Squaring a circle
|Squaring a circle, i.e. constructing
a square whose area equals that of a given circle, is one of the three famous
construction problems of antiquity. Its (negative) solution was eventually
obtained in 1882 from the following theorem by the German mathematician
Note that, in the simplest
case of a single term ea, the theorem implies that the only point on the
graph of y = ex with both x and y rational is (0,1). Although the set
of rational points is dense in the plane, the graph of y = ex somehow
manages to cut the plane without passing but through just one of them.
The theorem is also associated
with the French mathematician Hermite(1822-1901) who, in 1873, proved
the special case in which the coefficients and exponents were rational
numbers. Hermite applied his theorem to prove that e, the base of natural
logarithms, is transcendental. (Indeed, if P(e) = 0 for a polynomial with
integer coefficients, then we'll get an expression (*) with a's being
Indeed, on the left we have
an expression in the form (*). Since it equals zero, the exponent i can't
be algebraic. Hence, is transcendental.
Thus it's impossible to square
a circle using a straightedge and a compass; but like the problem of angle
trisection, this one can be solved by other means. Have a look at the
diagram on the right. Assume a circle of unit radius is rolled half a
turn on a straight line. Then the distance between the points A and B
will be exactly . If we draw a semicircle on AC = AB+1 as a diameter,
and continue the vertical radius of the right circle to the intersection
with the semicircle at a point D, then AB*BC = BD2. Which, of course,
solves the famous problem because AB = and BC = 1.
squaring a circle
R.Courant and H.Robbins, What
is Mathematics?, Oxford University Press, 1996
H.Dorrie, 100 Great Problems
Of Elementary Mathematics, Dover Publications, NY, 1965.
W.Dunham, Journey through Genius,
Penguin Books, 1991
M.Kac and S.M.Ulam, Mathematics
and Logic, Dover Publications, NY, 1968.
R.B.Nelsen, Proofs Without
Words, MAA, 1993