36 
SECTION A. 
real. Thus eS — e or —\/e; but the logarithm of e’ is i ; therefore 
the logarithm of —\/e as well as of -\-\/e is h- 
These opposing views arise from different conceptions of the negative 
symbol and of the magnitude treated by algebra. The magnitudes con 
sidered in elementary algebra are, first, a mere number or ratio; second, 
a magnitude which may have a given direction, or the opposite, and third, 
a geometric ratio which combines a number with a certain amount of 
change of direction. The logarithm of a ratio is itself a ratio, and is 
unique. If a directed magnitude has a logarithm, it is difficult to see how 
the direction of the logarithm, if it has any direction, can be different 
from that of the magnitude. It is of number in the sense of a geometric 
ratio that Euler’s proposition is true. This conception of number imme 
diately transcends representation by a single straight line; consequently a 
part of the ratio generally appears as impossible. 
In his Geometrie de Position, Carnot asks the following among other 
questions: “If two quantities, of which the one is positive and the other 
negative, are both real, and do not differ excepting in position, why 
should the root of the one be an imaginary quantity, while that of the 
otheras real? Why should |/—a not be as real as i/+a?” In this ques 
tion it is assumed that — a and -f- a denote directed magnitudes, the one 
being opposite to the other; and if such a quantity has a square root, it is 
difficult to understand why the one direction should differ from the other. 
But the — a which has the imaginary square roots, while -f- a has real, do 
not differ in direction; they differ in the amount of change of direction. 
In 1806, M. Buee published in the Philosophical Transactions a memoir on 
Imaginary Quantities, and in it he endeavors to answer some of the ques- 
tions raised by Carnot. His 
main idea is that -f-, —, and |/—1 are purely 
descriptive signs ; that is, signs which indi 
cate direction. Suppose three equal lines 
AB, AC, AD, drawn from a point A (fig. 1), 
of which AC is opposite to AB, and AD 
perpendicular to BAG ; then if the line AB 
is designated by -f-1, the line AC will be —1, 
and the line AD will be ~\/—1. Thus |/—1 
is the sign of perpendicularity. It follows 
from this view of v"—1 that it does not in 
dicate a unique direction, the opposite line 
AD', or any line in the plane as AD" is also 
indicated by j/—1- Bueé admits the conse 
quence. But it may be asked : If every 
perpendicular is represented by l/ —1, what meaning is left for —y'—1? 
Bue6 applies his theory to the interpretation of the solution of a quad 
ratic equation which had been considered by Carnot, namely: To divide a 
line A B into two parts such that the product of the segments shall be equal 
to half the square of the line.