FUNDAMENTAL THEOREM.—The crossratio of four points in a line is equal to the crossratio of their projections on any other line which lies in the same plane with it.
§ 14. Before we draw conclusions from this result, we must investigate the meaning of a crossratio somewhat more fully.
If four points A, B, C, D are given, and we wish to form their crossratio, we have first to divide them 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 and C being the first points in each pair. The crossratio is then the ratio AC:CB divided by AD: DB. This will be denoted by (AB, CD), so that
(AB, CD)= CB:ACADDB
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 B in their places, thus, P'B : PIB; and then fill
up, crosswise, the first by C and the other by D.
§ 15. If we take the points in a different order, the value of the crossratio will change. We can do this in twentyfour different ways by forming all permutations of the letters. But of these twentyfour crossratios groups of four are equal, so that there are really only six different ones, and these six are reciprocals in pairs.
We have the following rules:
I. If in a crossratio the two groups be interchanged, its value remains unaltered, i.e.
(AB, CD) = (CD, AB) re (BA, DC) = (DC, BA).
II. If in a crossratio the two points belonging to one of the two groups be interchanged, the crossratio changes into its reciprocal, i.e.
(AB, CD) = I/(AB, DC) = i/(BA, CD) = I/(CD, BA) = 'ADC, AB).
From I. and II. we see that eight crossratios are associated with (AB, CD).
[§ i6. If X.= (AB, CD), A= (AC, DB), v= (AD, BC), then A, At, v and their reciprocals i/A, I/µ, qv are the values of the total number of twentyfour crossratios. Moreover, A, ti, v are connected by the relations
P'
AiIiµ=,s+I/v=v+IIA= —Aµv= 1;
this proposition may be Droved by substituting for A, is, v andreducing to a common origin. There are therefore four equations between three unknowns; hence if one crossratio be given, the remaining twentythree are determinate. Moreover, two of the quantities A, µ, v are positive, and the remaining one negative.
The following scheme shows the twentyfour crossratios expressed in terms of A, is, v.]
(AB, CD) = (AB, DC) = (BA, CD).
For four harmonic points the six crossratios become equal two and two:
A=—I,I—A=2,AI=2, =—I,IIA,Ar1=2.
Hence if we get four points whose crossratio is 2 or , then they are harmonic, but not arranged so that conjugates are paired. If this is the case the crossratio = — I.
§ 19. If we equate any two of the above six values of the crossratios, we get either X= I, o, oo, or X= 1, 2, 1, or else A becomes a root of the equation A2—A+I =o, that is, an imaginary cube root of —I. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are allimportant.
§ 20. From the definition of 'harmonic points, and by aid of § I I, the following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are "separated " by
C and D.
For every position of C there will be one and only one point
D which is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, " if of four harmonic conjugates two
coincide, then a third coincides with them, and the fourth may be any point in the line."
If C is the middle point between A and B, then D is the point at infinity; for AC: CB=+1, hence AD:DB must be equal to —i.
The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.
This important property gives a first example how metric properties are connected with projective ones.
[§ 21. Harmonic properties of the complete quadrilateral and quadrangle.
(AB, CD) ' (AC, DB)'
(CD DC)  A t—µ I/(I—v) (CB CA I( —X — 1)/v
A,' BD)
(DC, BA) , (DB, AC)_
(AB, DC) ' (AD, BC)1
(BA, CD) . IjA IJ(I —ti) C, AB) Bv (~ —I)/A t+ l(µ—I)
A) I(BC,AD)
, D (CB, DA)
(DA, CB) J L
(AC, BD) ' (AD, CB)
(BD , AC) 1 —A µ v/(v—1) (BC, DD)~ ) IµI)/i I/ V
(
DB) (DA, BC) A/(A—
'
(DB, ..
§ 17. If one of the points of which a crossratio is formed is the point at infinity in the line, the crossratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the crossratio. Thus if I is a point at infinity, we have (AB, CI) = —AC/CB, because AI : IB = —i.
Every common ratio of three points in a line may thus be expressed as a crossratio, by adding the point at infinity to the group of points.
End of Article: FUNDAMENTAL THEOREM 

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