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

DETERMINANTS IN GENERAL (CAUCHY, 1844)

27

that therefore
2P = x2DP + y2 D2P+z 2DP+.
+ 2/yDDP+ 2zDDP+....
and
P,=~DP, Py=i DyP, P=DP,....
After this the second part of the memoir, consisting of
geometrical applications, is entered upon.
GRASSMANN, H. (June 1844).
[DIE WISSENSCHAFT DER EXTENSIVEN GROSSE, oder die Ausdehnungslehre, eine neue mathematische Disciplin dargestellt
und durch Anwendungen erlautert. Erster Theil, die lineale
Ausdehnungslehre enthaltend. xxxii + 279 pp. Leipzig,
1844. Abstract in Archiv d. Math. u. Phys. vi. (1845), pp.
337-369.]
A quite peculiar form of the law of formation of a determinant had its origin with Grassmann. Grassmann, it will be
remembered, was one of the most distinguished of the mathematicians who occupied themselves with the search for an Algebra
of directed quantities, or with the allied problem of the
geometrical interpretation of the so-called imaginary expressions
of ordinary algebra. By the beginning of the third decade of
the century, the way had been gradually, though intermittingly,
prepared for important discoveries on the subject by the writings
of Wallis (1685), Buee (1805), Argand (1806), Servois (1813),
Mourey (1828), Warren (1828), and Gauss (1831).* With
Hamilton and Grassmann important discoveries came. Hamilton,
whose writings of 1833 and 1835 show that even then he had
meditated to some purpose on the matter, announced in 1843 his
great invention of Quaternions. In 1844 Grassmann followed
with the first part of the Ausdehnungslehre.
In his preface Grassmann explains the steps by which he had
been led to his theory. First, there was the question of the
addition of directed straight lines (Strecken), or vectors, to use
* See art. "Quaternions," by Professor Tait, in Encyclopaedia Britannica; or
Hamilton's Lectures on Quaternions.