by E. T. Bell
This is enough on the debit side of the ledger. The last extract does indeed suggest one trait in which Laplace overtopped all courtiers—his moral courage where his true convictions were questioned. The story of Laplace’s encounter with Napoleon over the Mécanique céleste shows the mathematician as he really was. Laplace had presented Napoleon with a copy of the work. Thinking to get a rise out of Laplace, Napoleon took him to task for an apparent oversight. “You have written this huge book on the system of the world without once mentioning the author of the universe.” “Sire,” Laplace retorted, “I had no need of that hypothesis.” When Napoleon repeated this to Lagrange, the latter remarked “Ah, but that is a fine hypothesis. It explains so many things.”
It took nerve to stand up to Napoleon and tell him the truth. Once at a session of the Institut when Napoleon was in one of his most insultingly bad tempers he caused poor old Lamarck to burst into tears with his deliberate brutality.
Also on the credit side was Laplace’s sincere generosity to beginners. Biot tells how as a young man he read a paper before the Academy when Laplace was present, and was drawn aside afterward by Laplace who showed him the identical discovery in a yellowed old manuscript of his own, still unpublished. Cautioning Biot to secrecy, Laplace told him to go ahead and publish his work. This was but one of several such acts. Beginners in mathematical research were his stepchildren, Laplace liked to say, but he treated them as well as he did his own son.
As it is often quoted as an instance of the unpracticality of mathematicians we shall give Napoleon’s famous estimate of Laplace, of which he is reported to have delivered himself while he was a prisoner at St. Helena.
“A mathematician of the first rank, Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought subtleties everywhere, had only doubtful ideas, and finally carried the spirit of the infinitely small into administration.”
This sarcastic testimonial was inspired by Laplace’s short tenure—only six weeks—of the Ministry of the Interior. However, as Lucien Bonaparte needed a job at the moment and succeeded Laplace, Napoleon may have been rationalizing his well-known inclination to nepotism. Laplace’s testimonial for Napoleon has not been preserved. It might have run somewhat as follows.
“A soldier of the first rank, Napoleon quickly revealed himself as only a mediocre politician; from his first exploits we saw that he was deceived. Napoleon saw all questions from the obvious point of view; he suspected treachery everywhere but where it was, had only a childlike faith in his supporters, and finally carried the spirit of infinite generosity into a den of thieves.”
Which, after all, was the more practical administrator? The man who could not hang onto his gains and who died a prisoner of his enemies, or the other who continued to gather wealth and honor to the day of his death?
Laplace spent his last days in comfortable retirement at his country estate at Arcueil, not far from Paris. After a short illness he died on March 5, 1827, in his seventy eighth year. His last words have already been reported.
CHAPTER TWELVE
Friends of an Emperor
MONGE AND FOURIER
I cannot tell you the efforts to which I was condemned to understand something of the diagrams of Descriptive Geometry, which I detest.
—CHARLES HERMITE
Fourier’s Theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.
—WILLIAM THOMSON AND P. G. TAIT
THE CAREERS OF GASPARD MONGE (1746-1818) and Joseph Fourier (1768-1830) are curiously parallel and may be considered together. On the mathematical side each made one fundamental contribution: Monge invented descriptive geometry (not to be confused with the projective geometry of Desargues, Pascal, and others); Fourier started the current phase of mathematical physics with his classic investigations on the theory of heat-conduction.
Without Monge’s geometry—originally invented for use in military engineering—the wholesale spawning of machinery in the nineteenth century would probably have been impossible. Descriptive geometry is the root of all the mechanical drawing and graphical methods that help to make mechanical engineering a fact.
The methods inaugurated by Fourier in his work on the conduction of heat are of a similar importance in boundary-value problems—a trunk nerve of mathematical physics.
Monge and Fourier between them are thus responsible for a considerable part of our own civilization, Monge on the practical and industrial side, Fourier on the purely scientific. But even on the practical side Fourier’s methods are indispensable today; they are in fact a commonplace in all electrical and acoustical engineering (including wireless) beyond the rule of thumb and handbook stages.
A third man must be named with these mathematicians, although we shall not take space to tell his life: the chemist Count Claude-Louis Berthollet, (1748-1822), a close friend of Monge, Laplace, Lavoisier, and Napoleon. With Lavoisier, Berthollet is regarded as one of the founders of modern chemistry. He and Monge became so thick that their admirers gave up trying to distinguish between them in their nonscientific labors and called them simply Monge-Berthollet.
Gaspard Monge, born on May 10, 1746, at Beaune, France, was a son of Jacques Monge, a peddler and knife grinder who had a tremendous respect for education and who sent his three sons through the local college. All the sons had successful careers; Gaspard was the genius of the family. At the college (run by a religious order) Gaspard regularly captured the first prize in everything and earned the unique distinction of having puer aureus inscribed after his name.
At the age of fourteen Monge’s peculiar combination of talents showed up in the construction of a fire engine. “How could you, without a guide or a model, carry through such an undertaking successfully?” he was asked by the astonished citizens. Monge’s reply is a summary of the mathematical part of his career and of much of the rest. “I had two infallible means of success: an invincible tenacity, and fingers which translated my thought with geometric fidelity.” He was in fact a born geometer and engineer with an unsurpassed gift for visualizing complicated space-relations.
At the age of sixteen he made a wonderful map of Beaune entirely on his own initiative, constructing his own surveying instruments for the purpose. This map got him his first great chance.
Impressed by his obvious genius, Monge’s teachers recommended him for the professorship of physics at the college in Lyon run by their order. Monge was appointed at the age of sixteen. His affability, patience, and lack of all affectation, added to his sound knowledge, made him a great teacher. The order begged him to take their vows and cast his lot for life with them. Monge consulted his father. The astute knife grinder advised caution.
Some days later, on a visit home, Monge met an officer of engineers who had seen the famous map. The officer begged Jacques to send his son to the military school at Mézières. Perhaps fortunately for Monge’s future career the officer omitted to state that on account of his humble birth Monge could never get a commission. Not knowing this, Monge eagerly accepted and proceeded to Mézières.
Monge quickly learned where he stood at Mézières. There were only twenty pupils at the school, of whom ten were graduated each year as lieutenants in engineering. The rest were destined for the “practical” work—the dirty jobs. Monge did not complain. He rather enjoyed himself, as the routine work in surveying and drawing left him plenty of time for mathematics. An important part of the regular course was the theory of fortification, in which the problem was to design the works so that no part should be exposed to the direct fire of the enemy. The usual calculations demanded endless arithmetic. One day Monge handed in his solution of a problem of this sort. It was turned over to a superior officer for inspection.
Skeptical that anyone could have solved the problem in
the time, the officer declined to check the solution. “Why should I give myself the trouble of subjecting a supposed solution to tedious verifications? The author has not even taken the time to group his figures. I can believe in a great facility in calculation, but not in miracles!” Monge persisted, saying he had not used arithmetic. His tenacity won; the solution was checked and found correct.
This was the beginning of descriptive geometry. Monge was at once given a minor teaching position to instruct the future military engineers in the new method. Problems which had been nightmares before—sometimes solved only by tearing down what had been built and beginning all over again—were now as simple as ABC. Monge was sworn not to divulge his method, and for fifteen years it was a jealously guarded military secret. Only in 1794 was he allowed to teach it publicly, at the École Normale in Paris, where Lagrange was among the auditors. Lagrange’s reaction to descriptive geometry was like M. Jourdain’s when he discovered that he had been talking prose all his life. “Before hearing Monge,” Lagrange said after a lecture, “I did not know that I knew descriptive geometry.”
* * *
The idea behind it all now seems as ridiculously simple to us as it did to Lagrange. Descriptive geometry is a method for representing solids and other figures in ordinary three-dimensional space on one plane. Imagine first two planes at right angles to one another, like two pages of a thin book opened at a ninety degree angle; one plane is horizontal, the other vertical. The figure to be represented is projected onto each of these planes by rays perpendicular to the plane. There are thus two projections of the figure; that on the horizontal plane is called a plan of the figure, that on the vertical plane an elevation. The vertical plane is now turned down (“rabbatted”) till it and the horizontal plane lie in one plane (that of the horizontal plane)—as if the book were now opened out flat on a table.
The solid or other figure in space is now represented by two projections on one plane (that of the drawing board). A plane, for instance, is represented by its traces—the straight lines in which it cut the vertical and horizontal planes before the former was rabbatted; a solid, say a cube, is represented by the projections of its edges and vertices. Curved surfaces cut the vertical and horizontal planes in curves; these curves, or traces of the surface, represent the surface on the one plane.
When these and other equally simple remarks are developed we have a descriptive method which puts on one flat sheet of paper what we ordinarily visualize in space of three dimensions. A short training enables the draughtsman to read such representations as easily as others read good photographs—and to get a great deal more out of them. This was the simple invention that revolutionized military engineering and mechanical design. Like many of the first-rate things in applied mathematics its most conspicuous feature is its simplicity. There are many ways in which descriptive geometry can be developed or modified, but they all go back to Monge. The subject is now so thoroughly worked out that it is not of much interest to professional mathematicians.
To finish with Monge’s contributions to mathematics before continuing with his life, we recall that his name is familiar to every student in the second course in the calculus today in connection with the geometry of surfaces. Monge’s great step forward was a systematic (and brilliant) application of the calculus to the investigation of the curvature of surfaces. In his general theory of curvature Monge prepared the way for Gauss, who in his turn was to inspire Riemann, who again was to develop the geometry known by his name in the theory of relativity.
It seems rather a pity that a born geometer like Monge should have lusted after the fleshpots of Egypt, but so he did. His work in differential equations, closely connected with that in geometry, also showed what he had in him. Years after he left Mézières, where these great things were done, Monge lectured on his discoveries to his colleagues at the Ecole Polytechnique. Lagrange again was an auditor. “My dear colleague,” he told Monge after the lecture, “you have just explained some very elegant things; I should have liked to have done them myself.” And on another occasion: “With his application of analysis to geometry this devil of a man will make himself immortal!” He did; and it is interesting to note that although more urgent calls on his genius distracted him from mathematics, he never lost his talent. Like all the great mathematicians Monge was a mathematician to the last.
* * *
In 1768, at the age of twenty two, Monge was promoted to the professorship of mathematics at Mézières, and three years later, on the death of the professor of physics, stepped into his place also. The double work did not bother him at all. Powerfully built and as strong of body as he was of mind, Monge was always capable of doing three or four men’s work and frequently did.
His marriage had a touch of eighteenth century romance. At a reception Monge heard some noble bounder slandering a young widow to get even with her for having rejected him. Shouldering his way through the cackling crowd, Monge demanded to know whether he had heard aright. “What is it to you?” Monge demonstrated with a punch on the jaw. There was no duel. A few months later at another reception Monge was very much taken by a charming young woman. On being introduced he recognized her name—Madame Horbon—as that of the unknown lady he had tried to fight a duel for. She was the widow, only twenty, and somewhat reluctant to marry before her late husband’s affairs were straightened out. “Never mind all that,” Monge reassured her, “I’ve solved lots of more difficult problems in my time.” Monge and she were married in 1777. She survived him and did what she could to perpetuate his memory—unaware that her husband had raised his own monument long before he ever met her. Monge’s wife was the one human being who stuck to him through everything. Even Napoleon at the very last would have let him down on account of his age.
At about this time Monge began corresponding with D’Alembert and Condorcet. In 1780 these two had induced the Government to found an institute at the Louvre for the study of hydraulics. Monge was called to Paris to take charge, on the understanding that he spend half his time at Mézières. He was then thirty four. Three years later he was relieved of his duties at Mézières and appointed examiner of candidates for commissions in the navy, a position which he held till the outbreak of the Revolution in 1789.
In looking back over the careers of all these mathematicians of the Revolutionary period we cannot help noticing how blind they and everyone else were to what now seems so obvious to us. Not one of them suspected that he was sitting on a mine and that the train was already sputtering. Possibly our successors in 2036 will be saying the same about us.
For the six years he held the naval job Monge proved himself an incorruptible public servant. Disgruntled aristocrats threatened him with dire penalties when he unmercifully disqualified their incompetent sons, but Monge never gave in. “Get someone else to run the job if you don’t like the way I am doing it.” As a consequence the navy was ready for business in 1789.
His birth and his experiences with snobs seeking unmerited favors made Monge a natural revolutionist. By first-hand experience he knew the corruption of the old order and the economic disabilities of the masses, and he believed that the time had come for a new deal. But like the majority of early liberals Monge did not know that a mob which has once tasted blood is not satisfied till no more is forthcoming. The early revolutionists had more faith in Monge than he had in himself. Against his better judgment they forced him into the Ministry of the Navy and the Colonies on August 10, 1792. He was the man for the position, but it was not healthy to be a public official in the Paris of 1792.
The mob was already out of hand; Monge was put on the Provisional Executive Council to attempt some measure of control. A son of the people himself, Monge felt that he understood them better than did some of his friends—Condorcet, for instance, who had wisely declined the naval job to save his head.
But there are people and people, all of whom together comprise “the people.” By February, 1793 Monge found himself suspect of being not quite radical enoug
h, and on the 13th he resigned, only to be re-elected on the 18th to a job which stupid political interference, “liberty, equality, and fraternity” among the sailors, and approaching bankruptcy of the state had made impossible. Any day during this difficult time Monge might have found himself on the scaffold. But he never truckled to ignorance and incompetence, telling his critics to their faces that he knew what was what while they knew nothing. His only anxiety was that dissension at home would lay France open to an attack which would nullify all the gains of the Revolution.
At last, on April 10, 1793, Monge was allowed to resign in order to undertake more urgent work. The anticipated attack was now plainly visible.
With the arsenals almost empty the Convention began raising an army of 900,000 men for defense. Only a tenth of the necessary munitions existed and there was no hope of importing the requisite materials—copper and tin for the manufacture of bronze cannon, saltpetre for gunpowder, and steel for firearms. “Give us saltpetre from the earth and in three days we shall be loading our cannon,” Monge told the Convention. All very well, they retorted, but where were they to get the saltpetre? Monge and Berthollet showed them.
The entire nation was mobilized. Under Monge’s direction bulletins were sent to every town, farmstead, and village in France telling the people what to do. Led by Berthollet the chemists invented new and better methods for refining the raw material and simplified the manufacture of gunpowder. The whole of France became a vast powder factory. The chemists also showed the people where to find tin and copper—in clock metal and church bells. Monge was the soul of it all. With his prodigious capacity for work he spent his days supervising the foundries and arsenals, and his nights writing bulletins for the direction of the workers, and throve on it. His bulletin on The Art of Manufacturing Cannon became the factory handbook.
