An Elementary Course in Synthetic Projective Geometry | Page 8

_2_, _4_, _6_,
_8_ .... Are we justified in saying that there are just as many even
numbers as there are numbers altogether?
9. Is the axiom "The whole is greater than one of its parts" applicable to
infinite assemblages?

10. Make out a classified list of all the infinitudes of the first, second,
third, and fourth orders mentioned in this chapter.

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._
[Figure 3]
FIG. 3
Let the lines _AA'_, _BB'_, and _CC'_ meet in the point M (Fig. 3).
Conceive of the figure as in space, so that M is the vertex of a trihedral
angle of which the given triangles are plane sections. The lines AB and
_A'B'_ are in the same plane and must meet when produced, their point
of intersection being clearly a point in the plane of each triangle and
therefore in the line of intersection of these two planes. Call this point
P. By similar reasoning the point Q of intersection of the lines BC and
_B'C'_ must lie on this same line as well as the point R of intersection
of CA and _C'A'_. Therefore the points P, Q, and R all lie on the same

line m. If now we consider the figure a plane figure, the points P, Q,
and R still all lie on a straight line, which proves the theorem. The
converse is established in the same manner.

*26. Fundamental theorem concerning two complete quadrangles.*
This theorem throws into our hands the following fundamental theorem
concerning two complete quadrangles, a complete quadrangle being
defined as the figure obtained by joining any four given points by
straight lines in the six possible ways.
_Given two complete quadrangles, __K__, __L__, __M__, __N__ and
__K'__, __L'__, __M'__, __N'__, so related that __KL__, __K'L'__,
__MN__, __M'N'__ all meet in a point __A__; __LM__, __L'M'__,
__NK__, __N'K'__ all meet in