PREFACE. vil 
leaves the plane at infinity invariant. However, the quater- 
nion caleulus has not been extensively applied to the theory 
of collineations in ordinary space, probably because it has not 
been found to be a suitable instrument for the purpose. 
We mention next an analytic method whose most natural 
and obvious geometrical application is to the theory of colline- 
ations. I refer to Cayley’s theory of matrices. This theory 
was set forth in his memoir on this subject in 1858. This 
subject has never become a popular one among mathematicians 
in the sense that it has attracted a large number of independ- 
ent investigators. It did not lead its founder to the general 
theory of collineation groups, although it has contributed 
largely, through the labors of Frobenius and others, to some 
phases of group theory. 
In his Geometrie der Lage, Nuremberg, 1847-60, Von Staudt 
laid the foundations of pure projective geometry in a form in- 
dependent of the assumptions of measurement, mechanics or 
congruence, and without quantitative notions of any sort. He 
distinguishes sharply between Geometrie der Lage and Geom- 
etrie des Masses. Pure projective geometry and the theory of 
collineations may be considered in a certain sense as mutually 
inclusive sciences. My conception of the distinction between 
them is expressed by saying that projective geometry deals 
chiefly with the projective properties of figures, while the 
theory of collineations considers especially the properties of 
the projection itself. 
About the year 1870 there appeared upon the mathematical 
stage a new personality, Sophus Lie, from the land of Abel. 
He brought with him a new and original idea, the notion of a 
continuous group of transformations. Lie broadened and 
deepened the already existing notions of a transformation, and 
developed a complete theory of all continuous groups of trans- 
formations, a thirty years’ task. Among the many transfor- 
mations studied by Lie, the first, the simplest, the most 
centrally situated, and the most far-reaching in its theoretica! 
