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In 1545 a book by Gerolamo Cardano appeared whose title began with the Latin words . It was essentially devoted to solving third-and fourth-order equations, but its value for the history of mathematics far surpassed the limits of this specific problem. Even in the 20th century, Felix Klein, evaluating this book, wrote, “This most valuable work contains the germ of modern algebra, surpassing the bounds of ancient mathematics.”

Vincenzio Galilei, a well-known Florentine musician, had reflected for a long time on what field to choose for his oldest son Galileo. The son was undoubtedly talented in music, but the father preferred something more reliable. In 1581, when Galileo turned seventeen, the scales were leaning in the direction of medicine. Vincenzio understood the expenses of instruction would be great, but that his son’s future would be assured. The place of instruction was chosen to be the University of Pisa, perhaps a bit provincial but familiar to Vincenzio. He had lived for a long time in Pisa, and Galileo was born there.

We have told how Galileo laid the foundation for classical mechanics almost at the beginning of the 17th century. Christiaan Huygens (1629–1695) was Galileo’s immediate scientific successor. In Lagrange’s words, Huygens “was destined to improve and develop most of Galileo’s important discoveries.” There is a story about how Huygens, at age 17, first came into contact with Galileo’s ideas: He planned to prove that a projectile moves horizontally along a parabola after launch, but discovered a proof in Galileo’s book and did not want “to write the after Homer.” It is striking how close Huygens and Galileo were in scientific spirit and interests.

The curve “described so often before everyone’s eyes” was first noticed by Galileo in Italy and Marin Mersenne (1588–1648) in France. It was called a cycloid in Italy and a roulette in France. The term cycloid means “coming from a circle” and is due to Galileo. It prevailed, and the term roulette now denotes a more general kind of curve that we will discuss later. Seventeenth-century mathematicians, creating general methods of studying curves, were very interested in new “experimental” curves. The cycloid occupied a special place among these curves. It turned out to be one of the first curves (curves that cannot be derived algebraically), for which the problems of constructing tangents and calculating areas were solved beautifully and explicitly. But what was most striking was that the cycloid appeared again and again in the solution of very different problems where it was not a part of the original formulation. All this made the cycloid the most popular curve of the 17th century: the most powerful scientists in Italy and France (Torricelli, Vincenzo Viviani (1622–1703), Pierre de Fermat (1601–1665), René Descartes (1596–1650), Gilles de Roberval (1602–1675)) solved a variety of problems about the cycloid, and in 1673 Huygens said that “the cycloid is studied more carefully and thoroughly than all other curves.”

Blaise Pascal was inherently multifaceted, a characteristic of the Renaissance that had almost become passé in the 17th century. The natural sciences (say, physics and mathematics) had not yet completely separated from the humanities, but studies in the humanities and the natural sciences were already no longer commonly combined.

In 1684, a seven-page article by Gottfried Wilhelm Leibniz (1646–1716) was published in the journal (roughly meaning ), which had first appeared in 1682 in Leipzig. The article was entitled, . . ., or . This was the first publication on differential calculus, although calculus had arisen some twenty years earlier and the first steps were fifty years older and belong to the start of the 17th century.

In early 1783, Princess Ekaterina Romanovna Dashkova was named director of the Petersburg Academy of Sciences. Twenty years earlier she had been the closest associate of Catherine II (“Catherine the Great”) when the latter ascended to the Russian throne. Known for her ingenuity, the princess had thought of the perfect way to convince the academicians of her devotion to science. She persuaded the elderly Euler, who for a long time had not been on good terms with the academic establishment and had not attended Academy conferences, to accompany her on her first visit there. The blind Euler appeared with his son and grandson. Dashkova later recalled, “I said to them that I asked Euler to take me to the session since, regardless of my own ignorance, I consider that by such a ceremonial act I was testifying to my respect for science and enlightenment.”

In August 1755 the great Euler (1707–1783) received a letter from Turin, from the 19 year-old Lagrange who had written to him before. Euler no doubt had already formed the opinion that his correspondent was a talented and mature mathematician despite his youth. But all the same, the contents of the letter astonished the scientist.

On March 5, 1827 at nine o’clock in the morning the Marquis Laplace died, a peer of France, one of the first chevaliers of the Legion of Honor and worthy of its highest decoration, the Grand Cross. “What we know is nothing in comparison with what we do not know” were his last words. Laplace was called “the French Newton” and he died exactly one hundred years after Newton, who had been his idol.

In 1854, the health of Privy Councillor Gauss, as his colleagues at the University of Göttingen called him, worsened decisively. There was no question of continuing the daily walks from the observatory to the literary museum, a habit of over twenty years. They managed to convince the professor, who was nearing eighty, to go to the doctor! He improved during the summer and even attended the opening of the Hannover-Göttingen railway. In January 1855, Gauss agreed to pose for a medallion by the artist Heinrich Hesemann. After the scientist’s death in February 1855, a medal was prepared from the medallion, by order of the Hannover court. Beneath a bas-relief of Gauss, these words were written: (Prince of Mathematicians). The story of every real prince should begin with his childhood, embroidered with legends. Gauss is no exception.