An Elementary Course in Synthetic Projective Geometry | Page 8

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 a __ point __Q__; and __LN__, __L'N'__ meet in a point __B__ on the line __AC__; then the lines __KM__ and __K'M'__ also meet in a point __D__ on the line __AC__._
[Figure 4]
FIG. 4
For, by the converse of the last theorem, _KK'_, _LL'_, and _NN'_ all meet in a point S (Fig. 4). Also _LL'_, _MM'_, and _NN'_ meet in a point, and therefore in the same point S. Thus _KK'_, _LL'_, and _MM'_ meet in a point, and so, by Desargues's theorem itself, A, B, and D are on a straight line.

*27. Importance of the theorem.* The importance of this theorem lies in the fact that, A, B, and C being given, an indefinite number of quadrangles _K'_, _L'_, _M'_, _N'_ my be found