PREFACE. 
In laying before the public the present work on the Theory 
of Collineations, I wish to say a word on the historical de- 
velopment of the subject and the genesis of my own interest 
in it, as well as a word on the point of view I have adopted 
and the methods | have used. 
The concept and term collineation* were introduced into 
geometry by Mobius in his Barycentrische Calcul published at 
Leipzig in 1827. According to his definition of a collineation 
points correspond to points and straight lines to straight lines, 
i. e., collinear points to collinear points, whence the name. 
We owe to Mobius not only the first clear-cut notion of a 
collineation and its name, but also the fundamental theorem 
underlying all his work on this subject, viz., that the cross- 
ratios of four corresponding elements of two collinear figures 
are always equal. He also gives us methods for constructing 
collineations on a line, in a plane, and in space. He shows 
that three points on a line, four points in a plane, five points 
in ordinary space, in general +2 points in a space of nm di- 
mensions, determine a collineation in these spaces, respectively. 
He points out that two conics in a plane are always collinear 
to one another in o* ways; and that a curve of the nth de- 
gree corresponds to a curve of the same degree. But | find no 
hint anywhere in Mobius’s work that there are any self-corre- 
sponding points, lines, or planes in a collineation. 
With the introduction of homogeneous coordinates into 
analytic geometry there came in a generalized form the old 
problems connected with the transformation of coordinate 
axes. Such a transformation is a linear transformation, and 
hence the theory of linear transformations came to be studied 
* Mobius tells us in his Vorrede, p. xii, that the name was suggested to him by 
his friend, Professor Weiske. 
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