TITLE: Why are integrals over cycles called periods?
QUESTION [13 upvotes]: In the statement of de Rham theorem, a pairing is defined as follows:
$H_i(X, \mathbb R) \times H^i_{\mathrm{de Rham}}(X) \rightarrow \mathbb R$
It is given by
$\left( \left( \sum a_i \gamma_i \right) , \omega \right) \mapsto \sum a_i \int_{\gamma_i} \omega $.
Here $a_i$ takes real values, and $\gamma_i$ are integral homology cycles, and $\omega$ is an $i$-form.
The integral given above is called the period of the above integral, and the isomorphism given by this pairing is sometimes called the period isomorphism.
Question:

Why is the above integral of a closed form over a cycle called a period?

My peeve with this terminology of "period" is that it does not agree at all with anything else I know about this word. They are the following: 1) The period of a periodic function. 2) Periods as generalizations of algebraic numbers, as integrals of algebraic(rational) expressions over domains in Euclidean spaces defined by algebraic inequalities.
Indeed, this strange use of the word "period" is used even by Ahlfors in his book on complex analysis, for integrals of holomorphic(rather, meromorphic? depending on the domain ...) functions over loops.
I do not understand why on earth the word period appears in this setting of integrating on abstract manifolds. True, the cycles capture some underlying geometry of the space. But why is something a "period" when you integrate a form over a cycle? Why is the integral of a $1$-form over a line segment not a period?(Or is it also period, in some definition I am not aware of?)

REPLY [12 votes]: I must admit that this is the first time I heard the suggestion that
the word "period" came into complex analysis/algebraic geometry from
the period of the pendulum. The word was certainly used in astronomy
before the theory of the pendulum was known -- e.g. Kepler stated his
third law in 1618 as "The squares of the periodic times are to each
other as the cubes of the mean distances" -- and uniform circular motion
gives a "period" that was known before the period of the pendulum.
In any case, the concept of "period" is not a big deal until one
discovers functions with two (or more) periods, which first happened
when Gauss discovered this property of the lemniscate sine function in
1797. The lemniscatic sine, $sl(u)$,is the inverse function of the 
elliptic integral
$
u=\int^{x}_{0} \frac{dt}{\sqrt{1-t^4}},
$
and Gauss called its real period $2\varpi$, using the variant form of
the Greek letter pi. Presumably this was to stress its analogy with the
period $2\pi$ of the sine function.