*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.