On the Basel Problem and the Square of Gregory’s Series

the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler's formula, or its equivalent formulation ζ(2)=π2/6, based on the idea of squaring 1−13+15−⋯=π4, including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of ζ(2)=π2/6 that also relies on squaring Gregory's series.