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.
2. Lagrange
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 unpleas
ant.… 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.
III. PHYSICS
1. Matter, Motion, Heat, and Light
Mathematics grew because it was the basic and indispensable tool of all science, reducing experience and experiment to quantitative formulations that made possible precise prediction and practical control. The first step was to apply it to matter in general: to discover the regularities and establish the “laws” of energy, motion, heat, sound, light, magnetism, electricity; here were mysteries enough waiting to be explored.
Pierre Louis Moreau de Maupertuis abandoned a career in the French army to devote himself to science. He preceded Voltaire in introducing Newton to France, and in appreciating and instructing Mme. du Châtelet. In 1736, as we shall see, he directed an expedition to Lapland to measure a degree of the meridian. In 1740 he accepted an invitation to visit Frederick II; he followed Frederick into the battle of Mollwitz (1741), was captured by the Austrians, but was soon released. In 1745 he joined the Academy of Sciences at Berlin, and a year later he became its president. To the Paris Académie des Sciences in 1744, and to the Berlin Academy in 1746, he expounded his principle of least action: “Whenever any change occurs in nature, the quantity of action employed for this change is always the least possible.” This, he thought, proved a rational order in nature, and therefore the existence of a rational God.12 Euler and Lagrange developed the principle, and in our own time it played a part in the quantum theory. In an Essai de cosmologie (1750) Maupertuis revived an indestructible heresy: while still recognizing design in nature, he confessed to seeing in it also signs of stupidity or evil, as if a demon were competing with a benevolent deity in the management of the cosmos.13 Maupertuis might have agreed with his merciless enemy Voltaire that St. Augustine should have remained a Manichaean.
We have noted the birth of d’Alembert as the unpremeditated issue of a passing contact between an artilleryman and an ex-nun. The Paris police found him, a few hours old, on the steps of the Church of St.-Jean-le-Rond (1717); they had him baptized Jean Baptiste Le Rond, and sent him to a nurse in the country. His father, the Chevalier Destouches, claimed him, gave him (for reasons unknown to us) the name d’Arembert, and paid Mme. Rousseau, a glazier’s wife, to adopt the child. She proved a model stepmother, and Jean a model and precocious boy. When he was seven the father proudly displayed him to the mother, Mme. de Tencin, but she decided that her career as mistress and salonnière would be impeded by accepting him. She contributed nothing to his support, so far as we know, but the Chevalier, before dying in 1726, left him an annuity of twelve hundred livres.
Jean studied at the Collège des Quatre-Nations, then at the University of Paris, where he received the degree in law. There, about 1738, he changed his name from d’Arembert to d’Alembert. Tiring of law, he turned to medicine; but an incidental interest in mathematics became a passion: “mathematics,” he said, “was for me my mistress.”14 He continued till he was forty-eight to live with Mme. Rousseau, looking upon her gratefully as his only mother. She thought it disgraceful that a man should so abandon himself to study and show no economic itch. “You will never be anything better than a philosopher,” she mourned, adding, “And what is a philosopher? ’Tis a madman who torments himself all his life so that people may talk about him when he is dead.”15
Probably his inspiring motives were not a desire for posthumous fame but a proud rivalry with established savants, and that beaver instinct which takes delight in building, in forging order upon a chaos of materials or ideas. In any case he began at twenty-two to submit papers to the Académie des Sciences: one on integral calculus (1739), another on the refraction of light (1741); this gave the earliest explanation of the bending of light rays in passing from one fluid to another of greater density; for this the Académie admitted him to “adjoint” membership. Two years later he published his main scientific work, Traité de dynamique, which sought to reduce to mathematical equations all problems of matter in motion; this anticipated by forty-two years Lagrange’s superior Mécanique analytique; it keeps historical significance because it formulated the basic theorem now known as “D’Alembert’s principle,” too technical for our general digestion, but immensely helpful in mechanical calculations. He applied it in a Traité de l’équilibre et du mouvement des fluides (1744); this so impressed the Académie that it awarded him a pension of five hundred livres, which must have appeased Mme. Rousseau.
Partly from his principle, partly from an original equation in calculus, d’Alembert arrived at a formula for the motion of winds. He dedicated his Réflexions sur la cause générale des vents (1747) to Frederick the Great, who responded by inviting him to settle in Berlin; d’Alembert refused, showing at thirty more wisdom than Voltaire was to show at fifty-six. In an Essai d’une nouvelle théorie de la résistance des fluides (1752) he tried to find mechanical formulas for the resistance of water to a body moving on it; he failed, but in 1775, under a commission from Turgot, he and Condorcet and the Abbé Bossut made experiments that helped to determine the laws of fluid resistance to surface-moving bodies. Late in life he studied the motion of vibrating chords, and issued (1779) Eléments de musique théorique et pratique, following and modifying the system of Rameau; this book won the praise of the famous musicologist Charles Burney. All in all, d’Alembert had one of the keenest minds of the century.
When Maupertuis resigned as president of the Berlin Academy Frederick the Great offered the post to d’Alembert. The mathematician-physicist-astronomer-encyclopedist was poor, but he courteously refused; he cherished his freedom, his friends, and Paris. Frederick respected his motives, and, with the permission of Louis XV, sent him a modest pension of twelve hundred livres. In 1762 Catherine the Great invited him to Russia and the Academy of St. Petersburg; he declined, for he was now in love. Perhaps informed of this, Catherine persisted, bade him come “avec tous vos amis,” with all his friends, and offered him a salary of 100,000 francs a year. She took his refusals graciously, and continued to correspond with him, discussing with him her mode and problems of government. In 1763 Frederick urged him at least to visit Potsdam; d’Alembert went, and dined with the King for two months. He again declined the presidency of the Berlin Academy; instead he induced Frederick to raise the salary of Euler, who had a large family.16 We hope to meet d’Alembert again.
The amazing Bernoullis
made some incidental contributions to mechanics. Johann I formulated (1717) the principle of virtual velocities: “In all equilibrium of forces whatsoever, in whatever manner they are applied, and in whatever directions they act upon one another, whether directly or indirectly, the sum of the positive energies will be equal to the sum of the negative energies taken positively.” Johann and his son Daniel (1735) proclaimed that the sum of vis viva (living force) in the world is always constant; this principle was reformulated in the nineteenth century as the conservation of energy. Daniel applied the conception to good effect in his Hydrodynamics (1738), a modern classic in an especially difficult field. In that volume he founded the kinetic theory of gases: a gas is composed of tiny particles moving about with great rapidity, and exerting pressure upon the container by their repeated impacts; heat increases the velocity of the particles and therefore the pressure of the gas; and the lessening of the volume (as Boyle had shown) proportionately increases the pressure.
In the physics of heat the great name for the eighteenth century is Joseph Black. Born in Bordeaux to a Scot born in Belfast, he studied chemistry in the University of Glasgow, and at the age of twenty-six (1754) made experiments in what we would now call oxidation or corrosion; these indicated the action of a gas distinct from common air; he detected this in the balance, and called it “fixed air” (now called carbon dioxide); Black had come close to the discovery of oxygen. In 1756, as lecturer in chemistry, anatomy, and medicine at the university, he began observations that led him to his theory of “latent heat”: when a substance is in process of changing from a solid to a liquid state, or from a liquid to a gas, the changing substance absorbs from the atmosphere an amount of heat not detectable as a change of temperature; and this latent heat is given back to the atmosphere when a gas changes into a liquid or a liquid into a solid. James Watt applied this theory in his improvement of the steam engine. Black, like nearly all predecessors of Priestley, thought of heat as a material substance (“caloric”) added to or subtracted from matter rising or falling in warmth; not until 1798 did Benjamin Thompson, Count Rumford, show that heat is not a substance but a mode of motion, now conceived as an accelerated motion of a body’s constituent parts.