  In this paper we prove Garvan's conjectured formula for the square of the
modular discriminant $\Delta$ as a 3 by 3 Hankel determinant of classical
Eisenstein series $E_{2n}$. We then obtain similar formulas involving minors of
Hankel determinants for $E_{2r}\Delta^m$, for $m=1,2,3$ and $r=2,3,4,5,7$, and
$E_{14}\Delta^4$. We next use Mathematica to discover, and then the standard
structure theory of the ring of modular forms, to derive the general form of
our infinite family of formulas extending the classical formula for $\Delta$
and Garvan's formula for $\Delta^2$. This general formula expresses the
$n\times n$ Hankel determinant $\det(E_{2(i+j)}(q))_{1\leq i,j\leq n}$ as the
product of $\Delta^{n-1}(\tau)$, a homogeneous polynomial in $E_4^3$ and
$E_6^2$, and if needed, $E_4$. We also include a simple verification proof of
the classical 2 by 2 Hankel determinant formula for $\Delta$. This proof
depends upon polynomial properties of elliptic function parameters from
Jacobi's Fundamenta Nova. The modular forms approach provides a convenient
explanation for the determinant identities in this paper.
