Men of Mathematics (77 page)

where the
rows
of the array on the right are obtained, in an obvious way, by applying the
rows
of the first array on the left onto the columns of the second. Such arrays (of any number of rows and columns) are called
matrices.
Their algebra follows from a few simple postulates, of which we need cite only the following. The matrices
and
are
equal
(by definition) when, and only when,
a
=
A, b
=
B, c
= C,
d = D.
The
sum
of the two matrices just written is the matrix
The result of multiplying
by
m
(any
number)
is the matrix
The rule for “multiplying,” X, (or “compounding”) matrices is as exemplified for
above.

A distinctive feature of these rules is that multiplication is
not
commutative,
except for
special
kinds of matrices. For example, by the rule we get

and the matrix on the right is not equal to that which arises from the multiplication

All this detail, particularly the last, has been given to illustrate a phenomenon of frequent occurrence in the history of mathematics: the necessary mathematical tools for scientific applications have often been invented decades before the science to which the mathematics is the key was imagined. The bizarre rule of “multiplication” for matrices, by which we get different results according to the order in which we do the multiplication (unlike common algebra where
x
×
y
is always equal
to y
×
x),
seems about as far from anything of scientific or practical use as anything could possibly be. Yet sixty seven years after Cayley invented it, Heisenberg in
1925
recognized in the algebra of matrices exactly the tool which he needed for his revolutionary work in quantum mechanics.

Cayley continued in creative activity up to the week of his death, which occurred after a long and painful illness, borne with resignation and unflinching courage, on January
26, 1895.
To quote the closing sentences of Forsyth's biography: “But he was more than a mathematician. With a singleness of aim, which Wordsworth would have chosen for his 'Happy Warrior,' he persevered to the last in his nobly lived ideal. His life had a significant influence on those who knew him [Forsyth was a pupil of Cayley and became his successor at Cambridge]: they admired his character as much as they respected his genius: and they felt that, at his death, a great man had passed from the world.”

Much of what Cayley did has passed into the main current of mathematics, and it is probable that much more in his massive
Collected Mathematical Papers
(thirteen large quarto volumes of about
600
pages each, comprising
966
papers) will suggest profitable forays to adventurous mathematicians for generations to come. At present the fashion is away from the fields of Cayley's greatest interest, and the
same may be said for Sylvester; but mathematics has a habit of returning to its old problems to sweep them up into more inclusive syntheses.

*  *  *

In 1883 Henry John Stephen Smith, the brilliant Irish specialist in the theory of numbers and Savilian Professor of Geometry in Oxford University, died in his scientific prime at the age of fifty seven. Oxford invited the aged Sylvester, then in his seventieth year, to take the vacant chair. Sylvester accepted, much to the regret of his innumerable friends in America. But he felt homesick for his native land which had treated him none too generously; possibly also it gave him a certain satisfaction to feel that “the stone which the builders rejected, the same is become the head of the corner.”

The amazing old man arrived in Oxford to take up his duties with a brand-new mathematical theory (“Reciprocants”—differential invariants) to spring on his advanced students. Any praise or just recognition always seemed to inspire Sylvester to outdo himself. Although he had been partly anticipated in his latest work by the French mathematician Georges Halphen, he stamped it with his peculiar genius and enlivened it with his ineffaceable individuality.

The inaugural lecture, delivered on December 12, 1885, at Oxford when Sylvester was seventy one, has all the fire and enthusiasm of his early years, perhaps more, because he now felt secure and knew that he was recognized at last by that snobbish world which had fought him. Two extracts will give some idea of the style of the whole.

“The theory I am about to expound, or whose birth I am about to announce, stands to this ['the great theory of Invariants'] in the relation not of a younger sister, but of a brother, who, though of later birth, on the principle that the masculine is more worthy than the feminine, or at all events, according to the regulations of the Salic law, is entitled to take precedence over his elder sister, and exercise supreme sway over their united realms.”

Commenting on the unaccountable absence of a term in a certain algebraic expression he waxes lyric.

“Still, in the case before us, this unexpected absence of a member of the family, whose appearance might have been looked for, made an impression on my mind, and even went to the extent of acting on my emotions. I began to think of it as a sort of lost Pleiad in an Algebraical Constellation, and in the end, brooding over the subject, my feelings
found vent, or sought relief, in a rhymed effusion, a
jeu de sottise,
which, not without some apprehension of appearing singular or extravagant, I will venture to rehearse. It will at least serve as an interlude, and give some relief to the strain upon your attention before I proceed to make my final remarks on the general theory.