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

DETERMINANTS IN GENERAL (HESSE, 1843)

17

HESSE, 0. (1843).
[Ueber die Bildung der Endgleichung, welche durch Elimination
einer Variabeln aus zwei algebraischen Gleichungen hervorgeht, und die Bestimmung ihres Grades. Crelle's Journ.,
xxvii. pp. 1-5; or Werke, pp. 83 —88.]
Hesse, at this time, must have been unaware of Richelot's
paper (dated from the same University), and Grunert's paper,
not to speak of writings published outside Germany, for the
method which he gives of finding the final equation is nothing
more nor less than Sylvester's dialytic method. His exposition,
to say the least, is not preferable to Grunert's, and the determinant of the (m+ t)th order which he prints is misleading
in points of detail.
CAYLEY, A. (1843).
[On the theory of determinants. Transac. Cambridge Philos.
Soc., viii. pp. 1-16; or Collected Math. Papers, i. pp. 63-79.]
Up to this point Cayley had dealt with determinants, only,
as it were, incidentally. Now, however, he devotes a memoir
of sixteen quarto pages to the study of them.
The introductory page shows a pretty wide acquaintance with
previous writings on the subject, the authors mentioned being
Cramer, Bezout (1764), Laplace, Vandermonde, Lagrange,* Bezout
(1779), Gauss, Binet, Cauchy (1812), Lebesgue, Jacobi (1841),
and Cauchy (1841).
The first section of the paper is said to deal with "the
properties of determinants considered as derivational functions."
*As the memoir of Lagrange which. Cayley refers to is not one of those
brought into notice in the early part of our history, but is one bearing the title
" Sur le probleme de la determination des orbites des cometes d'apres trois
observations," it may be well to nmention that the substance of the only
sentence in it which concerns us had already appeared in the memoir of 1773.
The sentence is
'"De l il s'ensuit aussi qu'on aura
(t"' -t'u")2 = (x' - x'z")2 + (y"z' - y'z")2 + (x"y' -x'y")2,
= (x'2 + y'2 + z'2) (x2 + yl2 + 2) - (x' t + yyt + z'z" )2."
-l-ouv. Mmn. de l'Acad. Roy.... (Berlin), ann. 1778, p. 160.
M.D. II. B