most splendid triumphs

Everybody can sympathise with Cauchy students who just wanted to pass their exams and with his colleagues who just wanted standard material taught in the standard way. Most people neither need nor want to know about rigorous analysis.
But there remains a small group for whom the ideas and methods of rigorous analysis represent one of the most splendid triumphs of the human intellect. T. W. Körner - A Companion to Analysis A Second First and First Second Course.

l'Hôpital controversy

Bernoulli was hired by Guillaume François Antoine de L'Hôpital to tutor him in mathematics. Bernoulli and L'Hôpital signed a contract which gave L'Hôpital the right to use Bernoulli’s discoveries as he pleased. L'Hôpital authored the first textbook on calculus, "l'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes", which mainly consisted of the work of Bernoulli, including what is now known as L'Hôpital's rule.

Irrazionali densi nei reali...

Irrazionali densi nei reali...come sempre nella vita.

Invention of the devil

Abel wrote in 1828 : "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever".
In the ensuing period of critical revision they were simply rejected. Then came a time when it was found that something after all could be done about them. This is now a matter of course, but in the early years of the past century the subject, while in no way mystical or unrigorous, was regarded as sensational, and about the term "divergent series", now colourless, there hung an aroma of paradox and audacity.

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)

merely a humbler variation

A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is a merely a humbler variation of the first, which is the only answer we need consider seriously.

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of satisfying them than a mathematician. His subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.

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

The proof of Fermat's Last Theorem

Professor Andrew Wiles in 1993 presented his proof to the public for the first time at a conference in Cambridge. In August 1993 however, it turned out that the proof contained a gap. In desperation, Andrew Wiles tried to fill in this gap, but found out that the error he had made was a very fundamental one. According to Wiles, the crucial idea for circumventing, rather than closing this gap, came to him on 19 September 1994. Together with his former student Richard Taylor, he published a second paper which circumvented the gap and thus completed the proof. Both papers were published in 1995 in a special volume of the Annals of Mathematics.

The Most Famous Problem in Mathematics (Fermat's Last Theorem)

Observations on Diophantus  had been written by Fermat in the margin next to Diophantus' Problem 8 in Book II, which asks 'Given a number which is a square, write it as a sum of two other squares.' Fermat's note said (in Latin) that:

On the other hand, it is impossible for a cube to be written as a sum
of two cubes or a fourth power to be written as a sum of two fourth
powers or, in general, for any number which is a power greater
than the second to be written as a sum of two like powers. I have
a truly marvellous demonstration of this proposition which this
margin is too narrow to contain.
In algebraic terms, Diophantus' problem asks for rational numbers
x, y, z satisfying the equation

x^2 + y^2 = Z^2.

This turns out to be a fairly easy task. What Fermat's marginal comment asserts is that if n is a natural number greater than 2, then the equation

x^n + y^n = z^n

has no rational solutions.

The Weierstrass Function July 18 1871 - II

The Gödel Incompleteness

At the turn of the century, the world-famous German mathematician David Hilbert proposed a programme for the development of all of mathematics within the strict formalization of the axiomatic method. According to Hilbert's belief, all of mathematics could be regarded as the formal, logical manipulation of symbols based on prescribed axioms. (This would mean that, in principle, a computer could be programmed to 'do all of mathematics'.) But in 1930, with two startling and totally unexpected theorems, the young Austrian mathematician Kurt Gödel demonstrated that Hilbert's programme could not possibly succeed.

Niccolò Tartaglia

Apart from his solution of the cubic equation, Tartaglia is remembered for other contribution to science. It was he who discovered that a projectile should fired at 45 degrees to achieve maximum range. His conclusion was based on incorrect theory, however, as is clear from Tartaglia's diagrams of trajectories, Tartaglia's Italian translation of the Elements was the first printed translation of Euclid in a modern language.

Bourbaki

J’apprends que Grothendieck n’est plus membre de Bourbaki. Je le regrette beaucoup, ainsi que les circonstances qui ont amené cette décision...Ce qui importait, c’est une opposition systématique, plus ou moins explicitée selon les uns ou les autres, contre son point de vue mathématique, ou plutôt son emploi par Bourbaki....C’est un scandale que Bourbaki, non seulement ne soit pas à la tête du mouvement functorial, mais encore n’y soit même pas à la queue. . . . Si certains membres fondateurs (e.g., Weil) désirent revenir sur leur décision de ne pas influencer Bourbaki dans la direction qu’il désire prendre, qu’ils le disent explicite- ment....Si Bourbaki refuse, non pas de se mettre dans le nouveau mouvement, mais d’en prendre la tête, alors les traités visant à la redaction des éléments des mathématiques (et pas seulment à ceux de la géométrie algébrique) seront rédigés par d’autres, qui s’inspireront non pas de l’esprit de Bourbaki 1960, mais de son esprit 1939. Ce serait dommage.

The Weierstrass Function July 18 1871 - I

Funzione continua ma non derivabile in nessun punto.

harmony and regularity

... In effect, if one extends these functions by allowing complex values for the arguments, then there arises a harmony and regularity which without it would remain hidden.

B. Riemann, 1851

plague of functions

I turn away in fright and horror from this lamentable plague of functions that do not have derivatives.

C. Hermite, 1893

Rolle’s theorem

Rolle’s theorem is a simple but important result, familiar to anyone who has moved just beyond elementary calculus into the beginnings of analysis. Essentially it tells us that if a differentiable function has equal values at a and b, then somewhere between those two points it must have a local maximum or a local minimum.

A more formal statement of the theorem, typical of those given in modern textbooks, is as follows.

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there exists a point c in (a, b) for which f ́(c) = 0.

Karl Weierstrass 2

........But Weierstrass did not leave Berlin. He wrote to Kova-

levskaya about Kronecker:

"I deeply regret that such a gifted person, possessing

such indisputable scientific merits, can also be so envious

and petty in his vanity."

Karl Weierstrass 1

Karl Weierstrass, Professor of Mathematics at Berlin

University, was famous as a great scientist, Leo Koenigsberger

being one of his students.

Weierstrass was a giant of thinking and has left an

indelible mark on mathematics. His name is familiar to

everyone who studied the theory of complex variables.

Proceeding from the theory of real numbers he had developed

he provided a logical substantiation for mathematical

analysis, while his theory of analytic functions is very

significant.

The name Weierstrass is deeply ingrained in many sections

and theorems of mathematics: the Bolzano-Weierstrass

theorem, the Weierstrass theory of elliptic functions; the

study of sufficient conditions for the maximum of an integral

(in the calculus of variations); the geodesic lines and minimal

surfaces in differential geometry; the theory of elementary

divisors in linear algebra; the application of series in the

theory of analytic functions (in 1841, when Weierstrass was

twenty-six, he knew of the theorem Laurent published in

1843); the theory of analytic continuation.