Derivation of certain sums "the hard way"

Ramanujan's original paper On certain arithmetical functions gives a direct proof. The ideas behind this proof are closely related to the usual modular forms proof, but the words Eisenstein, vector space, modular forms are not mentioned. Indeed Ramanujan says explicitly that his identities "are of course really results in the theory of elliptic functions" and that "the elementary proof of these formulae given in the preceding sections seems to be of interest in itself."

Briefly, Ramanujan writes down the power series for $P$ (which is $E_2$), $Q$ (which is $E_4$) and $R$ (which is $E_6$), and shows that $$ \Phi_{r,s}(x) = \sum_{n=1}^{\infty} x^n \Big(\sum_{ab=n} a^r b^s\Big) $$ can be expressed by polynomials in $Q$ and $R$. Then he multiplies $\Phi_{0,r}$ and $\Phi_{0,s}$ and compares the answer with $\Phi_{1,r+s}$ and $\Phi_{0,r+s+1}$.

A really different proof, and the one Zagier refers to, is by Nils Skoruppa: "A quick combinatorial proof of Eisenstein series identities", J. Number Theory 43 (1993), 68--73.