This is a question about a result from Newton's Principia. It says, roughly, that the if you intersect lines $ax + by + c$ with a smooth, closed, convex curve, then the area of the curve that the line cuts off cannot be expressed as an algebraic function of $a, b$ and $c$.
My question is regarding Newton's proof. There are a couple versions out there, but I'll summarize the one on Wikipedia.
Fix a point $P$ inside the curve, and fix a line $L$ through $P$. Construct a spiral, $f$ as follows: draw another line $M$ through $P$, and let $\theta$ be the angle this makes with $L$. Thus $L$ and $M$ close off a sector of the curve, with some area $A(\theta)$. If we plot $(A(\theta), \theta)$ in polar coordinates, we get a spiral. Newton observes that this spiral intersects any line through $P$ infinitely many times. Thus the spiral cannot be algebraic, for if it were then there would be infinitely many solutions to a polynomial.
Fine. I agree with Newton's argument that the spiral cannot be algebraic, but how does this imply the original statement? More specifically, the proof is only successful because we allow the spiral to continue infinitely. But why can't we make a periodic function $B(\theta)$ such that $B$ expresses the area of the sector with angle $\theta$ for $\theta \in [0, 2\pi]$?