**Cesàro** was an Italian mathematician whose main contributions were to differential geometry and divergent series.

- In 1860, the year after Ernesto was born, there was a revolution led by Giuseppe Garibaldi aimed at achieving Italian unification.
- On 17 March 1861, almost exactly two years after Ernesto's birth, the Kingdom of Italy was formally created.
- Luigi Cesàro strongly supported the move towards Italian unification but this was not an easy time for farmers in Italy (nor for many others) and Ernesto grew up difficult financial circumstances.
- The newly created country of Italy suffered many problems but it also had a new confidence in education from which Cesàro benefited in his early years.
- Cesàro entered the École des Mines as a student but, preferring to study in Italy, made application for a university place there.
- Sur diverses questions d'arithmétique Ⓣ(On various arithmetical problems) was the first of a series which Cesàro wrote on the theory of numbers.
- Cesàro visited Paris during the period of his studies at Liège and there he attended lectures by Hermite, Darboux, Serret Briot, Bouquet and Chasles at the Sorbonne.
- Hermite in particular was interested in the results which Cesàro had obtained and he quoted these in his own work of 1883.
- Cesàro was particularly interested in lectures he attended given by Darboux on geometry and this led him to make his own studies of intrinsic geometry along similar lines.
- Back in Liège after the trip to Paris, Cesàro fell out with one of the professors there and left for Italy without completing his studies.
- After one month at the Lycée Terenzio Mamiani, however, Cesàro was offered the chair of mathematics at Palermo and Cremona advised him to accept it.
- Cesàro's main contribution was to differential geometry.
- The Lezione di geometria intrinseca contains descriptions of curves which today are named after Cesàro.
- Cesàro later pointed out that in fact his geometry did not use the parallel axiom so constituted a study of non-euclidean geometry.
- In addition to differential geometry Cesàro worked on many topics such as number theory where, in addition to the topics we mentioned above, he studied the distribution of primes trying to improve on results obtained in this area by Chebyshev.
- Cesàro's interest in mathematical physics is also evident in two very successful calculus texts which he wrote.
- Cesàro died in tragic circumstances.

Born 12 March 1859, Naples, Kingdom of the Two Sicilies (now Italy). Died 12 September 1906, Torre Annunziata, Italy.

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Analysis, Geometry, Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive