Men of Mathematics (37 page)

*  *  *

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 enough, 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.

Monge was not without enemies as the Revolution continued to fester. One day Monge's wife heard that Berthollet and her husband were to be denounced. Frantic with fear she ran to the Tuileries to learn the truth. She found Berthollet sitting quietly under the chestnut trees. Yes; he had heard the rumor, but believed nothing would happen for a week. “Then,” he added with his habitual composure, “we shall certainly be arrested, tried, condemned, and executed.”

When Monge came home that evening his wife told him Berthollet's prediction. “My word!” Monge exclaimed; “I know nothing of all that. What I do know is that my cannon factories are going forward marvelouslly!”

Shortly after this Citizen Monge was denounced by the porter at his lodgings. This was too much, even for Monge. He prudently left Paris till the storm blew over.

*  *  *

The third stage of Monge's career opened in 1796 with a letter
from Napoleon. The two had already met in 1792, but Monge was unaware of the fact. Monge at the time was fifty, Napoleon twenty three years younger.

“Permit me,” Napoleon wrote, “to thank you for the cordial welcome that a young artillery officer, little in favor, received from the Minister of the Navy in
1792;
he has preciously preserved its memory. You see this officer in the present general of the Army [of invasion] of Italy; he is happy to extend you a hand of recognition and friendship.”

Thus began the long intimacy between Monge and Napoleon. Commenting on this singular alliance, Arago
^I
reports Napoleon's words “Monge loved me as one loves a mistress.” On the other side Monge seems to have been the only man for whom Napoleon ever had an unselfish and abiding friendship. Napoleon knew of course that Monge had helped to make his career possible; but that was not the root of his affection for the older man.

The “recognition” mentioned in Napoleon's letter was the appointment of Monge and Berthollet by the Directory as commissioners sent to Italy to select the paintings, sculpture, and other works of art “donated” by the Italians (after being bled white of money) as part of their contribution to the expenses of Napoleon's campaign. In picking over the loot Monge developed a keen appreciation of art and became quite a connoisseur.

The practical implications of the looting, however, disturbed him somewhat, and when enough to furnish the Louvre half a dozen times over had been lifted and shipped to Paris, Monge counselled moderation. It would not do, he said, in governing a people either for their own good or for that of the conquerors to beggar them completely. His advice was heeded, and the goose continued laying its golden eggs.

After the Italian adventure Monge joined Napoleon at his chateau near Udine. The two became great cronies, Napoleon revelling in Monge's conversation and inexhaustible fund of interesting information, and Monge basking in the commander-in-chief's genial humor. At public banquets Napoleon always ordered the band to strike up
the
Marseillaise
—“Monge is an enthusiast for it!” Indeed he was, shouting it at the top of his lungs before sitting down to meals,

“Allons, enfants de la patrie,

Le jour de gloire est arrivé!”

It will be our special privilege to see the day of glory arriving in the company of another great Napoleonic mathematician—Poncelet.

In December, 1797, Monge made a second trip to Italy, this time as a member of the commission to investigate the “great crime” of General Duphot's assassination. The General had been shot down in Rome while standing near Lucien Bonaparte. The commission (rudely anticipated by one of the martyred General's brothers in arms) somewhat lamely prescribed a republic modelled on the French for the obstreperous Italians. “There must be an end of everything, even of the rights of conquest,” as one of the negotiators remarked when the matter of further extortions came up.

How right this canny diplomat was came out eight months later when the Italians scrapped their republic to the great embarrassment of Napoleon, then in Cairo, and to the greater embarrassment of Monge and Fourier who happened to be with him.

Monge was one of the dozen or so to whom Napoleon in 1798 confided his plan for the invasion, conquest, and civilization of Egypt. As Fourier enters naturally here we shall go back and pick him up.

*  *  *

Jean-Baptiste-Joseph Fourier, born on March 21, 1768, at Auxerre, France, was the son of a tailor. Orphaned at the age of eight, he was recommended to the Bishop of Auxerre by a charitable lady who had been captivated by the boy's good manners and serious deportment—little did she dream what he was to become. The Bishop got Fourier into the local military college run by the Benedictines, where the boy soon proved his genius. By the age of twelve he was writing magnificent sermons for the leading church dignitaries of Paris to palm off as their own. At thirteen he was a problem child, wayward, petulant, and full of the devil generally. Then, at his first encounter with mathematics, he changed as if by magic. He knew what had ailed him and cured himself. To provide light for his mathematical studies after he was supposed to be asleep he collected candle-ends in the kitchen and wherever he could find them in the college. His secret study was an inglenook behind a screen.