An introduction to the modern theory of equations, by Florian Cajori.

THEORY OF EQUATIONS

y', 8', we obtain by means of these expressions the following
relation: (' - ')(8'-y') (__(_ Y)
(''-y')(8'- ) (a - -)(8- )
The geometrical significance of each of these fractions becomes
apparent, if taking 0 as origin, we put a= OC, / == OA,
y= OB, 8 - OD. Then
A B C D......... - t -= AC, t —/ = B7C,
01 I
8 - 1 D = AI), 8 - y = BD),
and the fraction on the right-hand side is equal to AC AD
BC BD
This is the cross-ratio (anharmonic ratio) of the points C and D
with respect to the points A and B. See Ex. 10, ~ 113.
Similarly, the left-hand fraction expresses the cross-ratio of
points C' and D' with respect to points A' and B'. Hence, if
the roots a, /, p, 8 represent distances on a line, measured from
an origin 0, then the cross-ratio of the four points thus determined is the same as the cross-ratio, similarly formed, of the
points, determined in the same manner by the corresponding
roots a', 8', y', 8', of the transformed equation.
Thus, we have on the same line two ranges of points,
a,, y, 8,... and a(', ', y', 6',.** such that the cross-ratio of any
four points of one range is equal to the cross-ratio of the
corresponding four points on the other. Such ranges are called
homographic; hence the name, homographic transformation.
To a point in one range corresponds one, and only one, point in
the other. In other words, there is a one-to-one correspondence
between the two ranges of points. The homographic transformation is the most general transformation in which this
correspondence holds. We proceed to consider transformations
which are not usually homographic.
79. The Most General Transformation. The most general
rational algebraicc tracsfobrmation of the roots of ai equation.f(x)= 0 of the nth degree ca)n, be reduced to an integral transfor'mation of a degree not higher than the (n - l)tih.