The integrability of ovals: Newton's Lemma 28 and its counterexamples (Q5944758)

This lemma has excited discussion since its appearance in the \textit{Principia} in 1687, especially because of its cryptic and thereby incoherent formulation. The author considers the history of the discussion and possible plausible interpretations of the section's intensions, especially the status of proposed counter-examples. He concludes that Newton had a nice argument for the transcendence of expressions defining the area of sections of certain ovals determined by two arbitrary linear non-intersecting sections but not a rigorous proof.