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

DETERMINANTS IN GENERAL (GRASSMAN, 1861)

7

members of such a system are called 'extensive Grosse.' Determinants are no longer ignored. Early in the chapter on the 'Combinatorial Product' (chap. iii. pp. 31-107) the ordinary definition
and the E notation are given; and thereafter the functions are
repeatedly referred to. They are even found to be needed for the
statement of theorems in the 'Ausdehnungslehre' itself, for
example, the theorem on the inner product of two magnitudes each
of the mntl 'Stufe ' and consisting of m simple factors. On p. 130
this is expressed in the form
[abc...].l'b'c'...] = Determ. [[ala'] [alb'] [ala]...
[bla'] [bib']
[cla'] [omb'
The solution of a set of simultaneous linear equations which
formerly came at the end of a chapter on 'Outer Multiplication'
now occupies a similar position in chapter iii. In this matter,
too, there is a new procedure which is of considerable interest
'when brought into comparison with the old. Using as multipliers
with the equations
ajllX + a12X2 +... + a,,l = bn
a2lxl + a22X2 +.. + a2nt = b2
al4xl + cn2X2 +. annXn = bn,
the 'extensive magnitudes ' e,, e2,.., en, whose ' combinatorial
product' is 1, and performing addition, we obtain an equation
which may be written
Xl1 + X2(a2 +..+ Xnan = /3,
it we put
a. for a11ej + a2e2 +... + ale,, etc.
This equation, Grassmann says, takes the place of the given
equations, and in order to find the value of any one of the x's, xl
say, we apply the factor 2a2 3... a to both sides, and on
account of the vanishing of [a2fa2a3 ~ ~. an], [a3a2a3... an],. * * the
result is
X1 [a12... an] = [L2 2(. * (-]