Today we credit Pierre Wantzel with the first proof (1837) of the impossibility of doubling a cube and trisecting an arbitrary angle by ruler and compass. However two centuries earlier Descartes had ...put forward what probably counts as the first proof of these impossibilities. In this paper I analyze this proof, as well as the later related proof given by Montucla (1754) and the brief version of this proof published by Condorcet (1775). I discuss the many novelties of these early arguments and highlight the problematic points addressed by Gauss (1801) and Wantzel. In particular I show that although Descartes developed many of the algebraic techniques used in later proofs he failed to provide an algebraic impossibility proof and resorted to a geometric argument. Montucla and Condorcet turned this proof into an algebraic one. I situate the analysis of the early proof of the impossibility of the two classical problems in the general context of early modern mathematics where mathematics was primarily viewed as a problem solving activity. Within such a paradigm of mathematics impossibility results arguably do not play the role of proper mathematical results, but rather the role of meta‐results limiting the problem solving activity.
