Change was now slow in mathematics because so much had already been done in that field through five millenniums that Newton seemed to have left no other regions to conquer. For a while, after his death (1727), a reaction set in against the assumptions and abstruseness of calculus. Bishop Berkeley, in a vigorous critique (The Analyst, 1734), assailed these as quite equaling the mysteries of metaphysics and theology, and taunted the followers of science with “submitting to authority, taking things on trust, and believing points inconceivable,” precisely as had been charged against the followers of religious faith. Mathematicians have been as hard put to answer him on this head as materialists have been to refute his idealism.
However, mathematics built bridges, and the pursuit of numbers continued. In England Abraham Demoivre, Nicholas Saunderson, and Brook Taylor, and in Scotland Colin Maclaurin, developed the Newtonian form of calculus. Demoivre advanced the mathematics of chance and of life annuities; being of French birth and English residence, he was chosen by the Royal Society of London (1712) to arbitrate the rival claims of Newton and Leibniz to the invention of infinitesimal calculus. Saunderson became blind at the age of one; he learned to carry on long and complicated mathematical problems mentally; he was appointed professor of mathematics at Cambridge at the age of twenty-nine (1711), and wrote an Algebra that won international acclaim; we shall see how his career fascinated Diderot. Taylor left his name on a basic theorem of calculus, and Maclaurin showed that a liquid mass rotating on its axis would take an ellipsoidal form.
In Basel the Bernoulli family continued through three generations to produce distinguished scientists. Protestant in faith, the family had fled from Antwerp (1583) to avoid the atrocities of Alva. Two of seven Bernoulli mathematicians belong to the age of Louis XIV; a third, Johann I (1667–1748), overspread two reigns. Daniel (1700–82) became professor of mathematics at St. Petersburg at the age of twenty-five, but returned eight years later to teach anatomy, botany, physics, and finally philosophy in the University of Basel; he left works on calculus, acoustics, and astronomy, and almost founded mathematical physics. His brother Johann II (1710–90) taught rhetoric and mathematics, and left his mark on the theory of heat and light. Daniel won prizes of the Académie des Sciences ten times, Johann thrice. Of Johann’s sons, Johann III (1744–1807) became astronomer royal at the Berlin Academy, and Jakob II (1758?—89) taught physics at Basel, mathematics at St. Petersburg. This remarkable family spanned the curriculum, the century, and the Continent.
Leonhard Euler, pupil of Johann Bernoulli I and friendly rival of Daniel, stands out as the most versatile and prolific mathematician of his time. Born at Basel in 1707, dying at St. Petersburg in 1783, eminent in mathematics, mechanics, optics, acoustics, hydrodynamics, astronomy, chemistry, and medicine, and knowing half the Aeneid by heart, he illustrated the uses of diversity and the scope of the human mind. In three major treatises on calculus he freed the new science from the geometric placenta in which it had been born, and established it as algebraic calculus—“analysis.” To these classics he added works on algebra, mechanics, astronomy, and music; however, his Tentamen novae Theoriae Musicae (1729) “contained too much geometry for musicians, and too much music for geometers.”2 With all his science he retained his religious faith to the end.
When Daniel Bernoulli moved to St. Petersburg he promised to get Leonhard a post in the Academy there. The youth went, aged twenty; and when Daniel left Russia (1733) Euler succeeded him as head of the mathematical section. He astonished his fellow Academicians by computing in three days astronomical tables that were expected to require several months. On this and other tasks he worked so intensely, night and day and by poor light, that in 1735 he lost the sight of his right eye. He married, and began at once to add and multiply, while death subtracted; of his thirteen children eight died young. His own life was not safe in a capital racked with political intrigue and assassinations. In 1741 he accepted an invitation from Frederick the Great to join the Berlin Academy; there, in 1759, he succeeded Maupertuis in charge of mathematics. Frederick’s mother liked him, but found him strangely reticent. “Why don’t you speak to me?” she asked. “Madame,” he replied, “I come from a country where if you speak you are hanged.”3 The Russians, however, could be gentlemen. They continued his salary for a long time after his departure; and when a Russian army, invading Brandenburg, pillaged Euler’s farm, the general indemnified him handsomely, and the Empress Elizabeth Petrovna added to the sum.
The history of science honors Euler first for his work in calculus, and especially for his systematic treatment of the calculus of variations. He advanced both geometry and trigonometry as branches of analysis. He was the first to distinctly conceive the notion of a mathematical function, which is now the heart of mathematics. In mechanics he formulated the general equations that still bear his name. In optics he was the first to apply calculus to the vibrations of light, and to formulate the curve of vibration as dependent upon elasticity and density. He deduced the laws of refraction analytically, and made those studies in the dispersion of light that prepared for the construction of achromatic lenses. He shared in the international enterprise of finding longitude at sea by charting the position of the planets and the phases of the moon; his approximate solution helped John Harrison to draw up successful lunar tables for the British Admiralty.
In 1766 Catherine the Great asked Euler to return to St. Petersburg. He did, and she treated him royally. Soon after his arrival he became totally blind. His memory was so accurate, and his speed of calculation so great, that he continued to produce almost as actively as before. Now he dictated his Complete Introduction to Algebra to a young tailor who, when this began, knew nothing of mathematics beyond simple reckoning; this book gave to algebra the form that it retained to our time. In 1771 a fire destroyed Euler’s home; the blind mathematician was saved from the flames by a fellow Swiss from Basel, Peter Grimm, who carried him on his shoulders to safety. Euler died in 1783, aged seventy-six, from a stroke suffered while playing with a grandson.
Only one man surpassed him in his century and science, and that was his protégé. Joseph Louis Lagrange was one of eleven children born to a French couple domiciled in Turin; of these eleven he alone survived infancy. He was turned from the classics to science by reading a memoir addressed by Halley to the Royal Society of London; at once he devoted himself to mathematics, and soon with such success that at the age of eighteen he was professor of geometry at the Turin Artillery Academy. From his students, nearly all older than himself, he organized a research society that grew into the Turin Academy of Science. At nineteen he sent to Euler a new method for treating the calculus of variations; Euler replied that this procedure solved difficulties which he himself had been unable to overcome. The kindly Swiss delayed making public his own results, “so as not to deprive you of any part of the glory which is your due.”4 Lagrange announced his method in the first volume issued by the Turin Academy (1759). Euler, in his own memoir on the calculus of variations, gave the younger man full credit; and in that year 1759 he had him elected a foreign member of the Berlin Academy, at the age of twenty-three. When Euler left Prussia he recommended Lagrange as his successor at the Academy; d’Alembert warmly seconded the proposal; and in 1766 Lagrange moved to Berlin. He greeted Frederick II as “the greatest king in Europe”; Frederick welcomed him as “the greatest mathematician in Europe.”5 This was premature, but it soon became true. The friendly relations among the leading mathematicians of the eighteenth century—Euler, Lagrange, Clairaut, d’Alembert, and Legendre—form a pleasant episode in the history of science.
During his twenty years at Berlin Lagrange gradually put together his masterpiece, Mécanique analytique. Incidentally to this basic enterprise he delved into astronomy, and offered a theory of Jupiter’s satellites and an explanation of lunar librations—alterations in the visible portions of the moon. In 1786 Frederick the Great died, and was succeeded by Frederick William II, who cared little for science. Lagrange accepted an invitation from Louis XVI to join the Académie des Sciences; he was given comfortable quarters in the Louvre, and became a special favorite of Marie Antoinette, who did what she could to lighten his frequent spells of melancholy. He brought with him the manuscript of Mécanique analytique, but he could find no publisher for so difficult a printing problem in a city seething with revolution. His friends Adrien Legendre and the Abbé Marie finally prevailed upon a printer to undertake the task, but only after the abbé had promised to buy all copies unsold after a stated date. When the book that summed up his life work was placed in Lagrange’s hands (1788), he did not care to look at it; he was in one of those periodic depressions in which he lost all interest in mathematics, even in life. For two years the book remained unopened on his desk.
The Mécanique analytique is rated by general consent as the summit of eighteenth-century mathematics. Second only to the Principia in their field, it advanced upon Newton’s book by using “analysis”—algebraic calculus—instead of geometry in the discovery and exposition of solutions; said the preface, “No diagrams will be found in this work.” By this method Lagrange reduced mechanics to general formulas—the calculus of variations—from which specific equations could be derived for each particular problem; these general equations still dominate mechanics, and bear his name. Ernst Mach described them as one of the greatest contributions ever made to the economy of thought.6 They raised Alfred North Whitehead to religious ecstasy: “The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things.”7
When the Revolution broke out with the fall of the Bastille (July 14, 1789), Lagrange, as a favorite of royalty, was advised to return to Berlin; he refused. He had always sympathized with the oppressed, but he had no faith in the ability of revolution to escape the results of the natural inequality of men. He was horrified by the massacres of September, 1792, and the execution of his friend Lavoisier, but his moody silence saved him from the guillotine. When the École Normale was opened (1795) Lagrange was put in charge of mathematics; when that school was closed and the École Poly-technique was established (1797), he was its first professor; the mathematical basis and bent of French education are part of Lagrange’s enduring influence.
In 1791 a committee was appointed to devise a new system of weights and measures; Lagrange, Lavoisier, and Laplace were among its first members; two of this trinity were “purged” after three months, and Lagrange became the leading spirit in formulating the metric system. The committee chose as the basis of length a quadrant of the earth—a quarter of the great circle passing around the earth at sea level through the poles; one ten-millionth of this was taken as the new unit of length, and was called a mètre—a meter. A subcommittee chose as the new unit of weight a gram: the weight of distilled water, at zero temperature centigrade, occupying a cube each side of which measured one centimeter—one hundredth of a meter. In this way all lengths and weights were based upon one physical constant, and upon the number ten. There were still many defenders of the duodecimal system, which took twelve as its base, as in England and generally in our measurement of time. Lagrange stood firmly for ten, and had his way. The metric system was adopted by the French government on November 25, 1792, and remains, with some modifications, as perhaps the most lasting result of the French Revolution.
Romance brightened Lagrange’s advancing age. When he was fifty-six a girl of seventeen, daughter of his friend the astronomer Lemonnier, insisted on marrying him and devoting herself to mitigating his hypochondria. Lagrange yielded, and became so grateful for her love that he accompanied her to balls and musicales. He had learned to like music—which is a trick that mathematics plays upon the ear—because “it isolates me. I hear the first three measures; at the fourth I distinguish nothing; I give myself up to my thoughts; nothing interrupts me; and it is thus that I solve more than one difficult problem.”8
As the fever of revolution subsided, France complimented itself on having exempted the supreme mathematician of the age from the guillotine. In 1796 Talleyrand was sent to Turin to wait in state upon Lagrange’s father and tell him, “Your son, whom Piedmont is proud to have produced, and France to possess, has done honor to all mankind by his genius.”9 Napoleon, between campaigns, liked to talk with the mathematician-become-philosopher.
The old man’s interest in mathematics revived when (1810–13) he revised and enlarged the Mécanique analytique for its second edition. But as usual he worked too hard and fast; spells of dizziness weakened him; once his wife found him unconscious on the floor, his head bleeding from a cut caused by his fall against the edge of a table. He realized that his physical resources were running out, but he accepted this gradual disintegration as normal and reasonable. To Monge and others who attended him he said:
“I was very ill yesterday, my friends. I felt I was going to die. My body grew weaker little by little; my intellectual and physical faculties were extinguished insensibly. I observed the well-graduated progression of the diminution of my strength, and I came to the end without sorrow, without regrets, and by a very gentle decline. Death is not to be dreaded, and when it comes without pain it is a last function which is not unpleasant.… Death is the absolute repose of the body.”10
He died on April 10, 1813, aged seventy-five, mourning only that he had to leave his faithful wife to the hazards of that age, when it seemed that all the world was in arms against France.
His friends Gaspard Monge and Adrien Legendre carried into the nineteenth century those mathematical researches which provided the foundations of industrial advance. The work of Legendre (1752–1833) belongs to the post-Revolution age; we merely salute him on our way. Monge was the son of a peddler and knife-grinder; our notion of French poverty is checked when we see this simple workingman sending three sons through college. Gaspard took all available prizes in school. At fourteen he built a fire engine; at sixteen he declined the invitation of his Jesuit teachers to join their order; instead he became professor of physics and mathematics in the École Militaire at Mézières. There he formulated the principles of descriptive geometry—a system of presenting three-dimensional figures on one descriptive plane. The procedure proved so useful in designing fortifications and other constructions that for fifteen years the French army forbade him to divulge it publicly. Then (1794) he was allowed to teach it at the École Normale in Paris. Lagrange, attending his lecture there, marveled like Molière’s Jourdain: “Before hearing Monge, I did not know that I knew descriptive geometry.”11 Monge served the embattled republic well, and rose to be minister of the marine. Napoleon entrusted many confidential missions to him. After the restoration of the Bourbons Monge was reduced to insecurity and poverty. When he died (1818) his students at the École Polytechnique were forbidden to attend his funeral. The next morning they marched in a body to the cemetery and laid a wreath upon his grave.