Comunicación síncrona: cita extendida

1 8 Thus, for example, if p ⁼ - 1,

q

⁼ 0, i.e., if equation (1.1) is of the form

x3 - x =

0 (obviously the latter has the roots

x1

⁼ 0,

x2

^,

3 = ±

1), then for-mula ( 1.2) unexpectedly yields

x

⁼

- 1/27

+

- 1/27 .

19

In order to reduce the general quartic

ax4

+

bx3

+

cx2 + dx

+

e =

0 or

x4

+

ocx3

+

f3x2

+

yx

+ [)

=

0 to the form (1.3), it suffices to put

x = x1 - oc/

4 (cf. Note 10).

20

"A rosy young man with a gentle voice, merry visage, tremendous abili­

ties and the temper of a devil," according to Tartaglia's description in

Questi

et inventioni diversi

(1546), which contains a detailed (though evidently biased) account of the discovery of formula (1.2). The pro-Cardano side of the story is told in the book

Cartelli

by L. Ferrari (1547-1548). The latter had first been Cardano's servant, but Cardano noticed his outstanding abilities and soon began to study with him.

142 Notes

21

There was a time in the memory of people of the older generation when the Middle Ages, also known as the Dark Ages, were regarded as a thousand­

year gap in European culture. By now it is widely held that this viewpoint is untenable. What is clear to most of us is that medieval culture was based on principles other than those of ancient culture, and that the Renaissance built on the heritage of antiquity. In particular, mathematics was a serious concern of the leading thinkers of antiquity, Plato and Aristotle, and played a most important part in the investigation of the universe by ancient Greek philoso­

phers, whose basic assumption, due to Thales of Miletus and Pythagoras of Samos (6th century B.C.), was that the laws of nature were knowable and that the universe was harmonious. Byzantine and medieval European culture were based on Christianity and differed in many respects from the pagan culture of Greece and Rome. Mathematics played no important part in it, and so progress in mathematics was modest over the centuries. The Renaissance marked a spiritual revolution in the history of European culture which again brought mathematics and the natural sciences to the forefront of European thought.

22

In the 18th and early 19th centuries, the centers for mathematics and the natural sciences in Britain, France, Italy and in other countries were not the universities, which were preoccupied with philosophy, theology, and the hu­

manities, but rather the military, engineering, and naval institutions of higher learning, (not quite appropriately) referred to here as military academies.

Examples of the latter are, in addition to the Turin artillery school, the French military school in Mezieres (see Chapter 3) and the British royal military school in Woolwich, London.

23

Leonhard Euler, the leading mathematician of the 18th century, was born in Basel (Switzerland) into the family of pastor Paul Euler. The conservative views and deep religious faith characteristic of his family assumed a rather naive form in Leonhard (as when he attempted to prove the existence of God mathematically). The mathematician Euler was to retain these views all his life.

The Euler family was on close terms with the "mathematical" Bernoulli family. The elder of the Bernoulli brothers, Jacob (1654-1705) had, rather reluctantly, taught Paul Euler mathematics. Euler Sr. wanted his son to become a clergyman like himself and, conforming to his father's desire, Leon­

hard diligently studied theology. However, unfortunately for himself and fortunately for the rest of humanity, Paul Euler also gave mathematics lessons to Leonhard; the latter turned to Johann Bernoulli (1667-1748), Jacob's younger brother, for help. Johann was amazed by Leonhard's rapid progress and aptitude. He agreed to teach him free of charge once a week. Leonhard devoted most of his spare time to mathematics, discussing his most recent lesson with Johann's sons, the future outstanding mathematicians Daniel Bernoulli (1700-1782) and Nikolaus Bernoulli (1695-1726), and preparing for

the next lesson. With great difficulty, Johann Bernoulli persuaded Paul Euler that his son would make a great mathematician and that it would be a sin to hold back the development of his remarkable talent in order to make him into an ordinary pastor.

At that time a mathematician could not count on a regular income in Switzerland. For example, Johann Bernoulli earned a living by medical prac­

tice until the death of his brother Jacob, who held the only mathematics chair at Basel University. (Incidentally, Johann made the first attempts to apply the newly created differential calculus to the medical problems of muscle contrac­

tion.) As for Johann's sons, they went to Russia, to the newly founded St.

Petersburg Academy of Sciences. They suggested that their friend Euler follow their example and informed him of a vacancy in physiology at the academy.

Euler industriously studied biology and medicine. However, when he arrived in St. Petersburg, he immediately began working in the mathematical sciences and never again returned to physiology, which was generally alien to him. (At one time, following the example of his tutor Johann, he intended to apply mathematical methods to physiology.)

Euler spent most of his long life in St. Petersburg, except for the period between 1741 and 1766, when he decided that Russia, then ruled by Biren, a favorite of the reigning empress Anna Ioannovna, was no longer a safe place.

At the invitation of Frederick II he moved to Berlin� where he headed the physics and mathematics department of the Prussian Academy of Sciences.

But even during his stay in Berlin Euler did not break with Russia. He continued to publish papers in editions of the St. Petersburg Academy of Sciences and even received a small stipend from the Russian government.

Euler was exceptionally prolific; it seems that no other scientist made a comparable contribution to mathematics (in volume

and

in content). We encounter

Euler formulas, Euler theorems,

and

Euler relations

in the differential and integral calculus, in differential equations, in the theory of functions of a complex variable, in the theory of series, in the theory of numbers, in geometry and, of course, in the calculus of variations (a branch of mathematics largely created by Euler and Lagrange). Euler published many long books and innumerable papers. He himself estimated that the articles unpublished in his lifetime (but ready for the press) would last the St. Petersburg Academy of Sciences for 20 years after his death. Actually, he underestimated his own legacy; publication continued until 1862, four times longer than he had ex­

pected. Even the blindness which affiicted him at the end of his life did not interrupt the stream of his works and apparently did not reduce his creative powers: he dictated his last works to his sons, pupils, and colleagues.

Leonhard Euler passed away instantly and easily, in his 77th year. Blind, he wrote out on a blackboard his calculation of the orbit of the newly discovered planet Uranus; he then had some tea, and was playing with his grandson when the stroke came-the old mathematician "ceased to calculate and live," in the words of de Condorcet, who wrote his obituary in a

publi-144 Notes

cation of the Paris Academy of Sciences (Euler was a member of the Paris Academy of Sciences and the London Royal Society, as well as the academies in St. Petersburg and Berlin).

In Switzerland, just before World War I, it was decided to publish Euler's complete works. They were issued in a small number of copies, just enough for each country taking part in the undertaking to receive a copy. The number of(hefty!) volumes grew from an estimated 40 to 70. Then new notes and letters by Euler were discovered in Leningrad-and the number of volumes became greater still. The enormous and extremely valuable

Collected Works

have not been completed to this day, so the total number of volumes is still unknown.

24 In his invitation to Lagrange the king of Prussia, Frederick II, who insisted on being called "the Great", wrote, overbearingly, that he would like the greatest of geometers to work near the greatest of kings (the word

"geometer" meant mathematician then).

Frederick II, who valued science highly and was well-disposed towards Lagrange, died in 1786. This immediately made Lagrange's position in Berlin intolerable. Matters were exacerbated by the fact that Lagrange had to so1ne extent exhausted the subject he had been working on during his stay in Berlin, and this led to a certain disenchantment with mathematics. All this resulted in a depression, which might have proved quite serious. Under the circum­

stances, the invitation to Paris was a blessing for Lagrange. The new flourish­

ing of his creative activity in the Paris period was linked to his public duties (in particular, his work with the commission which introduced the metric system), to his scientific and literary work (described below) and, especially, to his professorial post at the Ecole Polytechnique; this last he accepted at the suggestion of Napoleon, who held him in high esteem. At the Ecole Polytech­

nique Lagrange found a circle of friends and people with the same interests (for example, he won the admiration of Jean Baptiste Joseph Fourier, 1768-1830, one of the pillars of the analytic trend); he also found new topics for research. His lectures on the calculus, delivered at the Ecole Polytechnique, were a major event in the history of mathematics and served as a springboard for the revision of its foundations by another Ecole Polytechnique professor­

Augustin Cauchy. Another major event was the publication of Lagrange's textbooks on theoretical mechanics

(Mecanique Analytique

in two vo!umes, 1788) and on the calculus

(Theorie des fonctions analytiques,

1797; and

Lq:ons sur le calcul des fonctions,

1801). The principal ideas of , ·-e

Mecanique An­

alytique

date back to the Turin period in Lagrange's lue, but the book's publication was mainly due to the efforts of an outstanding mathematician and admirer of Lagrange named Adrien Marie Legendre (1752-1833). The book appeared before Lagrange became a staff member at the Ecole Poly­

technique, in the period when he was still disillusioned with mathematics;

Lagrange took so little interest in it that it remained unopened on his desk for two years. On the other hand, his two-volume calculus textbook (part one,

In document Guía de referencia básica Ada, 2005 (página 79-85)