ADDRESS. 13 
magnitudes may be added or multiplied together or dealt with in any 
manner; the result is always a like imaginary magnitude. They are 
thus the magnitudes which are considered in analysis, and analysis is the 
science of such magnitudes. Observe the leading character that the 
imaginary magnitude a + bi is a magnitude composed of the two real 
magnitudes a and b (in the case = 0 it is the real magnitude a, and in 
the case a = () it is the pure imaginary magnitude bi). The idea is that 
of considering, in place of real magnitudes, these imaginary or complex 
magnitudes a + bi. 
In the Cartesian geometry a curve is determined by means of the 
equation existing between the coordinates (x,y) of any point thereof. In 
the case of a right line this equation is linear; in the case of a circle, or 
more generally of a conic, the equation is of the second order; and gener- 
ally, when the equation is of the order n, the curve which it represents 
is said to be of a curve of the order n. In the case of two given curves 
there are thus two equations satisfied by the coordinates (2, y) of the 
several points of intersection, and these give rise to an equation of a 
certain order for the coordinate x or y of a point of intersection. In 
the case of a straight line and a circle this is a quadric equation; it has 
two roots, real or imaginary. There are thus two values, say of a, and 
to each of these corresponds a single value of y. There are therefore two 
points of intersection—viz. a straight line and a circle intersect always 
in two points, real or imaginary. It is in this way that we are led 
analytically to the notion of imaginary points in geometry. The conclu- 
sion as to the two points of intersection cannot be contradicted by expe- 
rience: take a sheet of paper and draw on it the straight line and 
circle, and try. But you might say, or at least be strongly tempted to 
say, that it is meaningless. The question of course arises, What is the 
meaning of an imaginary point ? and further, In what manner can the 
notion be arrived at geometrically ? 
There is a well-known construction in perspective for drawing lines 
through the intersection of two lines, which are so nearly parallel as not 
to meet within the limits of the sheet of paper. You have two given 
lines which do not meet, and you draw a third line, which, when the 
lines are all of them produced, is found to pass through the intersection 
of the given lines. If instead of lines we have two circular arcs not 
meeting each other, then we can, by means of these arcs, construct a 
line; and if on completing the circles it is found that the circles intersect 
each other in two real points, then it will be found that the line passes 
through these two points: if the circles appear not to intersect, then 
the line will appear not to intersect either of the circles. But the 
geometrical construction being in each case the same, we say that in the 
second case also the line passes through the two intersections of the circles. 
Of conrse it may be said in reply that the conclusion is a very natural 
one, provided we assume the existence of imaginary points; and that, 
this assumption not being made, then, if the circles do not intersect, it is 
