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 "continuous convex curve", then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate y² = x² − x⁴, while Arnold (1989) 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.

Vassiliev (2002) generalized Newton's theorem to higher dimensions.

Statement

The lemniscate of Gerono or Huygens; the area cut off by a secant is algebraic, but the lemniscate is not smooth at the origin

An English translation Newton's original statement (Newton 1966, lemma 28 section 6 book I) 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

If the oval is a circle centered at the origin, then the spiral constructed by Newton is an Archimedean spiral.

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.

References

Arnold, V. I. (1989), "Topological proof of the transcendence of the abelian integrals in Newton's Principia", Istoriko-Matematicheskie Issledovaniya (31): 7–17, ISSN 0136-0949, MR 0993175
Arnold, V. I.; Vasilev, V. A. (1989), "Newton's Principia read 300 years later", Notices of the American Mathematical Society 36 (9): 1148–1154, ISSN 0002-9920, MR 1024727
Newton, I. (1966), Principia Vol. I The Motion of Bodies, Translated by Andrew Motte (1729) and revised by Florian Cajori (1934) (based on Newton's 2nd edition (1713) ed.), Berkeley, CA: University of California Press, ISBN 978-0-520-00928-8 Alternative translation of earlier (2nd) edition of Newton's Principia.
Pesic, Peter (2001), "The validity of Newton's Lemma 28", Historia Mathematica 28 (3): 215–219, doi:10.1006/hmat.2001.2321, ISSN 0315-0860, MR 1849799
Pourciau, Bruce (2001), "The integrability of ovals: Newton's Lemma 28 and its counterexamples", Archive for History of Exact Sciences 55 (5): 479–499, doi:10.1007/s004070000034, ISSN 0003-9519, MR 1827869
Vassiliev, V. A. (2002), Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs 97, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2948-6, MR 1930577

Isaac Newton

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	Hypotheses non fingo Standing on the shoulders of giants

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose Newton (unit)

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