constructable numbers. Simple trigonometry showed that the sine and
cosine of a 20-degree angle were not constructable numbers.^14 Addition-
ally, Wantzel polished off the problem of which regular polygons were
constructable by showing that the only such regular polygons were those
with n sides, where n is a product of a power of 2 and any number of Fer-
mat primes.
Jean Claude Saint-Venant, who was one of the leading French mathema-
ticians of the period and a colleague of Wantzel’s, described his habits as
follows: “He usually worked during the evening, not going to bed until
late in the night, then reading, and got but a few hours of agitated sleep,
alternatively abusing coffee and opium, taking his meals, until his mar-
riage, at odd and irregular hours.” Saint-Venant further commented upon
Wantzel’s failure to achieve more than he had (even though his achieve-
ments would do credit to 99 percent of the mathematicians who have ever
lived) by further stating, “I believe that this is mostly due to the irregular
manner in which he worked, to the excessive number of occupations in
which he was engaged, to the continual movement and feverishness of
his thoughts, and even to the abuse of his own facilities.”^15
The Impossibility of Squaring the Circle
By the middle of the nineteenth century, the lengths of line segments
that could be constructed had been shown to be the result of applying ad-
dition, subtraction, multiplication, division, and the taking of square
roots to integers (since cube roots cannot be obtained by this process, the
cube could not be doubled nor the angle trisected). In order to square a
circle of unit radius, since the area of the circle is , one must be able to
construct a line segment whose length is the square root of , which can
only be done if one can construct a line segment whose length is .
By this time, mathematicians had shown that the real line consisted of
two types of numbers: the rational numbers such as^22 / 7 , which could be
viewed as either the quotient of two integers or the ratio of one integer to
another, and the irrational numbers, those which could not be expressed
as ratios. As we have seen, the Pythagoreans knew that the square root of
2 is irrational; this knowledge was so well known to educated Greeks that
a proof of it appears in one of the Socratic dialogues.^16 The irrational
numbers had been further subdivided into the algebraic numbers, those
numbers that were the roots of polynomials with integer coefficients, and
the transcendental numbers. In 1882, the German mathematician Ferdi-
nand von Lindemann wrote a thirteen-page paper showing that was
transcendental, thus showing that the circle could not be squared by
74 How Math Explains the World