Projective geometry, by Oswald Veblen and John Wesley Young.

82

THE FUNDAMENTAL THEOREM

[CHAP. IV

THEOREM 5. Given two harmonic sets H(12, 34) and H(1'2', 3'4'),
there exists a projectivity such that 1234 - 1'2'3'4'. (A, E)
Proof. Any projectivity 123 - 1'2'3' (Theorem 1, Chap. III) must
transform 4 into 4' by virtue of Theorem 3, Cor., and the fact that
the harmonic conjugate of 3 with respect to 1 and 2 is unique (Theorem 2). This is the converse of Theorem 3, Cor.
COROLLARY 1. If H (12, 34) and H (12', 3'4') are two harmonic sets
of different one-dimensional forms having the element 1 in common,
we have 1234= 12'3'4'. (A, E)
For under the hypotheses of the corollary the projectivity 123 - 1'2'3'
of the preceding proof may be replaced by the perspectivity 123 12'3t.
A
COROLLARY 2. If H (12, 34) is a harmonic set, there exists a projectivity 1234 - 1243. (A, E)
This follows directly from the last theorem and the evident fact
that if H(12, 34) we have also H (12, 43). The converse of this
corollary is likewise valid; the proof, however, is given later in this
chapter (cf. Theorem 27, Cor. 5).
We see as a result of the last corollary and Theorem 2, Chap. III,
that if we have H (12, 34), there exist projectivities which will transform 1234 into any one of the eight permutations
1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.*
In other words, if we have H (12, 34), we have likewise H(12, 43),
H (21, 34), H(21, 43), H(34, 12), H (34, 21), H (43, 12), H (43, 21).
THEOREM 6. The two sides of a complete quadrangle which meet in
a diagonal point are harmonic conjugates with respect to the two sides
of the diagonal triangle which meet in this point. (A, E)
Proof. The four sides of the complete quadrangle which do not
pass through the diagonal point in question form a quadrilateral
which defines the set of four lines mentioned as harmonic in the
way indicated (fig. 36).
It is sometimes convenient to speak of a pair of elements of a
form as harmonic with a pair of elements of a form of different
kind. For example, we may say that two points are harmonic with
two lines in a plane with the points, if the points determine two

* These transformations form the so-called eight-group.