Introduction

Abstract

Euclid Vindicated from Every Blemish, by the Jesuit mathematician Gerolamo Saccheri, appeared in print in Milan in 1733. The work enjoyed little success in the eighteenth century, was completely forgotten during the following century, and was rediscovered and circulated only in the early nineteen hundreds. Today it is rather well-known, at least in outline, and is usually considered to be the birthplace of research on non-Euclidean geometry. The strategy of Euclid Vindicated is also widely regarded as one of the largest misunderstandings in the whole history of mathematics − and the most felicitous error in eighteenth-century geometry − as Saccheri’s intention was in fact to demonstrate Euclid’s Fifth Postulate, the parallel axiom, and thus to prove the impossibility of the very non-Euclidean geometries of which he is today regarded as the father. He undertook to prove the Fifth Postulate per absurdum and sought to spot a contradiction in the vast geometric theory that he constructed, for the first time in history, on the negation of the Euclidean axiom – a geometric theory that we nowadays identify without doubt as a genuine and well-structured system of hyperbolic geometry. Saccheri did not find the supposed contradiction, as it was nowhere to be found, but he was unable to convince himself that the new geometry he had erected might in fact be a reasonable alternative to Euclid’s Elements rather than a green-eyed monster: consequently, he pointed to a contradiction of his own making, and thereby proved himself to be nothing more than a Jesuit. This effort notwithstanding, the sacrilege, so to speak, had already been committed, and Saccheri’s outstanding achievements towards the construction of hyperbolic geometry, while disowned by their author and relegated to a book printed in quite few copies, sneaked into European mathematical culture and poisoned the minds of certain more acute, unprejudiced, or simply more modern geometers. One century after the Jesuit’s death, these scholars eagerly welcomed Saccheri’s ‘monster’ in their writings, thus celebrating the triumph of non-Euclidean geometry. Following this widespread story, Saccheri unwittingly (yet brilliantly) anticipated one of the most momentous conceptual revolutions in the genesis of contemporary mathematics.