Newton's theorem about ovals
Encyclopedia
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 "continuous 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" means "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.
In modern mathematical language, Newton essentially proved the following theorem:
In other words, "oval" in Newton's statement should mean "convex smooth curve". The infinite differentiability at all points is necessary: For any positive integer n there are algebraic curves that are smooth at all but one point and differentiable n times at the remaining point for which the area cut off by a secant is algebraic.
Newton observed that a similar argument shows that the arclength of a (smooth convex) oval between two points is not given by an algebraic function of the points.
Newton took the origin P inside the oval, and considered the spiral of points (r, θ) in polar coordinates whose distance r from P is the area cut off by the lines from P with angles 0 and θ. He then observed that this spiral cannot be algebraic as it has an infinite number of intersections with a line through P, so the area cut off by a secant cannot be an algebraic function of the secant.
This proof requires that the oval and therefore the spiral be smooth; otherwise the spiral might be an infinite union of pieces of different algebraic curves. This is what happens in the various "counterexamples" to Newton's theorem for non-smooth ovals.
Secant line
A secant line of a curve is a line that intersects two points on the curve. The word secant comes from the Latin secare, to cut.It can be used to approximate the tangent to a curve, at some point P...
of a smooth convex oval is not an algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
of the secant.
Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
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 "continuous 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" means "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.
Statement
An English translation Newton's original statement is:- "There is no oval figure whose area, cut off by right lines at pleasure, can be universally found by means of equations of any number of finite terms and dimensions."
In modern mathematical language, Newton essentially proved the following theorem:
- There is no convex smooth (meaning infinitely differentiable) curve such that the area cut off by a line ax + by = c is an algebraic function of a, b, and c.
In other words, "oval" in Newton's statement should mean "convex smooth curve". The infinite differentiability at all points is necessary: For any positive integer n there are algebraic curves that are smooth at all but one point and differentiable n times at the remaining point for which the area cut off by a secant is algebraic.
Newton observed that a similar argument shows that the arclength of a (smooth convex) oval between two points is not given by an algebraic function of the points.
Newton's proof
This proof requires that the oval and therefore the spiral be smooth; otherwise the spiral might be an infinite union of pieces of different algebraic curves. This is what happens in the various "counterexamples" to Newton's theorem for non-smooth ovals.