Theorems about curves | Theorems in plane geometry
Newton's theorem about ovals
In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's Principia, and used it to show that the position of a planet moving in an orbit is not an algebraic function of time. There has been some controversy about whether or not this theorem is correct because Newton did not state exactly what he meant by an oval, and for some interpretations of the word oval the theorem is correct, while for others it is false. If "oval" means merely a continuous closed convex curve, then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate y2 = x2 − x4, while pointed that if "oval" an infinitely differentiable convex curve then Newton's claim is correct and his argument has the essential steps of a rigorous proof. generalized Newton's theorem to higher dimensions. (Wikipedia).
