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

DETERMINANTS IN GENERAL (TRUDI, 1862)

9

The result of course is that considerably more space is occupied.
The scheme of arrangement resembles Baltzer's, there being a
first part devoted to the theory and a second part to the so-called
applications. Whereas, however, Trudi apportions to the parts
112 and 156 pages respectively, the like apportionment in Houel's
Baltzer is 63 and 172; further, on account of the unequal treatment of geometry 250 pages of Trudi's 268 are algebraical as against
170 of Baltzer's 235.
As an instructive illustration of the growing solicitude for the
reader's convenience, we may point out that whereas Spottiswoode
(1851) merely notes that
A B = A+ B
B C B+C C,
and Brioschi (1854) with equal brevity extends the property to a
general n-line determinant, and Baltzer (1857) at little more length
but with more of a teacher's skill provides a formal enunciation,
like Jacobi's original, and two helpful examples, Trudi on the other
hand.not only prolongs Baltzer's treatment to half a page, but
follows this up usefully with a whole page of application to the
computation of a single arithmetical.determinant of the fourth
order, illustrating in particular how the order is reducible when
an element of a line is a submultiple of each of the other elements.
Similarly, instead of the few lines which Spottiswoode and Baltzer
give to Jacobi's theorem
dflaI = LA7,,dC,^,
hIt,k,
Trudi devotes two pages (pp. 53-55), and is the first to put the
development in the afterwards familiar form of the sum of n determinants.
Save this marked fullness and clearness of explanation there is
not much fresh to be noted so far as general determinants are
concerned. A number of additional expressions of more or less
importance are by definition given a special sense for the sake oi
definiteness or convenience; for example (p. 9), 'matrici simili,'
' linee omologhe,' etc., and (p. 17) 'caratteristica,' which is used
in connection with a minor to denote the sign-index, that is to say
the integer v where the sign-factor of the minor is (- 1)v. In the