trisecting the angle—constructing an angle whose
degree measure is one-third the degree measure of a given angle—can
easily be accomplished by making a mark on the straightedge that is being
used (a method that is also ascribed to Archimedes), which also lies outside
the framework of classical straightedge-and-compass constructions allowed
in Euclidean geometry. These constructions showed that the Greeks,
though recognizing the formal restrictions of Euclidean geometry, were
willing to search for solutions to problems even if those solutions could
only be found outside the system in which the problems were posed.
We do not know if the Greeks ever conjectured that these tasks could
not be accomplished within the framework of straightedge-and-compass
constructions. It is certainly easy to believe that a mathematician such as
Archimedes, having expended a good deal of effort on one of these prob-
lems, might well have reached such a conclusion. What we do know is
that even today, when the impossibility of such tasks has been proven to
the satisfaction of at least five generations of mathematicians, countless
man-hours are spent formulating “proofs” and sending the results to
mathematical journals. Some of the people who spend time on these
problems are unaware that mathematicians have proved that trisecting
the angle or squaring the circle is impossible.^9 Others are aware, but ei-
ther believe that mathematical impossibility is not an absolute, or the
proof of that impossibility is f lawed.
There are simple straightedge-and-compass constructions to construct
Impossible Constructions 71