The theory of determinants in the historical order of development, by Sir Thomas Muir.

414 HISTORY OF THE THEORY OF DETERMINANTS

The study was pursued by Hindenburg and his followers during
the last twenty years of the eighteenth century, but not with
any great profit; and, although in the first half of the nineteenth
century considerable attention was given to the theory of continued fractions as a whole, little advance was made in elucidating
the properties of the functions referred to.* Their connection
with determinants, after the awakening of interest in the latter
about 1841, was sure sooner or later to be detected: there is no
evidence, however, of the discovery having been made before the
year 1853.
SYLVESTER, J. J. (1853, May 13).
[On a remarkable modification of Sturm's theorem. Philos.
Magazine (4), v. pp. 446-457; or Collected Math. Papers,
i. pp. 609-619.]
The mention of Sturm's theorem in the title of a paper renders
not improbable the occurrence therein of matter connected
with continued fractions. Especially likely is this in the case
of a writer like Sylvester when in a characteristic mood; and,
assuredly, the present communication is in structure, style, and
originality redolent of its author. It must have been written in
the white heat of discovery. The main part of it consists of six
pages: this is followed by a "Remark" a page and a quarter
long; then comes a " Postscript" of three and a half pages; and
finally a small-page footnote as long as the " Remark."
It is the postscript which particularly concerns us. It begins
thus:
"Suppose that we have any series of terms, up, u2, u3,..., u,, where
U = A1, ' 2=A1A2-l, -1 3=A1A2A3-Al-A3.
and in general
Zt == Aui - i-2,
then ul i, '2, u,..,, will be the successive principal coaxal determi* The state of the theory in 1833 can best be gathered from Stern's monograph,
published in vol. x. of Crelle's Journal.