G. V. Schiaparelli and Plane Geometric Transformations: his Contribution and Fundamental Error

The Italian astronomer Giovanni Virginio Schiaparelli (1835- 1910), during his youthful period, also devoted himself to geometric research. Some of his studies were published in "Sulla trasformazione geometrica delle figure ed in particolare sulla trasformazione iperbolica" (1864) and mentioned by Luigi Cremona (1830-1903) in "Sulle trasformazioni geometriche delle figure piane. Nota I." (1863). This paper first reconstructs the genesis of Schiaparelli’s 1864 memoir, then goes on to analyze its main contents to define Schiaparelli’s contribution to research on geometric transformations and Cremona’s work. However, Cremona pointed out that the astronomer came to the incorrect conclusion that birational transformations of the plane are only linear or quadratic. Apparently, neither Cremona nor others cared about analyzing the reasons for the mistake the astronomer made. This work tries to fill this gap, pointing out the error, analyzing and explaining it.