Men of Mathematics (74 page)

In
1838,
at the age of twenty four, Sylvester got his first regular job, that of Professor of Natural Philosophy (science in general, physics in particular) at University College, London, where his old teacher De Morgan was one of his colleagues. Although he had studied chemistry at Cambridge, and retained a lifelong interest in it, Sylvester found the teaching of science thoroughly uncongenial and, after about two years, abandoned it. In the meantime he had been elected a Fellow of the Royal Society at the unusually early age of twenty five. Sylvester's mathematical merits were so conspicuous that they could not escape recognition, but they did not help him into a suitable position.

At this point in his career Sylvester set out on one of the most singular misadventures of his life. Depending upon how we look at it, this mishap is silly, ludicrous, or tragic. Sanguine and filled with his usual enthusiasm, he crossed the Atlantic to become Professor of Mathematics at the University of Virginia in 1841—the year in which Boole published his discovery of invariants.

Sylvester endured the University only about three months. The refusal of the University authorities to discipline a young gentleman who had insulted him caused the professor to resign. For over a year after this disastrous experience Sylvester tried vainly to secure a suitable position, soliciting—unsuccessfully—both Harvard and Columbia Universities. Failing, he returned to England.

Sylvester's experiences in America gave him his fill of teaching for the next ten years. On returning to London he became an energetic actuary for a life insurance company. Such work for a creative mathematician is poisonous drudgery, and Sylvester almost ceased to be a mathematician. However, he kept alive by taking a few private pupils, one of whom was to leave a name that is known and revered in every country of the world today. This was in the early 1850's, the “potatoes, prunes, and prisms” era of female propriety when young women were not supposed to think of much beyond dabbling in paints and piety. So it is rather surprising to find that Sylvester's most distinguished pupil was a young woman, Florence Nightingale, the first
human being to get some decency and cleanliness into military hospitals—over the outraged protests of bull-headed military officialdom. Sylvester at the time was in his late thirties, Miss Nightingale six years younger than her teacher. Sylvester escaped from his makeshift ways of earning a living in the same year (
1854)
that. Miss Nightingale went out to the Crimean War.

Before this however he had taken another false step that landed him nowhere. In
1846,
at the age of thirty two, he entered the Inner Temple (where he coyly refers to himself as “a dove nestling among hawks”) to prepare for a legal career, and in
1850
was called to the Bar. Thus he and Cayley came together at last.

Cayley was twenty nine, Sylvester thirty six at the time; both were out of the real jobs to which nature had called them. Lecturing at Oxford thirty five years later Sylvester paid grateful tribute to “Cayley, who, though younger than myself is my spiritual progenitor—who first opened my eyes and purged them of dross so that they could see and accept the higher mysteries of our common Mathematical faith.” In
1852,
shortly after their acquaintance began, Sylvester refers to “Mr. Cayley, who habitually discourses pearls and rubies.” Mr. Cayley for his part frequently mentions Mr. Sylvester, but always in cold blood, as it were. Sylvester's earliest outburst of gratitude in print occurs in a paper of
1851
where he says, “The theorem above enunciated [it is his relation between the minor determinants of linearly equivalent quadratic forms] was in part suggested in the course of a conversation with Mr. Cayley (to whom I am indebted for my restoration to the enjoyment of mathematical life) . . ..”

Perhaps Sylvester overstated the case, but there was a lot in what he said. If he did not exactly rise from the dead he at least got a new pair of lungs: from the hour of his meeting with Cayley he breathed and lived mathematics to the end of his days. The two friends used to tramp round the Courts of Lincoln's Inn discussing the theory of invariants which both of them were creating and later, when Sylvester moved away, they continued their mathematical rambles, meeting about halfway between their respective lodgings. Both were bachelors at the time.

*  *  *

The theory of algebraic invariants from which the various extensions of the concept of invariance have grown naturally originated in
an extremely simple observation. As will be noted in the chapter on Boole, the earliest instance of the idea appears in Lagrange, from whom it passed into the arithmetical works of Gauss. But neither of these men noticed that the simple but remarkable algebraical phenomenon before them was the germ of a vast theory. Nor does Boole seem to have fully realized what he had found when he carried on and greatly extended the work of Lagrange. Except for one slight tiff, Sylvester was always just and generous to Boole in the matter of priority, and Cayley, of course, was always fair.

The simple observation mentioned above can be understood by anyone who has ever seen a quadratic equation solved, and is merely this. A necessary and sufficient condition that the equation
ax
²
+ 2
bx
+
c
= 0 shall have two equal roots is that
b
²
– ac
shall be zero. Let us replace the variable
x
by its value in terms of
y
obtained by the transformation
y = (px
+
q)/(rx + s).
Thus
x
is to be replaced by the result of solving this for
x,
namely
x
= (
q
–
sy
)/(
ry
–
p
). This transforms the given equation into another in
y
; say the new equation is
Ay
²
+
2By
+
C
= 0. Carrying out the algebra we find that the new coefficients
A, B, C
are expressed in terms of the old
a, b, c
as follows,

A = as
²
–
2bsr
+
cr
²
,

B = –aqs + b(qr
-f
sp) – cpr,

C = aq
²
- 2bpq + cp
²
.

From these it is easy to show (by brute-force reductions, if necessary, although there is a simpler way of reasoning the result out, without actually calculating
A, B, C)
that

B
²
- AC
=
(ps - qr)
²
(b
²
- ac).

Now
b
²
– ac
is called the discriminant of the quadratic equation in
x
; hence the discriminant of the quadratic in
y
is
B
²
– AC,
and it has been shown that
the discriminant of the transformed equation is equal to the discriminant of the original equation, times the factor (ps – qr)
²
which depends only upon the coefficients p, q, r, s in the transformation
y
= (
px + q)/(rx + s)
by means of which x was expressed in terms of y.

Boole was the first (in 1841) to observe something worth looking at in this particular trifle. Every algebraic equation has a discriminant, that is, a certain expression (such as
b
²
—ac
for the quadratic) which is equal to zero if, and only if, two or more roots of the equation are
equal. Boole first asked, does the discriminant of every equation when its
x
is replaced by the related
y
(as was done for the quadratic) come back unchanged except for a factor depending only on the coefficients of the transformation? He found that this was true. Next he asked whether there might not be expressions other than discriminants constructed from the coefficients having this same property of
invariance
under
transformation.
He found two such for the general equation of the fourth degree. Then another man, the brilliant young German mathematician, F. M. G. Eisenstein (1823-1852) following up a result of Boole's, in 1844, discovered that certain expressions involving
both the coefficients and the x
of the original equations exhibit the same sort of invariance: the original coefficients and the original
x
pass into the transformed coefficients and
y
(as for the quadratic), and the expressions in question constructed from the originals differ from those constructed from the transforms only by a factor which depends solely on the coefficients of the transformation.

Neither Boole nor Eisenstein had any
general
method for finding such
invariant
expressions. At this point Cayley entered the field in
1845
with his pathbreaking memoir,
On the Theory of Linear Transformations.
At the time he was twenty four. He set himself the problem of finding uniform methods which would give him
all
the invariant expressions of the kind described. To avoid lengthy explanations the problem has been stated in terms of equations; actually it was attacked otherwise, but this is of no importance here.

As this question of invariance is fundamental in modern scientific thought we shall give three further illustrations of what it means, none of which involves any symbols or algebra. Imagine any figure consisting of intersecting straight lines and curves drawn on a sheet of paper. Crumple the paper in any way you please without tearing it, and try to think what is the most obvious property of the figure that is the same before and after crumpling. Do the same for any figure drawn on a sheet of rubber, stretching but not tearing the rubber in any complicated manner dictated by whim. In this case it is obvious that sizes of areas and angles, and lengths of lines, have
not
remained “invariant.” By suitably stretching the rubber the straight lines may be distorted into curves of almost any tortuosity you like, and at the same time the original curves—or at least some of them—may be transformed into straight lines. Yet
something
about the whole figure has remained unchanged; its very simplicity and obviousness
might well cause it to be overlooked. This is the order of the points on any one of the lines of the figure which mark the places where other lines intersect the given one. Thus, if moving the pencil along a given line from
A
to C, we had to pass over the point
B
on the line before the figure was distorted, we shall have to pass over
B
in going from
A
to
C
after distortion. The
order
(as described) is an
invariant
under the particular
transformations
which crumpled the sheet of paper into a crinkly ball, say, or which stretched the sheet of rubber.

This illustration may seem trivial, but anyone who has read a non-mathematical description of the intersections of “world-lines” in general relativity, and who recalls that an intersection of two such lines marks a physical
“point-event”
will see that what we have been discussing is of the same stuff as one of our pictures of the physical universe. The mathematical machinery powerful enough to handle such complicated “transformations” and actually to produce the invariants was the creation of many workers, including Riemann, Christoffel, Ricci, Levi-Civita, Lie, and Einstein—all names well known to readers of popular accounts of relativity; the whole vast program was originated by the early workers in the theory of algebraic invariants, of which Cayley and Sylvester were the true founders.

As a second example, imagine a knot to be looped in a string whose ends are then tied together. Pulling at the knot, and running it along the string, we distort it into any number of “shapes.” What remains “invariant,” what is “conserved,” under all these distortions which, in this case, are our transformations? Obviously neither the shape nor the size of the knot is invariant. But the “style” of the knot itself is invariant; in a sense that need not be elaborated, it is the
same sort
of a knot whatever we do to the string provided we do not untie its ends. Again, in the older physics, energy was “conserved”; the total amount of energy in the universe was assumed to be an invariant, the same under all transformations from one form, such as electrical energy, into others, such as heat and light.

Our third illustration of invariance need be little more than an allusion to physical science. An observer fixes his “position” in space and time with reference to three mutually perpendicular axes and a standard timepiece. Another observer, moving relatively to the first, wishes to describe the same physical event that the first describes. He also has his space-time reference system; his movement relatively
to the first-observer can be expressed as a transformation of his own coordinates (or of the other observer's). The descriptions given by the two may or may not differ in mathematical form, according to the particular kind of transformation concerned. If their descriptions do differ, the difference is not, obviously, inherent in the physical event they are both observing, but in their reference systems and the transformation. The problem then arises to formulate only those mathematical expressions of natural phenomena which shall be independent, mathematically, of any
particular
reference system and therefore be expressed by all observers in the same form. This is equivalent to finding the invariants of the transformation which expresses the most general shift in “space-time” of one reference system with respect to any other. Thus the problem of finding the mathematical expressions for the intrinsic laws of nature is replaced by an attackable one in the theory of invariants. More will be said on this when we come to Riemann.