Men of Mathematics (13 page)

The next step was to translate all this into algebraical or analytical language. Knowing the coordinates
x, y
of the point
P
on the graph, and those, say
x
+
a, y + b,
of
Q,
before
Q
started to slip along to coincidence with
P,
they inspected the graph and saw that the
slope
of the
chord PQ
was equal to
b/a
—obviously a measure of the “steepness” of the chord with relation to the x-axis (the line along which x-distances are measured); this “steepness” is precisely what is meant by slope. From this it was evident that the
required slope of the tangent at P
(after
Q
had slipped into coincidence with
P)
would be the
limiting value
of
b/a
as both
b
and
a
approached the value
zero
simultaneously; for
x + a, y
+
b,
the coordinates of
Q,
ultimately become
x, y,
the coordinates of P. This limiting value is the required slope. Having the slope and the point P they could now draw the tangent.

This is not exactly Fermat's process for drawing tangents but his own process was, broadly, equivalent to what has been described.

Why should all this be worth the serious attention of any rational or practical man? It is a long story, only a hint of which need be given here; more will be said when we discuss Newton. One of the fundamental ideas in dynamics is that of the
velocity
(speed) of a moving
particle. If we graph the number of units of distance passed over by the particle in a unit of time against the number of units of time, we get a line, straight or curved, which pictures at a glance the
motion
of the particle, and the
steepness
of this line at any given point of it will obviously give us the
velocity
of the particle at the instant corresponding to the point; the faster the particle is moving, the steeper the
slope
of the
tangent line.
This slope does in fact measure the velocity of the particle at any point of its path. The problem in
motion,
when translated into
geometry,
is exactly that of finding the slope of the tangent line at a given point of a curve. There are similar problems in connection with
tangent planes
to surfaces (which also have important interpretations in mechanics and mathematical physics), and all are attacked by the differential calculus—whose fundamental problem we have attempted to describe as it presented itself to Fermat and his successors.

Another use of this calculus can be indicated from what has already been said. Suppose some quantity
y
is a “function” of another,
t,
written
y = f(t),
which means that when any definite number, say 10, is substituted for
t,
so that we get
f
(10)—“function
f
of 10”—we can calculate, from the
algebraical expression off,
supposed given, the
corresponding
value of
y,
here
y
=
f
(l0). To be explicit, suppose
f(t)
is that particular “function” of
t
which is denoted in algebra by
t
²
,
or
t
×
t.
Then, when
t
= 10, we get
y = f(
10), and hence
here y =
10
²
, = 100, for
this
value of
t
; when
t
= ½,
y
= ¼, and so on, for
any
value of
t.

All this is familiar to anyone whose grammar-school education ended not more than thirty or forty years ago, but some may have forgotten what they did in arithmetic as children, just as others could not decline the Latin
mensa
to save their souls. But even the most forgetful will see that we could plot the graph of
y
=
f(t)
for any particular form of
f
(when
f
(
t
) is
t
²
the graph is a parabola like an inverted arch). Imagine the graph drawn. If it has on it
maxima
(highest) or
minima
(lowest) points—points higher or lower than those
in their immediate neighborhoods
—we observe that the tangent at each of these
maxima
or
minima
is
parallel
to the ¿-axis. That is, the
slope
of the tangent at such an
extremum
(maximum
or
minimum) of the
f(t)
we are plotting is
zero.
Thus if we were seeking the
extrema
of a given function
f
(
t)
we should again have to solve our slope-problem for the particular curve
y
=
f(t)
and, having found the slope for the
general
point
t, y,
equate to zero the algebraical expression of this slope in order to find the values of
t
corresponding to the extrema. This is substantially what Fermat did in his method of maxima and minima invented in 1628-29, but not made semipublic till ten years later when Fermat sent an account of it through Mersenne to Descartes.

The scientific applications of this simple device—duly elaborated, of course, to take account of far more complicated problems than that just described—are numerous and far reaching. In mechanics, for instance, as Lagrange discovered, there is a certain “function” of the positions (coordinates) and velocities of the bodies concerned in a problem which, when made an extremum, furnishes us with the
“equations of motion” of the system considered, and these in turn enable us to determine the motion—to describe it completely—at any given instant. In physics there are many similar functions, each of which sums up most of an extensive branch of mathematical physics in the simple requirement that the function in question must be an extremum;
^I
Hilbert in 1916 found one for general relativity. So Fermat was not fooling away his time when he amused himself in the leisure left from a laborious legal job by attacking the problem of maxima and minima. He himself made one beautiful and astonishing application of his principles to optics. In passing it may be noted that this particular discovery has proved to be the germ of the newer quantum theory—in its mathematical aspect, that of “wave mechanics”—elaborated since 1926. Fermat discovered what is usually called “the principle of least time.” It would be more accurate to say “extreme” (least
or
greatest) instead of “least.”
^II

According to this principle, if a ray of light passes from a point
A
to another point
B,
being reflected and refracted (“refracted,” that is, bent, as in passing from air to water, or through a jelly of variable density) in any manner during the passage, the path which it must take can be calculated—all its twistings and turnings due to refraction, and all its dodgings back and forth due to reflections—from the
single
requirement that the
time
taken to pass from
A
to
B
shall be an extremum (but see the preceding footnote).

From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of incidence (in reflection) is equal to the angle of reflection; the sine of the angle of incidence (in refraction) is a
constant
number times the sine of the angle of refraction in passing from one medium to another.

The matter of analytic geometry has already been mentioned; Fermat was the first to apply it to space of three dimensions. Descartes contented himself with two dimensions. The extension, familiar to all students today, would not be self-evident to even a gifted man from Descartes' developments. It may be said that there is usually greater difficulty in finding a significant extension of a particular
kind of geometry from space of two dimensions to three than there is in passing from three to four or five . . . , or
n.
Fermat corrected Descartes in an essential point (that of the classification of curves by their degrees). It seems but natural that the somewhat touchy Descartes should have rowed with the imperturbable “Gascon” Fermat. The soldier was frequently irritable and acid in his controversy over Fermat's method of tangents; the equable jurist was always unaflfectedly courteous. As usually happens the man who kept his temper got the better of the argument. But Fermat deserved to win, not because he was a more skilful debater, but because he was right.

In passing, we should suppose that Newton would have heard of Fermat's use of the calculus and would have acknowledged the information. Until 1934 no evidence to this effect had been published, but in that year Professor L. T. More recorded in his biography of Newton a hitherto unnoticed letter in which Newton says explicitly that he got the hint of the method of the differential calculus from Fermat's method of drawing tangents.

*  *  *

We now turn to Fermat's greatest work, that which is intelligible to all, mathematicians and amateurs alike. This is the so-called “theory of numbers,” or “the higher arithmetic,” or finally, to use the unpedantic name which was good enough for Gauss,
arithmetic.

The Greeks separated the miscellany which we lump together under the name “arithmetic” in elementary textbooks into two distinct compartments,
logistica,
and
arithmetica,
the first of which concerned the practical applications of reckoning to trade and daily life in general, and the second, arithmetic in the sense of Fermat and Gauss, who sought to discover the properties of numbers as such.

Arithmetic in its ultimate and probably most difficult problems investigates the mutual relationships of those common whole numbers 1, 2, 3, 4, 5, . . . which we utter almost as soon as we learn to talk. In striving to elucidate these relationships, mathematicians have been driven to the invention of subtle and abstruse theories in algebra and analysis, whose forests of technicalities obscure the initial problems—those concerning 1, 2, 3, . . . but whose real justification will be the solution of those problems. In the meantime the by-products of these apparently useless investigations amply repay those who undertake them by suggesting numerous powerful methods applicable to other fields of mathematics having direct contact with the physical universe.
To give but one instance, the latest phase of algebra, that which is cultivated today by professional algebraists and which is throwing an entirely new light on the theory of algebraic equations, traces its origin directly to attempts to settle Fermat's simple Last Theorem (which will be stated when the way has been prepared for it).

We begin with a famous statement Fermat made about prime numbers. A positive prime number, or briefly
a prime,
is any number greater than 1 which has as its divisors (without remainder) only 1 and the number itself; for example 2, 3, 5, 7, IS, 17 are primes, and so are 257,
65537.
But 4294967297 is not a prime, because it has 641 as a divisor, nor is the number 18446744073709551617, because it is exactly divisible by 274177; both 641 and 274177 are primes. When we say in arithmetic that one number has as divisor another number, or is divisible by another, we mean
exactly divisible, without remainder.
Thus 14 is divisible by 7, 15 is not. The two large numbers were displayed above with malice aforethought for a reason that will be apparent in a moment. To recall another definition, the
n
th
power
of a given number, say
N,
is the result of multiplying together
n N's,
and is written
N
ⁿ
; thus 5
²
= 5 × 5 = 25; 8
⁴
= 8 × 8 × 8 × 8 = 4096. For uniformity
N
itself may be written as
N
^l
.
Again, such a pagoda as 2
^3
5
means that we are first to calculate 3
⁵
( = 243), and then “raise” 2 to this power, 2
²⁴³
; the resulting number has seventy four digits.