which is convenient for the computation of the distance _AD_ when _AB_ and _AC_ are given numerically.

*46. Anharmonic ratio.* The corresponding relations between the trigonometric functions of the angles determined by four harmonic lines are not difficult to obtain, but as we shall not need them in building up the theory of projective geometry, we will not discuss them here. Students who have a slight acquaintance with trigonometry may read in a later chapter (-- 161) a development of the theory of a more general relation, called the _anharmonic ratio_, or _cross ratio_, which connects any four points on a line.

PROBLEMS

*1*. Draw through a given point a line which shall pa.s.s through the inaccessible point of intersection of two given lines. The following construction may be made to depend upon Desargues's theorem: Through the given point _P_ draw any two rays cutting the two lines in the points _AB'_ and _A'B_, _A_, _B_, lying on one of the given lines and _A'_, _B'_, on the other. Join _AA'_ and _BB'_, and find their point of intersection _S_. Through _S_ draw any other ray, cutting the given lines in _CC'_.

Join _BC'_ and _B'C_, and obtain their point of intersection _Q_. _PQ_ is the desired line. Justify this construction.

*2.* To draw through a given point _P_ a line which shall meet two given lines in points _A_ and _B_, equally distant from _P_. Justify the following construction: Join _P_ to the point _S_ of intersection of the two given lines. Construct the fourth harmonic of _PS_ with respect to the two given lines. Draw through _P_ a line parallel to this line. This is the required line.

*3.* Given a parallelogram in the same plane with a given segment _AC_, to construct linearly the middle point of _AC_.

*4.* Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made.

*5.* Given the middle point of a line segment, to draw a line parallel to the segment and pa.s.sing through a given point.

*6.* A line is drawn cutting the sides of a triangle _ABC_ in the points _A'_, _B'_, _C'_ the point _A'_ lying on the side _BC_, etc. The harmonic conjugate of _A'_ with respect to _B_ and _C_ is then constructed and called _A"_. Similarly, _B"_ and _C"_ are constructed. Show that _A"B"C"_ lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point.

CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS

[Figure 9]

FIG. 9

*47. Superposed fundamental forms. Self-corresponding elements.* We have seen (-- 37) that two projective point-rows may be superposed upon the same straight line. This happens, for example, when two pencils which are projective to each other are cut across by a straight line. It is also possible for two projective pencils to have the same center. This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pencils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen, indeed, that if _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_, then the point-row described by _B_ is projective to the point-row described by _D_, and that _A_ and _C_ are self-corresponding points.

Consider more generally the case of two pencils perspective to each other with axis of perspectivity _u'_ (Fig. 9). Cut across them by a line _u_.

We get thus two projective point-rows superposed on the same line _u_, and a moment's reflection serves to show that the point _N_ of intersection _u_ and _u'_ corresponds to itself in the two point-rows. Also, the point _M_, where _u_ intersects the line joining the centers of the two pencils, is seen to correspond to itself. It is thus possible for two projective point-rows, superposed upon the same line, to have two self-corresponding points. Clearly _M_ and _N_ may fall together if the line joining the centers of the pencils happens to pa.s.s through the point of intersection of the lines _u_ and _u'_.

[Figure 10]

FIG. 10

*48.* We may also give an ill.u.s.tration of a case where two superposed projective point-rows have no self-corresponding points at all. Thus we may take two lines revolving about a fixed point _S_ and always making the same angle a with each other (Fig. 10). They will cut out on any line _u_ in the plane two point-rows which are easily seen to be projective. For, given any four rays _SP_ which are harmonic, the four corresponding rays _SP'_ must also be harmonic, since they make the same angles with each other. Four harmonic points _P_ correspond, therefore, to four harmonic points _P'_. It is clear, however, that no point _P_ can coincide with its corresponding point _P'_, for in that case the lines _PS_ and _P'S_ would coincide, which is impossible if the angle between them is to be constant.

*49. Fundamental theorem. Postulate of continuity.* We have thus shown that two projective point-rows, superposed one on the other, may have two points, one point, or no point at all corresponding to themselves. We proceed to show that

_If two projective point-rows, superposed upon the same straight line, have more than two self-corresponding points, they must have an infinite number, and every point corresponds to itself; that is, the two point-rows are not essentially distinct._

If three points, _A_, _B_, and _C_, are self-corresponding, then the harmonic conjugate _D_ of _B_ with respect to _A_ and _C_ must also correspond to itself. For four harmonic points must always correspond to four harmonic points. In the same way the harmonic conjugate of _D_ with respect to _B_ and _C_ must correspond to itself. Combining new points with old in this way, we may obtain as many self-corresponding points as we wish. We show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points. Thus, let a point _P_ lie between _A_ and _B_. Construct now _D_, the fourth harmonic of _C_ with respect to _A_ and _B_. _D_ may coincide with _P_, in which case the sequence is closed; otherwise _P_ lies in the stretch _AD_ or in the stretch _DB_. If it lies in the stretch _DB_, construct the fourth harmonic of _C_ with respect to _D_ and _B_. This point _D'_ may coincide with _P_, in which case, as before, the sequence is closed. If _P_ lies in the stretch _DD'_, we construct the fourth harmonic of _C_ with respect to _DD'_, etc. In each step the region in which _P_ lies is diminished, and the process may be continued until two self-corresponding points are obtained on either side of _P_, and at distances from it arbitrarily small.

We now a.s.sume, explicitly, the fundamental postulate that the correspondence is _continuous_, that is, that _the distance between two points in one point-row may be made arbitrarily small by sufficiently diminishing the distance between the corresponding points in the other._ Suppose now that _P_ is not a self-corresponding point, but corresponds to a point _P'_ at a fixed distance _d_ from _P_. As noted above, we can find self-corresponding points arbitrarily close to _P_, and it appears, then, that we can take a point _D_ as close to _P_ as we wish, and yet the distance between the corresponding points _D'_ and _P'_ approaches _d_ as a limit, and not zero, which contradicts the postulate of continuity.

*50.* It follows also that two projective pencils which have the same center may have no more than two self-corresponding rays, unless the pencils are identical. For if we cut across them by a line, we obtain two projective point-rows superposed on the same straight line, which may have no more than two self-corresponding points. The same considerations apply to two projective axial pencils which have the same axis.

*51. Projective point-rows having a self-corresponding point in common.*

Consider now two projective point-rows lying on different lines in the same plane. Their common point may or may not be a self-corresponding point. If the two point-rows are perspectively related, then their common point is evidently a self-corresponding point. The converse is also true, and we have the very important theorem:

*52.* _If in two protective point-rows, the point of intersection corresponds to itself, then the point-rows are in perspective position._

[Figure 11]

FIG. 11

Let the two point-rows be _u_ and _u'_ (Fig. 11). Let _A_ and _A'_, _B_ and _B'_, be corresponding points, and let also the point _M_ of intersection of _u_ and _u'_ correspond to itself. Let _AA'_ and _BB'_ meet in the point _S_. Take _S_ as the center of two pencils, one perspective to _u_ and the other perspective to _u'_. In these two pencils _SA_ coincides with its corresponding ray _SA'_, _SB_ with its corresponding ray _SB'_, and _SM_ with its corresponding ray _SM'_. The two pencils are thus identical, by the preceding theorem, and any ray _SD_ must coincide with its corresponding ray _SD'_. Corresponding points of _u_ and _u'_, therefore, all lie on lines through the point _S_.

*53.* An entirely similar discussion shows that

_If in two projective pencils the line joining their centers is a self-corresponding ray, then the two pencils are perspectively related._

*54.* A similar theorem may be stated for two axial pencils of which the axes intersect. Very frequent use will be made of these fundamental theorems.