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.

Cauchy is mad

Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused . . .

(Abel 1826, Oeuvres, vol. 2, p. 259)

Johann Bernoulli rule (l'Hopital's rule) 2

Johann Bernoulli rule (l'Hopital's rule) 1

G. F. de l'Hopital (1661-1704) - French mathematician, a capable student of Johann Bernoulli, a marquis for whom the latter wrote the first textbook of analysis in the years 1691-1692. The portion of this textbook devoted to differential  calculus was published in slightlyvaltered form by l'Hopital under his own name.
Thus "l'Hopital's rule" is really due to Johann Bernoulli.

Extremum

Briefly, but less precisely, one can say that if the derivative changes sign in passing through the point, then the point is an extremum, while if the derivative does not change sign, the point is not an extremum.

Gödel's Incompleteness Theorem

In the summer of 1930, the twenty-four-year-old mathematician Kurt Gödel proved a strange theorem: mathematics is open-ended. There can never be a final, best system of mathematics. Every axiom-system for mathematics will eventually run into certain simple problems that it cannot solve at all. This is Gödel's Incompleteness Theorem.
The implications of this epochal discovery are devastating. The thinkers of the Industrial Revolution liked to regard the universe as a vast preprogrammed machine. It was optimistically predicted that soon scientists would know all the rules, all the programs. But if Godel's Theorem tells us anything, it is this: Man will never know the final secret of the universe.
Of course, anyone can say that science does not have all the answers.
What makes Gödel's achievement so remarkable is that he could rigorously prove this, stating his proof in the utterly precise language of symbolic iogic. To come up with a mathematical proof for the incompleteness of mathematics is a little like managing to stand on one's own shoulders. How did Godel come to think of such a proof? What kind of person was he?

L'histoire est un théorème indémontrable.

 L'histoire est un théorème indémontrable.

J.-F. Revel ., Mémoires. Le voleur dans la maison vide, Paris (PIon), 1997.

L'INTEGRALE DE STIELTJÈS.

En 1894, Stieltjès, à l'occasion de recherches relatives à des développements en fractions continues, a défini un nouveau mode d'intégration des fonctions continues. II importe de bien comprendre l' originalité de la généralisation de Stieltjès et en quoi elle diffère profondément de celles que nous avons examinées ju qu'ici.

Henri Lebesgue (1928)

Cauchy Cauchy

There is, of course, nothing new about Cauchy's calculations of derivatives. Nor is there anything particularly new about the theorems Cauchy was able to prove about derivatives. Lagrange had derived the same results from his own definition of the derivative. But because Lagrange's definition of a derivative rested on the false assumption that any function could be expanded into a power series, the significance of Cauchy's works lies in his explicit use of the modem definition of a derivative, translated into the language of inequalities through his definition of limit, to prove theorems. The most important of these results, in terms of its later use, was in Lesson 7.

The Riemann Integral

In 1853, Georg Bernhard Riemann (1826-1866) attempted to generalize Dirichlet's result by first determining precisely which functions were integrable according to Cauchy's definition of the integral ∫  f(x) dx. He began, in fact, by changing the definition somewhat.

many highly respected motives

There are many highly respected motives which may lead men to prosecute research, but three which are much more important than the rest. The first (without which the rest must come to nothing) is intellectual curiosity, desire to know the truth. Then, professional pride, anxiety to be satisfied with one’s performance, the shame that overcomes any self-respecting craftsman when his work is unworthy of his talent. Finally, ambition, desire for reputation, and the position, even the power or the money, which it brings. It may be fine to feel, when you have done your work, that you have added to the happiness or alleviated the sufferings of others, but that will not be why you did it. So if a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desired to benefit humanity, then I should not believe him (nor should I think the better of him if I did). His dominant motives have been those which I have stated, and in which, surely, there is nothing of which any decent man need be ashamed.

G. H. Hardy - A Mathematician's Apology.

Henry Lebesgue (1928)