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'__,