This all written up in Ogilvy, pp. 39-41 and Chapter 7, pp. 97-101.
Page 1. (CA/CB) = (OA/OB)(sin COA/sin COB)
Page 2. (DA/DB) = (OA/OB)(sin DOA/sin DOB)
These are the same formulas. The ratio for the figure on page 2 is negative (both sides). The cancellation of the other sines follows from sin t = sin (180 – t) for angle t measure in degrees.
Page 3.
(CA/CB)/ (DA/DB) = (sin COA/sin COB)/(sin DOA/sin DOB)
This ratio is called the cross-ratio R(C, D, A, B). This formula shows that the cross-ratio of the 4 points can be computed from the lines through O alone, without reference to which line the points are on.
Page 4. The figure for page 4 shows that R(A, B, C, D) = R(A', B', C', D'), since both double ratios are equal to the same double-ratio of sines.