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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

1−x2

4+x3

9−x4

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 ≤x≤π, 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, Στoιχια, 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, S¨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.

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