  It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric
function ${}_2F_1$ arises from a relation between modular curves, namely the
covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ the N-fold
cover of $X_0(N)$ by $X_0(N^2)$ gives rise to an algebraic hypergeometric
transformation. The N=2,3,4 transformations are arithmetic-geometric mean
iterations, but the N=5,6,7 transformations are new. In the final two the
change of variables is not parametrized by rational functions, since
$X_0(6),X_0(7)$ are of genus 1. Since their quotients $X_0^+(6),X_0^+(7)$ under
the Fricke involution (an Atkin-Lehner involution) are of genus 0, the
parametrization is by two-valued algebraic functions. The resulting
hypergeometric transformations are closely related to the two-valued modular
equations of Fricke and H. Cohn.
