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

DETERMINANTS IN GENERAL (CAYLEY, 1854)

87

and the binary cubics
ax3 + 3bx2y + 3cxy2 + dy3, ax3 + bx2y + cxy2 + dy3
by (a, b, c, d x, y)3, (a, b, c, d X, y)3
respectively.
We may suggest for consideration in passing the following order
of ideas, as leading up to Cayley's contracted mode of writing
a set of linear equations. First, a row of separate quantities, e.g.
(a, b, c,... ); second, the statement of the identity of two rows,
e.g. (a, b, c,... ) = (x, y, z,.. ), or simply a, b, c,... = x, y,,...;
third, the so-called product of two rows, e.g. (a, b, c,... x, y, z,...);
fourth, a square of separate quantities, i.e. a matrix; fifth, the
result of multiplying a matrix and a row being a row. It is
unfortunate that, from the point of view of notation merely,
this does not at once suggest, in the sixth place, the result of
multiplying two matrices, where, as Cayley is careful to point
out, the multiplication is row-by-column and not row-by-row.
BRIOSCHI, F. (1854).
[LA TEORICA DEI DETERMINANTI, E LE SUE PRINCIPALI APPLICAZIONI; del Dr. Francesco Brioschi; viii+116 pp.; Pavia.
Translation into French, by Combescure; ix+ 216 pp.; Paris,
1856. Translation into German, by Schellbach; vii +102 pp.;
Berlin, 1856.]
This, the second separately-published text-book on determinants, is mainly on the same lines as the first, but is marked
by greater attention to verbal and logical accuracy. It consists
of an historical preface and eleven short chapters or sections,
seven of the latter being devoted to determinants in general,
and the remaining four to special forms.
Sylvester's umbral notation is given in the form
C ( ' ' '... I
but is not afterwards employed. The same author's term
"minor" (nrinore) is adopted, this being represented in the
French translation by " mineur," and in the German by " Unter