306 
vague and disappointing. He describes 
/ —1as follows: 
V—1is the sign of perpendicularity v—l1 
is not the sign of an arithmetical operation, nor of an 
arithmetico-geometric operation, but of an operation 
purely geometric. It is a purely descriptive sign 
which indicates the direction of a line without regard 
to its length. 
Near the close of his paper he investi- 
gates what becomes of the conic sections 
when their coordinates become imaginary 
and decides that the circle passes into an 
equilateral hyperbola in the plane perpen- 
dicular to the plane of the circle and simi- 
larly for the other conics. 
A further discussion of the justly cele- 
brated epoch-making memoir of Argand 
and the contributions of himself, Frangais- 
Gergonne and Servois to the Annales de Ger- 
gonne from 1813 to 1815 is rendered the less 
necessary by reason of Houel’s republica- 
tion of all these papers in 1874 and their 
translation into English by Hardy in 1881. 
It is interesting to note the early view of 
imaginaries entertained by so distinguished 
a mathematician as Cauchy. In his Cowrs 
d’ Analyse, 1821, we read : 
Tn analysis we apply the term symbolic expression 
or symbol to every combination of algebraic signs 
which signifies nothing by itself or to which we at- 
tribute a value different from that which it naturally 
ought to have. * * * * Among the symbolic expres- 
sions whose consideration is of importance in analysis 
we ought especially to distinguish those which are 
called imaginary. * * * * We write the formula 
cos (a-+-b)+V—1 sin (a+))= 
The three expressions which the preceding equation 
eentains * * * * are three symbolic expressions 
which cannot be interpreted according to generally 
established conventions and represent nothing real. 
* * * * The equation itself, strictly speaking, is in- 
exact and has no meaning. 
In 1849, however, in a paper Sur les quan- 
tités géométriques, in which he gives suitable 
credit to Argand, Frangais and others, he 
acknowledges : 
SCIENCE. 
(N.S. Von. VI. No. 139. 
In my Analyse algébrique, published in 1821, I was 
content to show that the theory of imaginary expres- 
sions and equations could be rendered rigorous by 
considering these expressions and equations symbolic. 
But after new and mature reflections the better side 
to take seems to be to abandon entirely the use of the 
sign ;/__1 and to replace the theory of imagi- 
nary expressions by the theory of quantities which I 
shall call geometric. 
Having defined the term geometric quan- 
tity exactly as we now define the term vector 
and shown when two geometric quantities 
are equal, he continues: © 
The notion of geometric quantity will comprehend 
asa particular case the notion of algebraic quantity, 
positive or negative, and a fortiori the notion of arith- 
metic quantity. * * * We must further define the 
different functions of these quantities, especially their 
sums, their products and their integral powers by 
choosing such definitions as agree with those ad- 
mitted when we are dealing with algebraic quanti- 
ties alone. This condition will be fulfilled if we 
adopt the conventions now to be given. 
Then follow the definitions called for, to- 
gether with a treatment of the whole sub- 
ject fully up to modern demands. Cauchy 
observes that a large part of the results of 
the investigations of Argand and others 
would seem to have been discovered as early 
as 1786 by Henri Dominique Truel, who 
communicated them about 1810 to Augustin 
Normand, of Havre. 
In 1828 there appeared in Cambridge, 
England, a remarkable work by Rev. John 
Warren, entitled A Treatise on the Geomet- 
rical Representation of the Square Roots of Neg- 
ative Quantities. Though this book has lat- 
terly received scant credit, its merits were 
fully recognized by De Morgan and ac- 
knowledgments of indebtedness were frank- 
ly made by Hamilton. 
Throughout Warren’s work the term 
quantity, like Cauchy’s geometric quantity, 
indicates a line given in length and direc- 
tion. Some of his definitions are as follows: 
The sum of two quantities is the diagonal of the 
parallelogram whose sides are the two quantities. 
The first of four quantities is said to have to the 
