What is meant by Newton's Rules, if A+B=C, A=C and B=C?

What is it meant by Newton's Rules in point (vii)?

Have you looked up "newton's rules" in the index of the text where you found the problem? Certainly google has turned up nothing relevant, which suggests that it is not a currently or widely used term.

Fronthe context, I guess it has something to do with multiple branches at the origin.

I have never heard of this term and couldn't Google it either. I meditated on the technique and I suspect the point is if you're near the origin and x is not equal to y, one of them is smaller, and you can throw out the highest order term in that variable to see locally what the graph looks like. This A+B=C statement is hiding a lot of work if that's what is going on.

There's an asymptotic principle that if ##A(x) + B(x) = C(x)## and both sides tend to infinity, then one of the terms ##A(x)## or ##B(x)## should become negligible towards the limit. So, either ##A(x)## or ##B(x)## is asymptotic to ##C(x)##, but probably not both. That is, of course, only a practical rule of thumb.

PLAGUE said:

TL;DR Summary: Newton's Rules: if A+B=C, A=C and B=C?

What is it meant by Newton's Rules in point (vii)?
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Maybe you can paste in a copy of the problem statement and not just the conclusion?

WWGD said:

Maybe you can paste in a copy of the problem statement and not just the conclusion?

I think the problem statement is draw a graph of ##x^3+y^3=3axy## but I agree it would have been nice

It is a technique for determining the shape of an algebraic curve close to and far away from the origin. (https://en.wikipedia.org/wiki/Curve_sketching)