The mathematical problem that Viviani proposed to the analysts in 1692 elicited numerous reactions in Europe. The reasons that stimulated so many illustrious mathematicians, especially Leibniz, to consider this problem are discussed as are the consequences that followed in the field of mathematics.The Aenigma, born as a challenge to the Italian traditional-classical mathematics in opposition to the rising Leibnizian calculus, proved to be very effective in demonstrating the power and superiority of this new calculus. Leibniz's first solutions are here interpreted (...) on the basis of his first unpublished draft and the definitive publication. Leibniz derived his inspiration from the results of Archimedes on the sphere and cylinder, and from the studies of Pascal and Gregory of Saint Vincent on spherical triangles and cylindrical nails.Two of Leibniz's unpublished manuscripts on the Aenigma are transcribed in the Appendices; these give respectively the first draft of his solutions published in the Acta Eruditorum, and the correction of a miscalculation that was pointed out by Jacob Bernoulli. (shrink)