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PRESIDENTIAL ADDRESS TO THE 
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other root is = — i) and it is easily seen that every quadric equation (with real 
coefficients as before) has two roots, a + bi, where a and 6 are real magnitudes. We 
are thus led to the conception of an imaginary magnitude, a + hi, where a and b are 
real magnitudes, each susceptible of any positive or negative value, zero included. The 
general theorem is that, taking the coefficients of the equation to be imaginary magni 
tudes, then an equation of the order n has always n roots, each of them an imaginary 
magnitude, and it thus appears that the foregoing form a + hi of imaginary magnitude 
is the only one that presents itself. Such imaginary 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 6 = 0 it is the real magnitude a, and in the case a = 0 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 generally, when the equation is of the order n, 
the curve which it represents is said to be a curve of the order n. In the case of 
two given curves, there are thus two equations satisfied by the coordinates (x, 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 x, 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 conclusion as to the two points of 
intersection cannot be contradicted by experience: 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.