770 Proceedings of Royal Society of Edinburgh. [ sess . 
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 Hamilton’s 
widely accepted term. This it was unnecessary to linger over, as 
his predecessors had already dealt satisfactorily with it. Then came 
the question of multiplication of vectors. Seeing that when a and 
b represent two lines in magnitude only, in other words, are scalars 
and not vectors, the product ab represents the rectangle of which a 
and b are adjacent sides, Grassmann ventured to denote by the pro- 
duct ab, when a and b are vectors, a parallelogram having the 
vectors for adjacent sides. This definition of multiplication mani- 
festly entailed the result 
a 2 = 0 ; 
and along with the definition of addition required further that 
a(b + c) = ab + ac . 
These two again involved a third, viz., 
ab — - ba 
for from the two we have 
0 = (a + b) 2 , 
= (a + b)a 4- (a, + b)b , 
= a 2 + ba + ab + b 2 , 
= ba + ab . 
The remaining steps of the building up of the theory need not be 
told, as these laws of outer multiplication (“ dussere Multiplica- 
tion”) suffice for the purpose we have in view. 
The exposition of the theory itself is broken up into an introduc- 
tion and nine chapters, all of them marked by ability and much 
originality. It is the second chapter which deals specially with 
outer multiplication, and at the end of it (pp. 70--73) occurs the 
application which concerns determinants. The matter is introduced 
by a sentence or two pointing out that it is scarcely to be expected 
that outer multiplication can be so directly applied to ordinary 
algebra as to geometry and dynamics, because in ordinary algebra 
the quantities are essentially alike ( gleichartige , in the sense of the 
Ausdehnungslehre), and outer multiplication presupposes the idea 
