In fact, once the idea has been grasped that quotients of determinants are the fundamental invariants, and that the double quotient arises from removing unobservable quantities, the remainder of the search for invariants is pure routine. Even the symmetry groups of the cross ratios are seen to be a natural consequence of reducing the full permutation group relative to the ambiguity in arranging the cancellation.
The reason this topic got to be included in the category of ``summer research'' was dissatisfaction with the presentation of the cross ratio in textbooks. So, it was more historical research or a literature search than purely mathematical research, which is not to say that it is less interesting.
Years ago, when the <PLOT> programs were under development in the Institute for Nuclear Energy at Salazar, the convenience of including projective geometry for perspective views and hidden line drawings was apparent, and coincided with the recent publication of Maxwell's two books [4,5]. Several other texts on projective geometry were available, purchased, and consulted. They all had one feature in common, that the invariance of the cross ratio was a given, which the readers were invited to verify, as an exercise. It can hardly be claimed that the cross ratio was an advanced topic which had been introduced in more elementary books, because many of these selfsame references were the basic texts.
Speculating on how Pappus encountered the invariance has to take into account that it was done long ago, and that whatever surviving details there may be are not in the literature which one has readily at hand, not even in the histories of mathematics. On the other hand, it is easy to surmise that geometers worked with intersections of families of lines, projections and what not, and wondered about such things as whether the midpoint of a line fell at the midpoint of its shadow. Of course, wondering and calculating are two different things.
It is also an interesting question, whether knowing about determinants and the fact that changes of basis multiply volumes by the determinant of the transformation, can be considered either as common or as elementary knowledge. Nevertheless, the idea should be familiar to anyone exposed to a course on linear algebra. So in that sense, it wasn't hard to devise an invariant, leaving open the question of whether that was how the result arose historically.
At this point, we had access to the Internet, so searching for ``cross ratio'' seemed to be a way to get more information, as did searching Barnes and Noble's online book catalog, for descriptions and conteent listings of books on projective geometry. The somewhat surprising result, for someone who has been out of touch with the field for years, was the degree to which Robotics and Computer Vision have adopted generalizations of the cross ratio as their stock in trade. Even so, without direct and convenient access to all the literature, it is hard to reconstruct even this recent history. That is, Who had the first understanding of the utility of the cross ratio? and From what sources did they obtain this insight?
Given that these fields are of active interest, there have been several conferences dedicated to the subject, and books written. Joseph L. Mundy has been one of the active researchers, and has edited two books [10,11] which include articles on projective invariants among their contributions. Another good reference is Pattern Classification and Scene Analysis by Richard O. Duda and Peter E. Hart [9], dating from 1973.
One of the more interesting results of this literature search concerns the fact that determinantal ideas seem to have originated with A. F. Móbius as long ago as 1827. His book seems to have been reprinted, even to the extent of being listed by Barnes and Noble, but on trying to buy a copy, we found that even the reprint seems to be out of print. But to fix the chronology somewhat, his work antedates the use of matrices and the formalization of linear algebra, but falls in a time frame in which determinants and their uses were reasonably familiar.
Why this would not be mentioned in projective geometry books originating a century later may have had something to do with a change of emphasis, tending toward non-euclidean geometries and the axiomatic foundations of geometry.