Le soluzioni di Girolamo Saccheri e Giovanni Ceva al ‘Geometram quaero’ di Ruggero Ventimiglia: Geometria proiettiva italiana nel tardo seicento

Abstract

Girolamo Saccheri's first printed work, his Quaesita Geometrica of 1693, has always been considered a minor one. It is not our intention wholly to reverse this judgement, but it is our considered opinion that it is something a deal more than a mere student's exercise. Above all, the solutions of its first two problems gave Saccheri, helped by his friend and colleague Giovanni Ceva, the opportunity to rediscover, by a systematic employment of the properties of harmonic and anharmonic ratio, many important theorems of Desargues, La Hire, and others. We accordingly stress that this work is an extension of Ceva's most famous book, his De lineis rectis, written fifteen years earlier.

The recent edition, moreover, of Newton's manuscript work on “Geometria”, and Whiteside's careful notes upon it have given us a new slant on the “refound” interest in classical geometry during the latter seventeenth century. We have compared Saccheri's book with standard contemporary equivalent works, in Italy itsef (especially ones by Borelli and Viviani) and elsewhere in Europe (notably ones by La Hire and Newton). We have now verified to our satisfaction not only that the development of Ceva's and Saccheri's mathematical thinking derived in an organic way from the study of the pole-polar properties of conics in the Italian tradition, but that they were also greatly influenced (in their search for unifying theoretical principles) by the ideas of René Descartes even while refusing to follow his preference for algebraic technicalities.

We give, finally, a brief sketch of Ruggero Ventimiglia, a man now all but forgotten, but who is shown by his Enodationes duodecim problematum of 1690 to have been following a program of projective study of “Analysis Geometrica”, one which his premature death (at the age of 28) made impossible to pursue to completion.