AUGUST 27, 1897.] 
second the same ratio which the third has to the 
fourth ; when the first has in length to the second the 
same ratio which the third has in length to the fourth, 
according to Huclid’s definition ; and also the angle 
at which the fourth is inclined to the third is equal to 
the angle at which the second is inclined to the first, 
and is measured in the same direction. Unity is a 
positive quantity arbitrarily assumed from a compari- 
son with which the values of other quanties are de- 
termined. If there be three quantities such that unity 
is to the first as the second to the third, the third 
is called the product, which arises from the multiplica- 
tion of the second by the first. If there be three 
quantities such that the first is to unity as the second 
is tothe third, the first quantity is called the quotient, 
which arises from the division of the second by the 
third. 
The fundamental laws of algebra as gov- 
erning these quantities are established in 
their utmost generality with a rigor of 
reasoning that has probably not been sur- 
passed. The author even goes so far as to 
deduce the binomial formula, to develop 
many series and to apply the methods of 
the differential and integral calculus to 
quantities of the class defined. In form 
Warren’s work is intensely algebraic and 
fairly bristles with formule. 
To sum up: 
Caspar Wessel, in 1797, published the 
first clear, accurate and scientific treatment 
of directed lines in the same plane, as rep- 
resented by quantities of the form a + 
bV —1, establishing the laws governing their 
addition, subtraction, multiplication and 
division, and showing these quantities to be 
of practical value in the demonstration of 
theorems and solution of problems; he also 
worked out a partial theory of rotations in 
space, so far as they can be decomposed 
into rotations about two axes at right an- 
gles. 
Not very much later, 1799, Gauss indi- 
cated that he was in possession of a method 
of dealing with quantities of the form a+ 
b,/—1 which would consider them as 
equally possible with real quantities, but 
its fuller exposition was deferred till 1831. 
SCIENCE. 
307 
Buée’s paper of 1805 lays great empha- 
sis upon “—1 as the sign of perpendicu- 
larity, but fails to give any satisfactory in- 
terpretation of the product of directed 
lines. 
Argand’s famous memoir of 1806 is hard- 
ly in danger of receiving too much credit. 
Though written after Wessel’s paper there 
is not the slightest probability that Argand 
had any knowledge of the Norwegian sur- 
veyor, and, in fact, certain of his theorems 
are established less rigorously than by Wes- 
sel. Argand gave numerous applications 
of his theory to trigonometry, geometry 
and algebra, some of which are very note- 
worthy, especially his demonstrations of 
Ptolemy’s theorem regarding the inscribed 
quadrilateral and of the fundamental prop- 
osition of the theory of equations. 
The contributions of Francais, Gergonne 
and Servois, 1813-1815, served to do away 
with some of the errors into which Argand 
had fallen and thus to give a clearer insight 
into the fundamental notions of the subject. 
Though Warren’s book of 1828 contains 
definitions differing but little from those of 
Wessel and Francais and a notation which 
seems only a modification of that of Fran- 
gais, his generalized treatment of directed 
lines in the plane must be regarded as high- 
ly original. 
Cauchy’s work lay in the extension and 
development of the labors of his predeces- 
sors rather than in the introduction of new 
ideas. 
Such were the beginnings of the study of 
the geometric representation of the imagi- 
nary which has led in modern times to the 
establishment of such great bodies of doc- 
trine as the theory of functions on the one 
side and quaternions on the other, with the 
Ausdehnungslehre occupying a position be- 
tween. Who can tell what the next century 
will bring forth ? 
W. W. Beman. 
UNIVERSITY OF MICHIGAN. 
