mathematics

Newton's theorem about ovals

Newton's theorem about ovals is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Newton's theorem about ovals rather than just read about it. In short: 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.

Key takeaways

Newton's theorem about ovals belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Newton's theorem about ovals to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Newton's theorem about ovals from memory before moving on to harder problems.

Reference excerpt

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 Arnold (1989) pointed that if "oval" means an 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

An English translation Newton's original statement (Newton 1934, 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

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" (PDF), Notices of the American Mathematical Society, 36 (9): 1148–1154, ISSN 0002-9920, MR 1024727 Newton, I. (1934), Cajori, Florian (ed.), Principia Vol. I The Motion of Bodies, translated by Motte, Andrew, Berkeley: University of California Press. Copyright renewed 1962 and reprinted, ISBN 978-0-520-00928-8. 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, S2CID 119853564 Vassiliev, V. A. (2002), Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs, vol. 97, Providence, R.I.: American Mathematical Society, doi:10.1090/surv/097, ISBN 978-0-8218-2948-6, MR 1930577

Illustrations

Worked examples

Example 1 — a first encounter with Newton's theorem about ovals

Start with the simplest possible case. Write down what Newton's theorem about ovals claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Newton's theorem about ovals before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Newton's theorem about ovals ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Newton's theorem about ovals

In research: Newton's theorem about ovals appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Newton's theorem about ovals in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Newton's theorem about ovals is common in secondary-school and first-year university syllabi. It links to neighbouring topics Isaac Newton, Theorems about curves, Theorems in plane geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Newton's theorem about ovals outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Newton's theorem about ovals in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Newton's theorem about ovals means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Newton's theorem about ovals out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Newton's theorem about ovals in simple terms?

Why does Newton's theorem about ovals matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Newton's theorem about ovals?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Newton's theorem about ovals.