A direct evaluation of the periods of the Weierstrass zeta function

We show how to obtain the difference function of the Weierstrass zeta function very directly, by choosing an appropriate order of summation in the series defining this function. As a byproduct, we show how to obtain the quasi-modularity of the weight 2 Eisenstein series immediately from the fact that it appears in this difference function and the homogeneity properties of the latter.