114 IMAGINARY QUANTITIES—THEIR PHILOSOPHY. 
by, ¢. g., —1, the effect, it is said, is to revolve the line 
OB about the origin O to the position OA’, OA describ- 
multiplied by —1, passes in succession through all in- 
perpendicular to AA’. 
B 
Ca 
v1, a 
and 1x V —1, by OB’, the an- 
v —1 should =OA’, or—OA. 
way between+and-. In reality, the change is usually 
and forth ina straight line, its motion becoming alter- 
sliding it by that point while keeping it parallel to itself. 
[a 
O 
termediate positions. Hence, as —1=/ —1xv =I, on; 
in other words, as there are here two equal factors, one 
This line, OB, is said to be 
the graphical representation 
With perfect consistency, 1t — O A 
is said 1x V —1 is represented 
gle AOB" being + of 180°, and so on. 
But, since V — 1=-1, the result of multiplying OA by 
In this graphic representation it is assumed that, in 
changing from + to —, OA, describes a semi-circumference 
made in a different way. A man walking east does not 
go west by any such circuitous route. A point in a 
nately + and—. If 100 feet east is changed to 100 west, 
the result is attained, not necessarily by revolving my 
Again, itis an objection to sucha representation of im- 
64 
ing a semi-circle on AA’, and implying that OA, when 
of ‘them will cause the line to rotate half way, or become 
of a unit x 7-1, or simply 
by OB’, AOB’ being 4 of 180° ; 
either of them should be the same, and, therefore, OAx 
about O, so that it has at one time the position OB, half- 
chord vibrating under a single impulse, goes back 
tape around the fixed point, or “ origin,’’ but rather by 
