When someone in America once asked me who the three greatest mathematicians of all time were, I immediately responded: Archimedes, Newton and Gauss. “Goddamnit, why do you mathematicians always come up with the same three names?” demanded my interlocutor. Probably because we have all read Men of Mathematics by E.T. Bell, which features portraits of this trio on its cover. To pick a fourth, a name often mentioned is Leonhard Euler, and Bell’s chapter on this prolific 18th-century genius, whose collected works fill 80 volumes, quotes mathematician, physicist, astronomer and one-time president of France, François Arago: “Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves on the wing.”
Born in 1707 in Basel, where Johann Bernoulli, the greatest mathematician of the day worked, Euler got off to a good start and in 1726 was offered a position at the new Academy in St Petersburg founded by Peter the Great, whose aim was to draw Russia into the European Enlightenment, and away from 17th-century attitudes that branded arithmetic and geometry as magic. Still, it was not easy. The Church censored the idea of a solar system centred at the sun, and the Old Believer nobles treated foreign academicians — mainly German, Swiss and French — with suspicion, even hostility.
After Peter’s death in 1725, benevolent control of the Academy continued under his successors until Empress Anna died in 1740, but even in the 1730s censors blocked the astronomical idea of heliocentricity, and publications from outside Russia were considered heretical. Meanwhile, Euler threw himself wholeheartedly into mathematics both pure and applied, even dealing with cartography for expeditions to Kamchatka in the far east of Russia.
Newton had died in 1727 but his differential calculus and principles of dynamics still needed substantial development, and it fell to the young Euler to lead the 18th-century creation of what we now call Newtonian dynamics and mechanics. Indeed, his two-volume Mechanica, published in 1736, became a landmark in the history of physics, and although the book was in Latin, Euler was equally at home writing in French or his native German. During the early St Petersburg years he also learned to read and write Russian, publishing in all these languages across the entire span of mathematics, including applications to optics, mechanics, fluid dynamics, navigation, ship construction and even music, which for Euler formed part of the quadrivium: arithmetic, geometry, music and astronomy. His teaching was superb, and in later years his Letters to a German Princess formed one of the finest explanations of science ever written. Yet he also took administrative duties very seriously, and found ample time for family. In the mid-1730s he married Katharina, daughter of another Swiss gentleman, the Petersburg Court painter Georg Gsell, and would happily work with a child on his knee and a cat on his shoulder.
Under more stable circumstances Euler might have spent his entire career in the newly cosmopolitan surroundings of St Petersburg, but after Empress Anna died in 1740 society regressed intellectually, and political instability created an atmosphere that his wife could not think back on without fear. That same year Frederick II (later the Great) was crowned King of Prussia and immediately invited Euler to help revitalise the almost moribund Academy in Berlin. After negotiating a suitable deal, Euler began 25 years of work in Berlin, where he completed or wrote more than 380 articles and books, “whose combined depth, originality, difficulty, range and sheer numbers make his achievement unmatched in the history of the mathematical sciences”, according to Leonhard Euler: Mathematical Genius in the Enlightenment (Princeton, £40), an excellent new biography by Ronald S. Calinger.
Euler became the jewel of the Berlin Academy, a man who “had read all the Roman authors, knew perfectly the ancient history of mathematics, held in his memory the historical events of all times and peoples . . . and knew more about medicine, botany and chemistry” than anyone but an expert. Yet Frederick II never warmed to him, valuing aristocratic French wit over intellectual distinction, elevating French as the language of the Academy and wanting science only for its utility to the state. Despite exchanges of letters, Euler had no personal meeting with him until a 1749 invitation to the Potsdam Palace to advise on the water fountains, the same year he published his Scientia Navalis, which made floating navigation a complete science and applied mathematical principles to the optimal design of ships.
Lacking the aristocratic pedigree to be Academy president, Euler could only act as surrogate, which he did from 1750 until the end of the Seven Years War in 1763 when Frederick II began interfering in the administration, placing greater emphasis on ideas from the French Enlightenment. After losing control over the election of Academy members, Euler was open to a generous offer from the recently-installed Catherine II (another Great) of Russia, but only in 1766 did the Euler family finally move back to St Petersburg from Berlin. It had not been easy to get the king’s dispensation to leave, despite the fact that he treated Euler with a degree of contempt and later wrote a letter to the aristocratic French mathematician d’Alembert saying, “The one-eyed monster has been replaced by another who has both eyes.”
Euler had virtually lost the sight in one eye many years earlier, and in St Petersburg his eyesight deteriorated further. He lost most of the sight in the other eye, despite a cataract operation by the future oculist to George III, yet his energy and activity remained undimmed and more than half his publications date from after the return to St Petersburg, where Catherine the Great honoured him in a way that Frederick the Great never had. In 1783 after settling a fierce dispute between the academics and administrators of her Academy she appointed Princess Diashkova as director, for whose inauguration she entered on Euler’s arm.
At this time he was occasionally suffering from attacks of vertigo, and on September 18, after teaching mathematics to four grandchildren in the morning, making difficult mental calculations on the upward motion of hot-air balloons (news of the Montgolfier brothers’ success had just reached St Petersburg), followed by lunch-time conversations about improvements to Herschel’s new telescope, and a discussion with his assistant on the orbit of Uranus (discovered by Herschel in 1781), he felt faint and went to lie down. After waking and taking tea, he dropped his pipe, stood up, and suddenly clasping two hands to his forehead spoke his last words: “Ich sterbe.”