4 
ALGEBRAIC INVARIANTS 
These rigid motions (translations, rotations, and combinations 
of them) preserve angles and distances. But the transformation 
x'= 2x, y' = 2y is a stretching in all directions from the origin 
in the ratio 2:1; while x’ — 2x, y'=y is a stretching perpen 
dicular to the y-axis in each direction in the ratio 2:1. 
From the multiplicity of possible types of transformations, 
we shall select as the basis of our theory of invariants the very 
restricted set of transformations which have an interpretation 
in projective geometry and which suffice for the ordinary needs 
of algebra. 
2. Projective Transformations. All of the points on a 
straight line are said to form a range of points. Project the 
v 
points A, B, C, . . . of a range from a point V, not on their 
line, by means of a pencil of straight lines. This pencil is 
cut by a new transversal in a rangeai, B x , C x , , said to be 
perspective with the range A, B, C, ... . Project the points 
Ai, Bi, Ci, . . . from a new vertex v by a new pencil and cut it 
by a new transversal. The resulting range of points A', B' 
C, . . . is said to be projective with the range A, B, C, . . . 
Likewise, the range obtained by any number of projections 
and sections is called projective with the given range, and