Men of Mathematics (36 page)

The potential is merely the function
u
described in connection with fluid motion and Laplace's equation in the chapter on Newton. The function
u
is there a “velocity potential”; if it is a question of the force of Newtonian gravitational attraction,
u
is a “gravitational potential.” The introduction of the potential into the theories of fluid motion, gravitation, electromagnetism, and elsewhere was one of the longest strides ever taken in mathematical physics. It had the effect of replacing partial differential equations in two or three unknowns by equations in one unknown.

*  *  *

In 1785, at the age of thirty six, Laplace was promoted to full membership in the Academy. Important as this honor was in the career of a man of science, the year 1785 stands out as a landmark of yet greater significance in Laplace's career as a public character. For in that year Laplace had the unique distinction of examining a singular candidate of sixteen at the Military School. This youth was destined to upset Laplace's plans and deflect him from his avowed devotion to mathematics into the muddy waters of politics. The young man's name was Napoleon Bonaparte (1769-1821).

Laplace rode through the Revolution on horseback, as it were, and saw everything in comparative safety. But no man of his prominence and restless ambition could escape danger entirely. If De Pastoret knew what he was talking about in his eulogy, both Lagrange and Laplace escaped the guillotine only because they were requisitioned to calculate trajectories for the artillery and to help in directing the manufacture of saltpetre for gunpowder. Neither was forced to eat grass as some less necessary savants were driven to do, nor was either so careless as to betray himself, as their unfortunate friend Condorcet did, by ordering an aristocrat's omelet. Not knowing how many eggs go into a normal omelet Cordorcet ordered a dozen. The good cook asked Condorcet his trade. “Carpenter.”—“Let me see your hands. You're no carpenter.” That was the end of Laplace's close friend Condorcet. They either poisoned him in prison or let him commit suicide.

After the Revolution Laplace went in heavily for politics, possibly in the hope of beating Newton's record. The French refer politely to Laplace's “versatility” as a politician. This is too modest. Laplace's alleged defects as a politician are his true greatness in the slippery game. He has been criticized for his inability to hold public office under successive regimes without changing his politics. It would seem that a man who is sharp enough to convince opposing parties that he is a loyal supporter of whichever one happens to be in power at the moment is a politician of no mean order. It was his patrons who played the game like amateurs, not Laplace. What would we think of a Republican Postmaster General who gave all the fattest jobs to undeserving Democrats? Or the other way about? Laplace got a better job every time the government flopped. It cost him nothing to switch overnight from rabid republicanism to ardent royalism.

Napoleon shoved everything Laplace's way, including the portfolio of the interior—about which more later. All the Napoleonic orders of any note adorned the versatile mathematician's chest—including the Grand Cross of the Legion of Honor and the Order of the Reunion, and he was made a Count of the Empire. Yet what did he do when Napoleon fell? Signed the decree which banished his benefactor.

After the restoration Laplace had no difficulty in transferring his loyalty to Louis XVIII, especially as he now sat in the Chamber of Peers as the Marquis de Laplace. Louis recognized his supporter's merits and in 1816 appointed Laplace president of the committee to reorganize the École Polytechnique.

Perhaps the most perfect expressions of Laplace's political genius are those to be found in his scientific writings. It takes real genius to doctor science according to fluctuating political opinion and get away with it. The first edition of the
Exposition du système du monde,
dedicated to the Council of Five Hundred, closes with these noble words: “The greatest benefit of the astronomical sciences is to have dissipated errors born of ignorance of our true relations with nature, errors all the more fatal since the social order must rest solely on these relations.
Truth
and
justice
are its immutable bases. Far from us be the dangerous maxim that it may sometimes be useful to deceive or to enslave men the better to insure their happiness! Fatal experiences have proved in all ages that these sacred laws are never infringed with impunity.” In 1824 this is suppressed and the Marquis de Laplace substitutes: “Let us conserve with care and increase the store of this advanced knowledge, the delight of thinking beings. It has rendered important services to navigation and geography; but its greatest benefit is to have dissipated the fears produced by celestial phenomena and to have destroyed the errors born of ignorance of our true relations with nature, errors which will soon reappear if the torch of the sciences is extinguished.” In loftiness of sentiment there is but little to choose between these two sublime maxima.

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.

I cannot tell you the efforts to which I was condemned to understand something of the diagrams of Descriptive Geometry, which I detest.

—C
HARLES
H
ERMITE

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.

—W
ILLIAM
T
HOMSON AND
P. G. T
AIT

T
HE 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.”