CHAPTER II — RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER

*23. Seven fundamental forms.* In the preceding
chapter we have called attention to seven fundamental
forms: the point-row, the pencil of rays, the
axial pencil, the plane system, the point system, the
space system, and the system of lines in space.
These fundamental forms are the material which we
intend to use in building up a general theory which
will be found to include ordinary geometry as a special
case. We shall be concerned, not with measurement
of angles and areas or line segments as in the study
of Euclid, but in combining and comparing these fundamental
forms and in “generating” new forms by
means of them. In problems of construction we
shall make no use of measurement, either of angles
or of segments, and except in certain special applications
of the general theory we shall not find it necessary
to require more of ourselves than the ability to draw
the line joining two points, or to find the point of
intersections of two lines, or the line of intersection
of two planes, or, in general, the common elements
of two fundamental forms.

*24. Projective properties.* Our chief
interest in this chapter will be the discovery of
relations between the elements of one form which hold
between the corresponding elements of any other form
in one-to-one correspondence with it. We have
already called attention to the danger of assuming
that whatever relations hold between the elements of
one assemblage must also hold between the corresponding
elements of any assemblage in one-to-one correspondence
with it. This false assumption is the basis of
the so-called “proof by analogy” so much
in vogue among speculative theorists. When it
appears that certain relations existing between the
points of a given point-row do not necessitate the
same relations between the corresponding elements
of another in one-to-one correspondence with it, we
should view with suspicion any application of the
“proof by analogy” in realms of thought
where accurate judgments are not so easily made.
For example, if in a given point-row *u* three
points, *A*, *B*, and *C*, are taken
such that *B* is the middle point of the segment
*AC*, it does not follow that the three points
*A’*, *B’*, *C’* in
a point-row perspective to *u* will be so related.
Relations between the elements of any form which do
go over unaltered to the corresponding elements of
a form projectively related to it are called *projective
relations.* Relations involving measurement of lines
or of angles are not projective.

*25. Desargues’s theorem.* We consider
first the following beautiful theorem, due to Desargues
and called by his name.

*If two triangles, *_A__, __B__, __C__ and __A’__,
__B’__, __C’__, are so situated that the
lines __AA’__, __BB’__, and __CC’__
all meet in a point, then the pairs of sides __AB__
and __A’B’__, __BC__ and __B’C’__,
__CA__ and __C’A’__ all meet on a straight
line, and conversely._