—% r CROSS-P.ATIOS.] the parallel to 32’ through S cuts )1. 'e thus see that every point in p is projected into a single point 111])’. _ _ A glance at fig. 5 shows that a segment A13 will be proJec-ted into a segment A'l5’ which is 11ot (-1111111 to it, at least not as a rule ; and also that the ratio AC :(.'B is not equal to the ratio A'C’:C'l}' fbrnn-«l lay the projections. These ratios will become equal only if p an.l 1;’ are parallel, for in this tase the triangle SAB is similar to the triangle b'A'l'}’. Between three points in a line and their projections there exists therefore in general no relation. But between four points a relation does exist. § 13. Let A, B, C, D be four points in p, A’, 13', C’, D’ their pro- jections in -1'/. then the ratio of the two ratios AC : C1’; and AD: l)l'» into whiz-h (_‘ and I) divide tl1e segment All is equal to the cor- respoinling (.'1|l't‘S>10]l between A’. 1}’, U’, D’. 111 symbols we ha'e—— JV‘ _ AD A'C’ _ A’D’ (‘Ii ‘ D1: = CB" ‘"171?’ 1) This is easily proved by aid of Fig. 5 similar triangles. Through the points A and 13 o11 p draw parallels to 1/, which cut the projecting rays in C2, D3, ll._. and A1, C1, D], as indicated in I':_-_:. G. ’l'he two triangles AC(‘._. and B(_'t,‘, will be similar, as will I-1.50 be the triangles ADD._. and ];]._).1)l 'e have therefore—— .C_At,'._, Al) _'D___ eT;’c,1; ’ lie’ ‘trfi’ wln-re an"-count is taken of tl1e sense. llence— _AL) _ _ Ag, AC1, _ C‘,B _ C15 ‘ms “ L',B '1),15 .»),_,'1),I’.’ 1 ut AC, A’L" C 1’; CT)’ -= —,—— and —‘—=—,_; A1), A D’ D,l’» D 15' 2) that the above expression becomes A’C’ _ C'B' A’D’ ' D'J5” ' 1! I I
- as required.
This result is of fuinlamental importance. The expression formed has been called by Chasles the anharmonie ratio of the four points A, 1}, C, 1). Instead of this Professor Clitlord has proposed the shorter and more expressive name of “cross- ratio." We shall adopt the latter. 'e have then the Ft-'.‘D..Il-‘.."l‘.[. THeon1:.r.—T/tc cross-ratio of four points in a I inc is equal to the c1'os9-mtz'0 Qf I/zeir projections on any other line 'u;Iu'ch lies in t/zc same plane with it. § 14. ll:-fore we draw conclusions from this result, we must in- vestigate the meaning of a cross—ratio somewhat more full_v. ll‘ four points A, B, C, D are given, and we wish to form their cross—ratio, we have first to divide then1 into two groups of two, the points in each group being taken in a definite order. Thus, let A, B be the first, C. D the second pair, A aml C being the first points in each pair. The cross-ratio is then the ratio AC : CB divided by Al) : DB. This will be denoted by (ABCD), so that , AC AI) (A1}CD,=fi3 : DB. rshieh is I:-rlual to GEOMETRY 391 This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these the letters A and 13 in their places, thus, AT, 2 J—B ; and then fill up, crosswise, the first by C and the other by D. § 15. If we take the points in a difl'ercnt order, the value of the cross-ratio will change. “'0 can do this in twcnty—four dif1'eren1: ways by forming all permutations of the letters. ’;ut of these tventy-four cross-ratios groups of four are equal, so that there are really only six diiii-rent ones. 'e have the following rules :— I. If in a cross-ratio the two groups be interchanged, its value remains unaltered. (ABCD) = (CDAIE). II. If in a ct‘os:s-1-atio the two points belonging to one of the two groups be intereltanged, the cross-ratio changes into its reciprocal. (M-)’CD)=(.;1>c;' 111. If in a cross—ratio the two middle letters be interchanged, the cross-ratio It changes into its complement 1 —;c. (Al}C‘D) = 1 —(ACBD). The first two are easily proved by writing out their values. The third is pioved by aid of" the formula (3), § 10, 11C . Al)+CA . BD+AB . CD=0. If we divide this by (‘B . AD we get CA . DD AB . CD —1+eTm+T.t1‘)=°~ 01‘, £0 . 9:‘? J“ 239.1. L‘l3'1)B BC ' DL‘_ ’ that is, ' (ABCD) + (ACBD) = 1, which was ts be proved. IV. From 11. it follows at once that if we interchange the elements in each pair, the cross—ratio remains unaltered; and thus we see thft I (Al’»(‘D)=(C‘DAl3) by I. =(l}ADC)=(I)Cl3A) by II. § 16. By aid of these rules we get the following results :- (ABCD) = (BADC) = (CDAI3) = (DCBA) = K. (ABDC) = ( BACD) = (CDBA) = (DCA B) = (ACBD) = (BDAC) = (CADB) = (DBCA) = 1 — rc. 1 (ACDB) = (coca) = (CABD) = (DBAC = 1-3. 1 (ADB(‘)= (13e-u)) = (com) =(DACB) ="—;-. (ADCB)=(B(‘])-)=(('BAD)=(DABC‘)=;§i. In the theorem that the cross—ratio of four points equals that of the projections, the points have, of course, to be taken in the same order. § 17. 11' one of the points of which a cross—ratio is formed is the point at infinity in the line, the cross—ratio changes into a simple ratio. It is convenient to let the point at: infinity occupy the last place in the symbolic expression for the cross—ratio. Thus if 1 is a point at infinity, we have AC (ABCI)= —-L.—B, because AI ; IB= — 1. Every common ratio of three points in a line may thus he ex- pressed as a cross—ratio, by adding the point at infinity to the group of points. § 18. If the points have special positions, the cross-1 atios K may have such a value that, of the six different one:, two and two I become cqual. If the first two shall be equal, we get rc=,}s 01' rc'-‘=1, IC= il- AC AD . If we take rc= +1, we have (AI3CD)=1, or a;=m-3; that is, the points C and D eoineitle, provided that A and B are ditl‘erent. This is too special a case to be of much interest. If, however, we take :c= — 1, so that (Al3CD)= -1, we l1aV0 AC AD elf D1:' in the samc 7'al2'n. _ _ The four points are in this case said to be ha7'7_12071lC]70“?[S. “W1 C and D are said to be harmonic cruIju_I/ates 'zL'z¢'IL regard to A and B.
Ilcncc C and D tli2'1'(le Al} z'ntcrn(:ll_1/ and cJ:tcrm1lI_1/