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A Solution to the Basel Problem that Uses Euclid’s Inscribed Angle Theorem

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Abstract

We present a short, rigorous solution to the Basel Problem that uses Euclid's Inscribed Angle Theorem (Proposition 20 in Book III of the Elements) and can be seen as an elaboration of an idea of Leibniz communicated to Johann Bernoulli in 1696.
A solution to the Basel Problem that uses Euclid’s
Inscribed Angle Theorem
David Brink
20 December 2013
Abstract. We present a short, rigorous solution to the Basel Problem that uses Euclid’s
Inscribed Angle Theorem (Proposition 20 in Book III of the Elements ) and can be seen as
an elaboration of an idea of Leibniz communicated to Johann Bernoulli in 1696.
We present a short, rigorous solution to the Basel Problem in the form
X
n=0
1
(2n+ 1)2=π2
8.
Proof. Mercator’s Formula
log(1 + z) =
X
n=1
(1)n+1
nzn
converges uniformly on the closed unit disk minus a neighborhood of –1. To see this, multiply
by 1 + zand note that the resulting series converges uniformly on the entire closed unit disk.
In particular
log 1 + eix=
X
n=1
(1)n+1
neinx for π < x < π, ()
convergence being uniform on closed subintervals. We may therefore integrate termwise to
obtain the antiderivative
F(x) =
X
n=1
(1)n+1
in2einx.
This series even converges uniformly on the entire real axis. Continuity at x=πthus gives
the (improper) integral
Zπ
0
log 1 + eixdx =F(π)F(0) = 2i
X
n=0
1
(2n+ 1)2.
The imaginary part of the integrand is
Im log 1 + eix= arg 1 + eix =x
2,
where the second equality follows from the Inscribed Angle Theorem:
In a circle the angle at the centre is double of the angle at the circumference, when
the angles have the same circumference as base.
This is Proposition 20 in Book III of Euclid’s Elements, quoted here from Heath’s translation
[2]; see also Figure 1. Hence
Im Zπ
0
log 1 + eixdx =π2
4,
and the claim follows.
1
Figure 1.
According to Heath [3, pp. 201–202], the main propositions of Book III of the Elements were
in fact known already to Hippocrates of Chios, a contemporary of Socrates.
Euler gave four solutions to the Basel Problem, using Taylor series and product formulas
for various trigonometric functions, cf. [1]. Myriads of other solutions have been found since
then, involving techniques from many different areas of mathematics. For an overview, the
reader is encouraged to see the highly readable paper of Kalman [4]. We now discuss the
relations between three previous solutions and the one given here.
Russell [7] essentially integrates the right-hand side of the real part
Re log 1 + eix= log
1 + eix
= log 2 cos x
2.
Note, however, that, contrary to this proof, we have avoided appealing to Abel’s Limit
Theorem—which would in any case have been unjustified since we are in effect integrating
along the circle of convergence of the Mercator Series.
The reader might have noticed the Fourier series
x
2=
X
n=1
(1)n+1
nsin(nx) for π < x < π
lurking in the background. In fact this classical formula follows directly from () by taking
imaginary parts. A standard textbook proof now proceeds by invoking Parseval’s Identity.
In two letters to Johann Bernoulli dated 6 and 9 November 1696, Leibniz discussed the
Basel Problem and showed
Zlog(1 + x)
xdx =x
1x2
4+x3
9x4
16 etc.,
but his subsequent attack on the integral was unsuccessful.1Of course, the Basel Problem was
to remain open until Euler’s first solution in 1734. Ultimately, however, Leibniz was vindicated
as there is a direct way to evaluate the integral, using a suitable functional equation. The
details and history of this proof are given by Kalman and McKinzie [5]. Alternatively, we can
equip Leibniz’s integral with the limits ±1 and view the result as a contour integral along a
straight line. If we then instead integrate along the semicircle γ(x) = eix, 0 xπ, thereby
quite literally circumventing the roadblock mentioned in [5], we get the following elaboration
1See Leibniz [6, pp. 174–180]. Ayoub [1], who gives much interesting information on the history of the Basel
Problem, quotes passages from the second of these letters but erroneously dates it to 1673.
2
of Leibniz’s idea:
2
X
n=0
1
(2n+ 1)2=Z1
1
log(1 + x)
xdx = Im Zπ
0
log 1 + eixdx =π2
4.
From a philosophical point of view, the idea of solving the Basel Problem by replacing an inte-
gral along the diameter of the unit circle with one along the perimeter explains the surprising
appearance of πin a natural way. On the other hand, changing the path of integration obvi-
ously needs some justification which can be avoided simply by integrating along the perimeter
from the outset, as we have done here.
References
[1] Raymond Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067–1086.
[2] Euclid, Στoιχια, Alexandria, ca. 300 B.C. Translation: Thomas L. Heath, The Thirteen
Books of the Elements, second edition, volume 2 (Books III–IX), Dover, New York, 1956.
Originally published by Cambridge University Press, Cambridge, 1908.
[3] Thomas Heath, A History of Greek Mathematics. Volume I. From Thales to Euclid, Dover,
New York, 1981. Originally published by Clarendon Press, Oxford, 1921.
[4] Dan Kalman, Six ways to sum a series, College Math. J. 24 (1993), 402–421.
[5] Dan Kalman, Mark McKinzie, Another way to sum a series: generating functions, Euler,
and the dilog function, Amer. Math. Monthly 119 (2012), 42–51.
[6] Gottfried Wilhelm Leibniz, amtliche Schriften und Briefe, Reihe III, Band 7, Akademie-
Verlag, Berlin, 2011. Available electronically at www.leibniz-edition.de/Baende/
[7] Dennis C. Russell, Another Eulerian-type proof, Math. Mag. 60 (1991), 349.
3
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Available electronically at www.leibniz-edition
  • Gottfried Wilhelm Leibniz
  • Sämtliche Schriften Und Briefe
  • Iii Reihe
Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe III, Band 7, Akademie- Verlag, Berlin, 2011. Available electronically at www.leibniz-edition.de/Baende/
A History of Greek Mathematics. Volume I. From Thales to Euclid
  • Thomas Heath
Thomas Heath, A History of Greek Mathematics. Volume I. From Thales to Euclid, Dover, New York, 1981. Originally published by Clarendon Press, Oxford, 1921.
Translation: Thomas L. Heath, The Thirteen Books of the Elements
  • Στ Euclid
  • Alexandria Oιχχια
Euclid, Στ oιχχια, Alexandria, ca. 300 B.C. Translation: Thomas L. Heath, The Thirteen Books of the Elements, second edition, volume 2 (Books III–IX), Dover, New York, 1956. Originally published by Cambridge University Press, Cambridge, 1908.