In the second section, after eq. 3.33, Feynman says:

My question is - what is "the next approximation" he's referring to. I don't see where any higher order terms like (delta y)^2 terms would come from in this analysis.

Ok, yes - I see it. I wasn't thinking about power series approximation. I guess he had to use that since he used a generalized form for the vector function C(x,y) , so C could be about anything. Thanks.