vl PREFACE. 
as a subsection of modern analytic geometry. A forward step 
in the theory of collineations was taken by the English school 
of mathematicians who founded the invariant theory of linear 
transformations. This theory took its rise shortly after 1840, 
and the principal names associated with its early development 
are those of Boole, Cayley, Sylvester, Salmon.* 
Since a linear transformation is a projective transformation, 
every theorem concerning linear transformations has its bear- 
ing on the theory of collineations. The workers in projective 
invariant theory who considered the geometric applications 
of their science, looked more to the effect of a linear transfor- 
mation on a geometric figure than to the properties of the 
transformation itself. Thus we look in vain through the 
standard works on invariant theory for a classification of linear 
transformations or a discussion of their characteristic proper- 
ties. It was left to men with a different point of view to call 
the attention of the mathematical world from the effects of a 
collineation back to the properties of the collineation itself. 
In 1844 Hermann Grassmann published his Auwsdehnungs- 
lehre, or Calculus of Extension, and a second presentation of 
the same subject in 1862. The method of the Calculus of Ex- 
tension was not applied directly by Grassmann to the study of 
collineations, but it is capable of application to some phases 
of the subject. For example, by this method the various 
types of ecllineations in ordinary space have been determined. 
Although the contributions of Grassmann’s theory to the 
theory of collineations have been relatively small, they are 
perhaps sufficient to warrant the mention of it among the ana- 
lytic methods of treating the subject of collineations. 
The quaternion calculus invented by Sir William R. Ham- 
ilton, and published by him in his Lectures on Quaternions in 
1844, isan algebra founded on a complex number system of 
four units. One of its valuable applications is to the theory 
of homogeneous strains. A homogeneous strain is by definition 
a collineation, though of a very special kind, viz., one which 
*See note to Salmon’s Algebra, chapter XIII. 
