These questions are taken from Stillwell's The Four Pillars of Geometry. For $Q 5.7.3$, I am confused about what the question is asking. Is at asking me to:
1) Compute the cross ratio's between four points $P,$ $Q,$ $R,$ $S$, where the distance $PQ=1$, $QR=e$, $RS=e^2$
OR
2) Is it asking me to compute first the cross ratio between four point $P$,$Q$,$R$,$S$ each separated by distance $1$, then compute the cross ratio for another four points $A,B,C,D$ each separated by distance $e$, and then compute the cross ratio between another 4 points $W,X,Y,Z$ each separated by distance $e^2$.
I need clarification of what is being asked of me, but I would also really appreciate some sort of dumbed down intuition (I am a complete beginner at Projective Geometry) about the technique mentioned in the question would be incorrect. Because as it stands, I really don't see how either one of 1) or 2) would show that the technique is incorrect? In fact, what does it even mean for the technique to be incorrect? What does it do incorrectly?
In response to rschwieb's answer:
What I understood from your answer:
The points on the tiles in the real world are $P=0, Q=1, R=2, S=3$. When making his drawing, the artist thinks that right way to draw this would be to have each width a constant fraction $e$ of the one before it. So on this drawing board he puts the point points $P'=0, Q'=1, R'=1+ e, S'=1+ e + e^2$. If we calculate the cross ratio of $P',Q', R', S'$, we know that it should be $\frac{4}{3}$ because the original points $P,Q,R,S$ were equally spaced. But when we calculate $\frac{(1+e)(e^2+e)}{(e)(e^2+e+1)} =\frac{4}{3}$, we get $e=0,1$, which contradicts our original assumption that $e$ was a fraction. So we conclude the technique is wrong, and doesn't correctly project the original points $P, Q, R, S$ in terms of perspective.
Your first interpretation is fine. It's saying that along the line receding into the distance, the first gap has length $1$, the second $e$ and the third $e^2$. The artist is succeeding in showing the gaps are shrinking, but it does not reflect reality.
The cross-ratio is a quantity for the positions of the four points, one which is invariant under projective transformations. We know that a projective transformation can transform the line into one which has the three points equally spaced because we're supposed to assume a rectangular or square tiling.
We also know that equally spaced points always give a cross ratio of $\frac 43$, so if you compute the cross ratio of $P,Q,R,S$ as you have given, you should find it's $\frac{(1+e)^2}{1+e+e^2}$. If you equate this to $\frac43$ and solve, you should find the only solution is $e=1$, which of course isn't under consideration (we're using $e< 1$.)
You might follow up by computing, given the first gap $1$, the second gap $e$ what that third gap is in terms of $e$ for the cross-ratio to work out to $\frac43$.
In desmos, I plotted the artist's prediction (blue) and the cross-ratio's prediction (red) for $x$ in the interval $[0,1]$
For fun I picked the value $e=0.3$ and generated a few plots that demonstrate what the perspective looks like, when drawn that way. Keep in mind that they plot the same sequence of colored lines, just at different $y$ values.
Here's the naïve artist's formula:
And here's what projective geometry says:
With a little more work one should be able to predict the $y$ coordinate of the horizon. From what it looks like here, the artist's prediction looks like it might converge way too fast, below the true horizon.