Men of Mathematics (39 page)

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 2
π
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
=
a
₀
+ a
₁
cos
x
+
a
₂
cos
2x
+
a
₂
cos
3x
+ . . .

+
b
₁
sin
x
+
b
₂
sin 2
x
+
b
₃
sin 3
x
+ . . .

where the dots indicate that the two series are to continue indefinitely according to the rule shown, and the coefficients
a
₀
, a
₁
a
₂
,
 . . . ,
b
₁
,
b
₂
,
b
₃
, . . . 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 problems given in the chapter on Newton are solved by Fourier's method. In any given problem it is required to find the coefficients
a
₀
, a
₁
 . . . ,
b
₀
, b
₁
 . . . 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.

*  *  *

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.