One possible answer is that hypergeometric series were (and are) used to compute periods of elliptic integrals.
In modern terminology, take a smooth cubic $X \subset \mathbb{P}^2$ whose Weierstrass form is
$y^2z=x(x-z)(x-\lambda z)y^2w=x(x-w)(x-\lambda w), \quad \lambda \in \mathbb{P}^1-\{0, 1, \infty\}$.
Then $X$ is an elliptic curve, then it can be written as $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice. It turns out that the generators $\omega_1(\lambda)$, $\omega_2(\lambda)$ of $\Lambda$, i.e. the periods of the associated Weierstrass $\wp$-function
$\wp(z; \Lambda):=\frac{1}{z^2} + \sum_{l \in \Lambda-0} \big(\frac{1}{(z-l)^2}-\frac{1}{l^2} \big)$,
can be written in terms of the standard hypergeometric series $F$, namely
$\omega_1(\lambda)=i\pi F(\frac{1}{2},\frac{1}{2} , 1, 1-\lambda)$,
$\omega_2(\lambda)=i \pi F(\frac{1}{2}, \frac{1}{2}, 1, \lambda)$.
For further details see Chapter 1 of Kobliz's book "Introduction to elliptic curves and modular forms".
