by E. T. Bell
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It was while at Grenoble that Fourier composed the immortal Theorie analytique de la chaleur (The Mathematical Theory of Heat), a landmark in mathematical physics. His first memoir on the conduction of heat was submitted in 1807. This was so promising that the Academy encouraged Fourier to continue by setting a contribution to the mathematical theory of heat as its problem for the Grand Prize in 1812. Fourier won the prize, but not without some criticism which he resented deeply but which was well taken.
Laplace, Lagrange, and Legendre were the referees. While admitting the novelty and importance of Fourier’s work they pointed out that the mathematical treatment was faulty, leaving much to be desired in the way of rigor. Lagrange himself had discovered special cases of Fourier’s main theorem but had been deterred from proceeding to the general result by the difficulties which he now pointed out. These subtle difficulties were of such a nature that their removal at the time would probably have been impossible. More than a century was to elapse before they were satisfactorily met.
In passing it is interesting to observe that this dispute typifies a radical distinction between pure mathematicians and mathematical physicists. The only weapon at the disposal of pure mathematicians is sharp and rigid proof, and unless an alleged theorem can withstand the severest criticism of which its epoch is capable, pure mathematicians have but little use for it.
The applied mathematician and the mathematical physicist, on the other hand, are seldom so optimistic as to imagine that the infinite complexity of the physical universe can be described fully by any mathematical theory simple enough to be understood by human beings. Nor do they greatly regret that Airy’s beautiful (or absurd) picture of the universe as a sort of interminable, self-solving system of differential equations has turned out to be an illusion born of mathematical bigotry and Newtonian determinism; they have something more real to appeal to at their own back door—the physical universe itself. They can experiment and check the deductions of their purposely imperfect mathematics against the verdict of experience—which, by the very nature of mathematics, is impossible for a pure mathematician. If their mathematical predictions are contradicted by experiment they do not, as a mathematician might, turn their backs on the physical evidence, but throw their mathematical tools away and look for a better kit.
This indifference of scientists to mathematics for its own sake is as enraging to one type of pure mathematician as the omission of a doubtful iota subscript is to another type of pedant. The result is that but few pure mathematicians have ever made a significant contribution to science—apart, of course, from inventing many of the tools which scientists find useful (perhaps indispensable). And the curious part of it all is that the very purists who object to the boldly imaginative attack of the scientists are the loudest in their insistence that mathematics, contrary to a widely diffused belief, is not all an affair of grubbing, meticulous accuracy, but is as creatively imaginative, and sometimes as loose, as great poetry or music can be on occasion. Sometimes the physicists beat the mathematicians at their own game in this respect: ignoring the glaring lack of rigor in Fourier’s classic on the analytical theory of heat, Lord Kelvin called it “a great mathematical poem.”
As has already been stated Fourier’s main advance was in the direction of boundary-value problems (described in the chapter on Newton)—the fitting of solutions of differential equations to prescribed initial conditions, probably the central problem of mathematical physics. Since Fourier applied this method to the mathematical theory of heat conduction a crowded century of splendidly gifted men has gone farther than he would ever have dreamed possible, but his step was decisive. One or two of the things he did are simple enough for description here.
In algebra we learn to plot the graphs of simple algebraic equations and soon notice that the curves we get, if continued sufficiently far, do not break off suddenly and end for good. What sort of an equation would result in a graph like that of the heavy line segment (finite length, terminated at both ends) repeated indefinitely as in the figure?
Such graphs, made up of disjointed fragments of straight or curved lines recur repeatedly in physics, for example in the theories of heat, sound, and fluid motion. It can be proved that it is impossible to represent them by finite, closed, mathematical expressions; an infinity of terms occur in their equations. “Fourier’s Theorem” provides a means for representing and investigating such graphs mathematically: it expresses (within certain limitations) a given function continuous within a certain interval, or with only a finite number of discontinuities in the interval, and having in the interval only a finite number of turning-points, as an infinite sum of sines or cosines, or both. (This is only a rough description.)
Having mentioned sines and cosines we shall recall their most important property, periodicity. Let the radius of the circle in the figure be 1 unit in length. Through the center O draw rectangular axes as in Cartesian geometry, and mark off AB equal to 2π units of length; thus AB is equal in length to the circumference of the circle (since the radius is l). Let the point P start from A and trace out the circle in the direction of the arrow. Drop PN perpendicular to OA. Then, for any position of P, the length of NP is called the sine of the angle AOP, and ON the cosine; NP and ON are to have their signs as in Cartesian geometry (NP is positive above OA, negative below; ON is positive to the right of OC, negative to the left).
For any position of P, the angle AOP will be that fraction of four right angles (360°) that the arc AP is of the whole circumference of the circle. So we may scale off these angles AOP by marking along AB the fractions of 2π which correspond to the arcs AP. Thus, when P is at C, ¼ the whole circumference has been traversed; hence, corresponding to the angle AOC we have the point K at ¼ of AB from A.
At each of these points on AB we erect a perpendicular equal in length to the sine of the corresponding angle, and above or below A B according as the sine is positive or negative. The ends of these perpendiculars not on AB lie on the continuous curve shown, the sine curve. When P returns to A and begins retracing the circle the curve is repeated beyond B, and so on indefinitely. If P revolves in the opposite direction, the curve is repeated to the left. After an interval of 2π the curve repeats: the sine of an angle (here AOP) is aperiodic function, the period being 2π The word “sine” is abbreviated to “sin”; and, if x is any angle, the equation expresses the fact that sin x is a function of x having the period 2π
sin (x + 2π) = sin x
It is easily seen that if the whole curve in the figure is shifted to the left a distance equal to AK, it now graphs the cosine of AOP. As before
cos (x + 2π) = COS X,
“cos” being the short for “cosine”
Inspection of the figure shows that sin 2x will go through its complete period “twice as fast” as sin x, and hence that the graph for a complete period will be one half as long as that for sin x. Similarly sin 3x will require only 2π/3 for its complete period, and so on. The same holds for cos x, cos 2x, cos 3x, . . ..
Fourier’s main mathematical result can now be described roughly. Within the restrictions already mentioned in connection with “broken” graphs, any function having a well-determined graph can be represented by an equation of the type
y = a0 + a1 cos x + a2 cos 2x + a2 cos 3x + . . .
+ b1 sin x + b2 sin 2x + b3 sin 3x + . . .
where the dots indicate that the two series are to continue indefinitely according to the rule shown, and the coefficients a0, a1 a2, . . . , b1, b2, b3, . . . are determinable when y, any given function of x, is known. In other words, any given function of x, say f(x), can be expanded in a series of the type stated above, a trigonometric or Fourier series. To repeat, all this holds only within certain restrictions which, fortunately, are not of much importance in mathematical physics; the exceptions are more or less freak cases of little or no physical significance. Once more, Fourier’s was the first great attack on boundary value problems. The specimens of such problem
s given in the chapter on Newton are solved by Fourier’s method. In any given problem it is required to find the coefficients a0, a1 . . . , b0, b1 . . . in a form adapted to computation. Fourier’s analysis provides this.
The concept of periodicity (simple periodicity) as described above is of obvious importance for natural phenomena; the tides, the phases of the Moon, the seasons, and a multitude of other familiar things are periodic in character. Sometimes a periodic phenomenon, such for example as the recurrence of sunspots, can be closely approximated by superposition of a certain number of graphs having simple periodicity. The study of such situations can then be simplified by analysing the individual periodic phenomena of which the original is the resultant,
The process is the same mathematically as the analysis of a musical sound into its fundamental and successive harmonics. As a first very crude approximation to the “quality” of the sound only the fundamental is considered; the superposition of only a few harmonics usually suffices to produce a sound indistinguishable from the ideal (in which there is an infinity of harmonics). The like holds for phenomena attacked by “harmonic” or “Fourier” analysis. Attempts have even been made to detect long periods (the fundamentals) in the recurrence of earthquakes and annual rainfall. The notion of simple periodicity is as important in pure mathematics as it is in applied, and we shall see it being generalized to multiple periodicity (in connection with elliptic functions and others), which in its turn reacts on applied mathematics.
Fully aware that he had done something of the first magnitude Fourier paid no attention to his critics. They were right, he wrong, but he had done enough in his own way to entitle him to independence.
When the work begun in 1807 was completed and collected in the treatise on heat-conduction in 1822, it was found that the obstinate Fourier had not changed a single word of his original presentations, thus exemplifying the second part of Francis Galton’s advice to all authors: “Never resent criticism, and never answer it.” Fourier’s resentment was rationalized in attacks on pure mathematicians for minding their own proper business and not blundering about in mathematical physics.
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All was going well with Fourier and France in general when Napoleon, having escaped from Elba, landed on the French coast on March 1, 1815. Veterans and all were just getting comfortably over their headache when the cause of it popped up again to give them a worse one. Fourier was at Grenoble at the time. Fearing that the populace would welcome Napoleon back for another spree, Fourier hastened to Lyons to tell the Bourbons what was about to happen. With their usual stupidity they refused to believe him. On his way back Fourier learned that Grenoble had capitulated. Fourier himself was taken prisoner and brought before Napoleon at Bourgoin. He was confronted by the same old commander he had known so well in Egypt and had learned to distrust with his head but not with his viscera. Napoleon was bending over a map, a pair of compasses in his hand. He looked up.
“Well, Monsieur Prefect! You too; you have declared war against me?”
“Sire,” Fourier stammered, “my oaths made it a duty.”
“A duty, do you say? Don’t you see that nobody in the country is of your opinion? And don’t let yourself imagine that your plan of campaign frightens me much. I suffer only at seeing amongst my adversaries an Egyptian, a man who has eaten the bread of the bivouac with me, an old friend! How, moreover, Monsieur Fourier, have you been able to forget that I made you what you are?”
That Fourier, remembering Napoleon’s callous abandonment of him in Egypt, could swallow such tripe and like it says a great deal for the goodness of his heart and the toughness of his stomach but precious little for the soundness of his head.
Some days later Napoleon asked the now loyal Fourier:
“What do you think of my plan?”
“Sire, I believe you will fail. You will meet a fanatic on your road, and everything will be over.”
“Bah! Nobody is for the Bourbons—not even a fanatic. As for that, you have read in the papers that they have put me outside the law. I myself will be more indulgent: I shall content myself with putting them outside the Tuileries!”
The leopard’s spots and Napoleon’s swellhead should be wedded in one proverb instead of pining apart in two.
The second restoration found Fourier in Paris pawning his effects to keep alive. But before he could starve to death old friends took pity on him and got him appointed director of the Bureau of Statistics for the Seine. The Academy tried to elect him to membership in 1816, but the Bourbon government ordered that no friend of their late kicker was to be honored in any way. The Academy stuck to its guns and elected Fourier the following year. This action of the Bourbons against Fourier may seem petty, but beside what they did to poor old Monge it was princely. Noblesse oblige!
Fourier’s last years evaporated in clouds of talk. As Permanent Secretary of the Academy he was always able to find listeners. To say that he bragged of his achievements under Napoleon is putting it altogether too mildly. He became an insufferable, shouting bore. And instead of continuing with his scientific work he entertained his audience with boastful accounts of what he was going to do. However, he had done far more than his share for the advancement of science, and if any human work merits immortality, Fourier’s does. He did not need to boast or bluff.
Fourier’s experiences in Egypt were responsible for a curious habit which may have hastened his death. Desert heat, he believed, was the ideal condition for health. In addition to swathing himself like a mummy he lived in rooms which his uncooked friends said were hotter than hell and the Sahara desert combined. He died of heart disease (some say an aneurism) on May 16, 1830, in the sixty third year of his life. Fourier belongs to that select company of mathematicians whose work is so fundamental that their names have become adjectives in every civilized language.
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Monge’s decline was slower and more distressing. After the first restoration Napoleon felt embittered and vindictive toward the snobocracy of his own creation which, naturally, had let him down the moment his power waned. Once more in the saddle Napoleon was inclined to use the butt end of his crop on the skulls of the ungrateful. Monge, good old plebeian that he was, counselled mercy and common sense: Napoleon might some day find himself with his back to the wall (after an earthquake had cut off all means of flight), and be grateful for the support of the ingrates. Cooling off, Napoleon wisely tempered injustice with mercy. For this gracious dispensation Monge alone was responsible.
After Napoleon had run away from Waterloo, leaving his troops to get out of the mess as best they could, he returned to Paris. Fourier’s devotion cooled then; Monge’s boiled.
The school histories often tell of Napoleon’s last dream—the conquest of America. The Mongian version differs and is on a much higher—in fact, incredibly high—plane. Hemmed in by enemies and appalled at the thought of enforced idleness for lack of further European conquest, Napoleon turned his eagle eye West, and in one flashing glance surveyed America from Alaska to Cape Horn. But, like the sick devil he was, Bonaparte longed to become a monk. The sciences alone could satisfy him, he declared; he would become a second and infinitely greater Alexander von Humboldt.
“I wish,” he confessed to Monge, “in this new career to leave works, discoveries, worthy of me.”
What, precisely, are the works which could be worthy of a Napoleon? Continuing, the fallen eagle outlined his dream.
“I need a companion,” he admitted, “to first put me abreast of the present state of the sciences. Then you [Monge] and I will traverse the whole continent, from Canada to Cape Horn; and in this immense journey we shall study all those prodigious phenomena of terrestrial physics on which the scientific world has not pronounced its verdict.” Paranoia?
“Sire,” Monge exclaimed—he was nearly sixty seven—“your collaborator is already found; I will go with you!”
His old self once more, Napoleon curtly dismissed the thought of the willing veteran hampering
his lightning marches from Baffin Bay to Patagonia.
“You are too old, Monge. I need a younger man.”
Monge tottered off to find “a younger man.” He approached the fiery Arago as the ideal travelling companion for his energetic master. But Arago, in spite of all his eloquent rhetoric on the gloriousness of glory, had learned his lesson. A general who could desert his troops as Napoleon had done at Waterloo, Arago pointed out, was no leader to follow anywhere, even in easy America.
Further negotiations were rudely halted by the British. By the middle of October Napoleon was exploring St. Helena. The hoard of money which had been put aside for the conquest of America found its way into deeper pockets than those of the scientists, and no “American Institute” rose on the banks of the Mississippi or the Amazon to match its fantastic twin overlooking the Nile.
Having enjoyed the bread of imperialism Monge now tasted the salt. His record as a revolutionist and favorite of the upstart Corsican made his head an extremely desirable object to the Bourbons, and Monge dodged from one slum to another in an endeavor to keep his head on his shoulders. For sheer human pettiness the treatment accorded Monge by the sanctified Bourbons would take a lot of beating. Small enough for anything they stripped the old man of his last honor—one with which the generosity of Napoleon had had nothing whatever to do. In 1816 they commanded that Monge be expelled from the Academy. The academicians, tame as rabbits now, obeyed.
The final touch of Bourbon pettiness graced the day of Monge’s funeral. As he had foreseen he died after a prolonged stupor following a stroke. The young men at the Polytechnique, whom he had protected from Napoleon’s domineering interference, were the pride of Monge’s heart, and he was their idol. When Monge died on July 28, 1818, the Polytechnicians asked permission to attend the funeral. The King denied the request.