Vito Volterra

Volterra’s wide scientific production concerns various subjects of mathematics and mathematical physics – terrestrial mechanics, rational mechanics, elasticity, hydrodynamics, electrodynamics, differential equations, integral equations (we mention the so-called integral equation of Volterra type) and their applications to biology and economical models. He also introduced the concept of function of line (later called functional by Hadamard), which opened a new and fruitful research field – known as functional analysis – a part of which was connecting the theory of integral and integro-differential equations. Volterra soon became an important public and political figure. In 1887 he was elected corresponding member of the Accademia Nazionale dei Lincei (in 1899 he became an ordinary member); in 1897 he founded the Italian Physics Society (SIF, Societa` Italiana di Fisica); in 1905 he was nominated Senator of the Kingdom. 
At the outbreak of World War I, Volterra enlisted in the engineers corps (Genio) although aged 55! In 1922, when Fascism came to power in Italy, Volterra immediately opposed it. In 1925, he signed the intellectuals’ manifesto against fascism – drawn up by Benedetto Croce – and in 1931 refused to take an oath of loyalty to Fascism. As a consequence of his refusal, he was expelled from the University of Rome and, in 1932, from all Italian cultural institutions. In 1940 he died isolated and with no official recognition.

Ars Magna, sive de regulis algebraicis

La formula risolutiva per le equazioni algebriche di quarto grado fù trovata subito dopo di quella per le equazioni cubiche. La formula è dovuta a Ludovico Ferrari (1522-1565), uno studente di Cardano, e fù pubblicata per la prima volta nel trattato "Ars Magna, sive de regulis algebraicis".

Tartaglia a Cardano (lettera 1539) Sulle soluzione di una equazione cubica

Quando che'l cubo con le cose appresso
Se agguaglia a qualche numero discreto:
Trovan dui altri, differenti in esso.
Dapoi terrai, questo per consueto,
Che'l lor produtto, sempre sia eguale
Al terzo cubo delle cose neto;
El residuo poi suo genérale,
Delli lor lati cubi, bene sottratti
Varra la tua cosa principale.

Nicola Oresme

La prima e semplice dimostrazione della divergenza della serie armonica e' dovuta a Nicola Oresme (1350).

Since Newton and Leibniz

Since Newton and Leibniz, infinite series a_0 +a_1 +a_2 +a_3  +...have been the universal tool for all calculations.