CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER
23. Seven fundamental forms. In the preceding chapter we have called attention to seven fundamental forms: the point-row, the pencil of rays, the axial pencil, the plane system, the point system, the space system, and the system of lines in 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 [pg 15] 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.
[pg 16]
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 [pg 17] 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.
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' passing 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 [pg 18] joining K' and M', found in this way, must pass 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 pass 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 rôle 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 pass through B, two through D, and of the remaining pair, one through A and the other through C.
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 [pg 19] passes through A, one through C, and one remaining side through D; therefore the other remaining side must pass through B. Similarly, RS passes through B and PS and QR pass 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 pass 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 passing through S. This plane cuts out on the set of seven planes another [pg 20] 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 pass through a point and which pass 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 pass one through each of four harmonic points, and we may show that
Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points.
For let the planes α, β, γ, δ, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that α passes through A, etc. Then it is clear that any plane π through A, B, C, D will cut out four harmonic lines from the four planes, for they are [pg 21] lines through the intersection P of g with the plane π, and they pass through the given harmonic points A, B, C, D. Any other plane σ cuts g in a point S and cuts α, β, γ, δ in four lines that meet π 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 α, β, γ, δ 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 assemblages 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.
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.
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 [pg 22] 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, passing 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.
[pg 23]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.
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 [pg 24] 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.
[pg 25]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 repetition 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.
44. Let A, B, C, D be four harmonic points (Fig. 8), and let SA, SB, SC, SD be four harmonic lines. Assume 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 [pg 26] 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',
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 quantities 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
which is convenient for the computation of the distance AD when AB and AC are given numerically.
[pg 27]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 pass 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.
[pg 28]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 passing 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.