Harmonic quadruples and von Staudt's theorem

In the nineteenth century, some geometers where unsatisfied with the characterization of projective transformations given by the second fundamental theorem on the grounds that it was not sufficiently geometric. In this section we describe the characterization they came up with.

Definition 3.1   A quadruple of points on the projective line is said to be a harmonic quadruple if its cross-ratio equals $-1$.

Theorem 3.1   In the following figure the quadruple $x,y,z,t$ is harmonic.

Exercise 3.1 (05)   Find in the figure two perspectives such that their composition interchanges $x$ and $y$ while fixing both $z$ and $t$.

Exercise 3.2 (05)   Use that perspectives preserve cross-ratios to deduce the theorem from the previous exercise.

Theorem 3.2 (von Staudt's theorem)   A map from the projective line to itself is projective if and only if it sends harmonic quadruples onto harmonic quadruples.

The proof is given in the form of exercises.

Exercise 3.3 (20)   Let $f : {\Bbb R}\rightarrow {\Bbb R}$ be a function satisfying the following properties:

• $f(0) = 0, f(1) = 1$.
• $f(x + y) = f(x) + f(y)$.
• $f(x \cdot y) = f(x) \cdot f(y)$.

Show that $f(x) = x$ for all $x \in {\Bbb R}$.

Exercise 3.4 (00)   Show that if $x$ and $y$ are any real numbers, then

• $[x,y,(x+y)/2,\infty] = -1$.
• $[1,x^2,x,-x] = -1$

Exercise 3.5 (15)   Let $T$ be a map from the real projective line to itself that preserves harmonic quadruples and fixes the lines with slopes $\infty, 0$ and $1$. If $f : {\Bbb R}\rightarrow {\Bbb R}$ is the function such that $f(x)$ is the slope of the image under $T$ of the line of slope $x$, show that $f$ satisfies all hypotheses in exercise 3.3 and is, therefore, equal to the identity. Use this to prove von Staudt's theorem.