An Elementary Course in Synthetic Projective Geometry Part 3

If, now, two forms are perspectively related to a third, any four harmonic elements of one must correspond to four harmonic elements in the other. We take this as our definition of projective correspondence, and say:

*36. Definition of projectivity.* _Two fundamental forms are protectively related to each other when a one-to-one correspondence exists between the elements of the two and when four harmonic elements of one correspond to four harmonic elements of the other._

[Figure 6]

FIG. 6

*37. Correspondence between harmonic conjugates.* Given four harmonic points, _A_, _B_, _C_, _D_; if we fix _A_ and _C_, then _B_ and _D_ vary together in a way that should be thoroughly understood. To get a clear conception of their relative motion we may fix the points _L_ and _M_ of the quadrangle _K_, _L_, _M_, _N_ (Fig. 6). Then, as _B_ describes the point-row _AC_, the point _N_ describes the point-row _AM_ perspective to it. Projecting _N_ again from _C_, we get a point-row _K_ on _AL_ perspective to the point-row _N_ and thus projective to the point-row _B_.

Project the point-row _K_ from _M_ and we get a point-row _D_ on _AC_ again, which is projective to the point-row _B_. For every point _B_ we have thus one and only one point _D_, and conversely. In other words, we have set up a one-to-one correspondence between the points of a single point-row, which is also a projective correspondence because four harmonic points _B_ correspond to four harmonic points _D_. We may note also that the correspondence is here characterized by a feature which does not always appear in projective correspondences: namely, the same process that carries one from _B_ to _D_ will carry one back from _D_ to _B_ again.

This special property will receive further study in the chapter on Involution.

*38.* It is seen that as _B_ approaches _A_, _D_ also approaches _A_. As _B_ moves from _A_ toward _C_, _D_ moves from _A_ in the opposite direction, pa.s.sing through the point at infinity on the line _AC_, and returns on the other side to meet _B_ at _C_ again. In other words, as _B_ traverses _AC_, _D_ traverses the rest of the line from _A_ to _C_ through infinity. In all positions of _B_, except at _A_ or _C_, _B_ and _D_ are separated from each other by _A_ and _C_.

*39. Harmonic conjugate of the point at infinity.* It is natural to inquire what position of _B_ corresponds to the infinitely distant position of _D_. We have proved (-- 27) that the particular quadrangle _K_, _L_, _M_, _N_ employed is of no consequence. We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem. We choose the point _L_ so that the triangle _ALC_ is isosceles (Fig. 7). Since _D_ is supposed to be at infinity, the line _KM_ is parallel to _AC_. Therefore the triangles _KAC_ and _MAC_ are equal, and the triangle _ANC_ is also isosceles. The triangles _CNL_ and _ANL_ are therefore equal, and the line _LB_ bisects the angle _ALC_. _B_ is therefore the middle point of _AC_, and we have the theorem

_The harmonic conjugate of the middle point of __AC__ is at infinity._

[Figure 7]

FIG. 7

*40. Projective theorems and metrical theorems. Linear construction.* This theorem is the connecting link between the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry. Up to this point we have said nothing about measurements, either of line segments or of angles. Desargues's theorem and the theory of harmonic elements which depends on it have nothing to do with magnitudes at all. Not until the notion of an infinitely distant point is brought in is any mention made of distances or directions. We have been able to make all of our constructions up to this point by means of the straightedge, or ungraduated ruler. A construction made with such an instrument we shall call a _linear_ construction. It requires merely that we be able to draw the line joining two points or find the point of intersection of two lines.

*41. Parallels and mid-points.* It might be thought that drawing a line through a given point parallel to a given line was only a special case of drawing a line joining two points. Indeed, it consists only in drawing a line through the given point and through the "infinitely distant point" on the given line. It must be remembered, however, that the expression "infinitely distant point" must not be taken literally. When we say that two parallel lines meet "at infinity," we really mean that they do not meet at all, and the only reason for using the expression is to avoid tedious statement of exceptions and restrictions to our theorems. We ought therefore to consider the drawing of a line parallel to a given line as a different accomplishment from the drawing of the line joining two given points. It is a remarkable consequence of the last theorem that a parallel to a given line and the mid-point of a given segment are equivalent data.

For the construction is reversible, and if we are given the middle point of a given segment, we can construct _linearly_ a line parallel to that segment. Thus, given that _B_ is the middle point of _AC_, we may draw any two lines through _A_, and any line through _B_ cutting them in points _N_ and _L_. Join _N_ and _L_ to _C_ and get the points _K_ and _M_ on the two lines through _A_. Then _KM_ is parallel to _AC_. _The bisection of a given segment and the drawing of a line parallel to the segment are equivalent data when linear construction is used._

*42.* It is not difficult to give a linear construction for the problem to divide a given segment into _n_ equal parts, given only a parallel to the segment. This is simple enough when _n_ is a power of _2_. For any other number, such as _29_, divide any segment on the line parallel to _AC_ into _32_ equal parts, by a repet.i.tion of the process just described.

Take _29_ of these, and join the first to _A_ and the last to _C_. Let these joining lines meet in _S_. Join _S_ to all the other points. Other problems, of a similar sort, are given at the end of the chapter.

*43. Numerical relations.* Since three points, given in order, are sufficient to determine a fourth, as explained above, it ought to be possible to reproduce the process numerically in view of the one-to-one correspondence which exists between points on a line and numbers; a correspondence which, to be sure, we have not established here, but which is discussed in any treatise on the theory of point sets. We proceed to discover what relation between four numbers corresponds to the harmonic relation between four points.

[Figure 8]

FIG. 8

*44.* Let _A_, _B_, _C_, _D_ be four harmonic points (Fig. 8), and let _SA_, _SB_, _SC_, _SD_ be four harmonic lines. a.s.sume a line drawn through _B_ parallel to _SD_, meeting _SA_ in _A'_ and _SC_ in _C'_. Then _A'_, _B'_, _C'_, and the infinitely distant point on _A'C'_ are four harmonic points, and therefore _B_ is the middle point of the segment _A'C'_. Then, since the triangle _DAS_ is similar to the triangle _BAA'_, we may write the proportion

_AB : AD = BA' : SD._

Also, from the similar triangles _DSC_ and _BCC'_, we have

_CD : CB = SD : B'C._

From these two proportions we have, remembering that _BA' = BC'_,

[formula]

the minus sign being given to the ratio on account of the fact that _A_ and _C_ are always separated from _B_ and _D_, so that one or three of the segments _AB_, _CD_, _AD_, _CB_ must be negative.

*45.* Writing the last equation in the form

_CB : AB = -CD : AD,_

and using the fundamental relation connecting three points on a line,

_PR + RQ = PQ,_

which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written

_(CB - BA) : AB = -(CA - DA) : AD,_

or

_(AB - AC) : AB = (AC - AD) : AD;_

so that _AB_, _AC_, and _AD_ are three quant.i.ties in hamonic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third. Also, from this last proportion comes the familiar relation

[formula]