Problem with statement of Needham's Visual Complex analysis exercise 15 chapter 3 page 185 
The exercise 15 and accompanying diagram above show 3 clockwise points $q, r, s$ on a circle and a 4th point $z$ on the circle between $q$ and $s$, and a second diagram with the same three points $q, r, s$ but with $z$ on the circle again but this time between $q$ and $r$. $\angle qrs$ is shown as positive and equal to $\phi$ and $\angle qzs$, shown as positive in the first diagram and negative in the second diagram and equal to $\theta$.
The exercise says:
Show that in both of the figures Arg$[z,q,r,s]=\theta+\phi$.
where $[z,q,r,s]$ is the cross ratio.
This seems strange to me since in the first diagram $\theta+\phi$ is always equal to $\pi$ and in the second diagram $\theta$ is always equal to $\phi$.
As far as the last part about deducing (31) is concerned,  (31) in the book says:
A point $p$ lies on the circle $C$ through $q,r,s$ if and only if
Im$[p,q,r,s]=0$.
I don't think this last part is relevant to understanding the exercise as stated, but I may be wrong.
What do you think?
 A: 
This seems strange to me since in the first diagram $\theta+\phi$ is always equal to $\pi$

Right, and that's the point. The sum of the angles is $\pi$, and a complex number whose argument is $\pi$ is a negative real number.

and in the second diagram $\theta$ is always equal to $\phi$.

Except for the orientation. Look at the arrows to see that in the second configuration you have $\theta = -\phi$, so the assertion is that in this configuration the cross ratio is a positive real number.
Together, the two configurations give you one direction of $(31)$, if four points lie on a circle, then their cross ratio is real. I don't see how you would directly obtain the other direction (if the cross ratio is real, then the four points lie on a circle [or straight line]) from the two configurations. But viewing the cross ratio - rather the map $z \mapsto [z,q,r,s]$ - as a Möbius transformation (with fixed $q,r,s$) quickly gives that too.
To obtain the desired result that the argument of the cross ratio is $\theta+\phi$, write down the formula for the cross ratio, and look at the arguments of the two ratios that combine to the cross ratio.
A note of caution: There are two different conventions for the cross ratio, in one, $z \mapsto \operatorname{CR}(z,q,r,s)$ maps $q\mapsto 0$ and $r\mapsto 1$, while in the other, we have $q\mapsto 1$ and $r\mapsto 0$. Both agree that $s\mapsto \infty$. It is important to be aware which convention each author uses.
