*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 a.s.suming that whatever relations hold between the elements of one a.s.semblage must also hold between the corresponding elements of any a.s.semblage in one-to-one correspondence with it. This false a.s.sumption is the basis of the so-called "proof by a.n.a.logy" 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 a.n.a.logy" 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 s.p.a.ce, 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 such that _K'L'_ and _M'N'_ meet in _A_, _K'N'_ and _L'M'_ in _C_, with _L'N'_ pa.s.sing through _B_.

Indeed, the lines _AK'_ and _AM'_ may be drawn arbitrarily through _A_, and any line through _B_ may be used to determine _L'_ and _N'_. By joining these two points to _C_ the points _K'_ and _M'_ are determined.

Then the line joining _K'_ and _M'_, found in this way, must pa.s.s through the point _D_ already determined by the quadrangle _K_, _L_, _M_, _N_.

_The three points __A__, __B__, __C__, given in order, serve thus to determine a fourth point __D__._

*28.* In a complete quadrangle the line joining any two points is called the _opposite side_ to the line joining the other two points. The result of the preceding paragraph may then be stated as follows:

Given three points, _A_, _B_, _C_, in a straight line, if a pair of opposite sides of a complete quadrangle pa.s.s through _A_, and another pair through _C_, and one of the remaining two sides goes through _B_, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed.

*29. Four harmonic points.* Four points, _A_, _B_, _C_, _D_, related as in the preceding theorem are called _four harmonic points_. The point _D_ is called the _fourth harmonic of __B__ with respect to __A__ and __C_.

Since _B_ and _D_ play exactly the same role in the above construction, _B__ is also the fourth harmonic of __D__ with respect to __A__ and __C_.

_B_ and _D_ are called _harmonic conjugates with respect to __A__ and __C_. We proceed to show that _A_ and _C_ are also harmonic conjugates with respect to _B_ and _D_-that is, that it is possible to find a quadrangle of which two opposite sides shall pa.s.s through _B_, two through _D_, and of the remaining pair, one through _A_ and the other through _C_.

[Figure 5]

FIG. 5

Let _O_ be the intersection of _KM_ and _LN_ (Fig. 5). Join _O_ to _A_ and _C_. The joining lines cut out on the sides of the quadrangle four points, _P_, _Q_, _R_, _S_. Consider the quadrangle _P_, _K_, _Q_, _O_. One pair of opposite sides pa.s.ses through _A_, one through _C_, and one remaining side through _D_; therefore the other remaining side must pa.s.s through _B_. Similarly, _RS_ pa.s.ses through _B_ and _PS_ and _QR_ pa.s.s through _D_. The quadrangle _P_, _Q_, _R_, _S_ therefore has two opposite sides through _B_, two through _D_, and the remaining pair through _A_ and _C_.

_A_ and _C_ are thus harmonic conjugates with respect to _B_ and _D_. We may sum up the discussion, therefore, as follows:

*30.* If _A_ and _C_ are harmonic conjugates with respect to _B_ and _D_, then _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_.

*31. Importance of the notion.* The importance of the notion of four harmonic points lies in the fact that it is a relation which is carried over from four points in a point-row _u_ to the four points that correspond to them in any point-row _u'_ perspective to _u_.

To prove this statement we construct a quadrangle _K_, _L_, _M_, _N_ such that _KL_ and _MN_ pa.s.s through _A_, _KN_ and _LM_ through _C_, _LN_ through _B_, and _KM_ through _D_. Take now any point _S_ not in the plane of the quadrangle and construct the planes determined by _S_ and all the seven lines of the figure. Cut across this set of planes by another plane not pa.s.sing through _S_. This plane cuts out on the set of seven planes another quadrangle which determines four new harmonic points, _A'_, _B'_, _C'_, _D'_, on the lines joining _S_ to _A_, _B_, _C_, _D_. But _S_ may be taken as any point, since the original quadrangle may be taken in any plane through _A_, _B_, _C_, _D_; and, further, the points _A'_, _B'_, _C'_, _D'_ are the intersection of _SA_, _SB_, _SC_, _SD_ by any line. We have, then, the remarkable theorem:

*32.* _If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic._

*33. Four harmonic lines.* We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define _four harmonic rays_ as four rays which pa.s.s through a point and which pa.s.s one through each of four harmonic points, we have the theorem

_Four harmonic lines are cut by any transversal in four harmonic points._

*34. Four harmonic planes.* We also define _four harmonic planes_ as four planes through a line which pa.s.s one through each of four harmonic points, and we may show that

_Four harmonic planes are cut by any plane not pa.s.sing through their common line in four harmonic lines, and also by any line in four harmonic points._

For let the planes a, , ?, d, which all pa.s.s through the line _g_, pa.s.s also through the four harmonic points _A_, _B_, _C_, _D_, so that a pa.s.ses through _A_, etc. Then it is clear that any plane p through _A_, _B_, _C_, _D_ will cut out four harmonic lines from the four planes, for they are lines through the intersection _P_ of _g_ with the plane p, and they pa.s.s through the given harmonic points _A_, _B_, _C_, _D_. Any other plane s cuts _g_ in a point _S_ and cuts a, , ?, d in four lines that meet p in four points _A'_, _B'_, _C'_, _D'_ lying on _PA_, _PB_, _PC_, and _PD_ respectively, and are thus four harmonic hues. Further, any ray cuts a, , ?, d in four harmonic points, since any plane through the ray gives four harmonic lines of intersection.

*35.* These results may be put together as follows:

_Given any two a.s.semblages of points, rays, or planes, perspectively related to each other, four harmonic elements of one must correspond to four elements of the other which are likewise harmonic._