Ernesto Cesàro was an Italian mathematician. He worked as professor at Sapienza University almost his whole life.

Background

Cesàro was born on March 12, 1859, in Naples, Italy, the son of Luigi Cesàro and Fortunata Nunziante, his second wife. The elder Cesàro owned a farm and shop in Torre Annunziata; he was one of the first farmers in Italy to introduce agricultural machinery, a supporter of Italian unification, and a backer of Garibaldi’s revolution of 1860 - all of which led him into financial difficulties.

Education

Cesàro completed the first class of the Gymnasium in Naples, studied in the seminary of Nola for two years, then returned to Naples to finish the fourth class of the Gymnasium in 1872.

In 1873 his father sent him to Liège to join his older brother Giuseppe, who had gone there in 1867. Cesàro stayed for a year with his brother, and then entered the École himself on a scholarship. He matriculated there, then applied unsuccessfully for admission to an Italian university, following which he was forced to enter the University of Rome, from which he graduated in 1887.

He studied mathematics with Eugène Catalan, who noticed Cesàro’s talent and helped him publish his first mathematical paper in Nouvelle correspondance de mathématiques, of which Catalan was an editor.

His paper, “Sur diverses questions d’arithmétique,” brought Cesàro to the attention of the mathematical public. He returned to Torre Annunziata and sought to continue his work in Italy. His mathematical works and the recommendations of Cremona, Battaglini, and Dino secured him a scholarship to the University of Rome, which he entered in 1884 in the fourth form in pure mathematics. Here, in addition to attending a great many lectures, he wrote some eighty works - on infinite arithmetics, isobaric problems, holomorphic functions, theory of probability, and, particularly, intrinsic geometry - in the two years 1884-1886.

In 1886 Cesàro won a competition for the position of professor of mathematics at the Lycée Terenzio Mamiani in Rome; in similar competitions at the universities of Messina and Naples he placed first and second, respectively. On Cremona’s advice, Cesàro left the Lycée Mamiani after a month to fill the vacant chair of higher algebra at the University of Palermo. He stayed at Palermo until 1891, when he accepted the chair of mathematical analysis at Naples. He held this chair until his death, never realizing his intention of going over to the chair of theoretical mechanics.

Cesàro’s bibliography is extensive; indeed, the author of the most complete bibliography available, A. Perna, mentions 259 works and expresses doubt whether his list is complete. Cesàro’s topics are varied. In 1878, when he was nineteen, he attempted to master certain topological problems in a non-traditional way in his Forme poUedrichi regolari e semi - regolari in tutti gli spazii (published in Lisbon in 1888). The most prominent of his early works, however, deal with the sums of divergent series, for Cesàro, Bord, Fejér, and Voronoj were together creating the techniques for the elaboration of such problems. One of Cesàro’s first published works, Sur diverses questions d’arithmé tique (Liège, 1882), and, more importantly, his series of nine articles published in Annali di mathematica pura ed applicata are related to the theory of numbers. He was here concerned with such problems as the determination of the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series. Later he occupied himself with prime numbers of a certain type and tried to make Chebyshev’s formulas more precise.

Despite the generally sophisticated level of his mathematics, Cesaro reverted to such elementary problems as, for example, his work on constructions using limited geometrical means (1899) which repeats results already known. His textbooks, on the other hand, are rather exacting. They were successful and influential in their time; Corso di analisi algebrica con introduzione al calculo infinitesimale was published in Turin in 1894 and Elementa di calcolo infinitesimale appeared in Naples in 1899. Both texts were the outgrowth of Cesiro’s lectures in Palermo and Naples and both were distinguished by the pertinent and novel exercises that they contained. The two textbooks also reveal Cesaro’s interest in the problems of mathematical physics. In addition, his textbook had dealt with the theory of elasticity in an elementary way; there is no doubt that he planned to investigate mathematical physics in more detail, since he prepared two works on this subject, “Teoria matematica dels calore” and “Lezione sull'idrodinamica”; he died before he could publish these, however, and they remain unpublished to this day.

Cesaro’s most important contribution remains his intrinsic geometry. It has been noted that he began to develop it while he was in Paris in 1883; it occupied him, with interruptions, from that time on. He returned to this subject in the last years of his life and emphasized the independence of his geometry from the axiom of parallels. The special selection of the square of the linear element enabled him to extend the results to multidimensional spaces with constant curvature.

Throughout his life, the variety of Cesàro’s interests was always remarkable, ranging from elementary geometrical problems to the application of mathematical analysis; from the theory of numbers to symbolic algebra; from the theory of probability to differential geometry.

Connections

Around 1882 Cesàro returned to Italy, where he married his stepbrother’s daughter Angelina.