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 *y*^{2} = *x*^{2} − *x*^{4}, 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.

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 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.

**Brouwer's fixed-point theorem** is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself.

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In mathematics, **differential calculus** is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

In mathematics, a **curve** is an object similar to a line, but that does not have to be straight.

**Bézout's theorem** is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that *in general* the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout.

In topology, a **Jordan curve**, sometimes called a *plane simple closed curve*, is a non-self-intersecting continuous loop in the plane. The **Jordan curve theorem** asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes some ingenuity to prove it by elementary means. *"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."*. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

In geometry, a **curve of constant width** is a simple closed curve in the plane whose width is the same in all directions. The shape bounded by a curve of constant width is a **body of constant width** or an **orbiform**, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century. By the end of the 17th century, both Leibniz and Newton claimed that the other had stolen his work, and the Leibniz–Newton calculus controversy continued until the death of Leibniz in 1716.

In mathematics, a **rational variety** is an algebraic variety, over a given field *K*, which is birationally equivalent to a projective space of some dimension over *K*. This means that its function field is isomorphic to

The classical **four-vertex theorem** of geometry states that the curvature function of a simple, closed, smooth plane curve has at least four local extrema. The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion.

**Robert Ralph Phelps** was an American mathematician who was known for his contributions to analysis, particularly to functional analysis and measure theory. He was a professor of mathematics at the University of Washington from 1962 until his death.

In differential geometry, a **hedgehog** or **plane hedgehog** is a type of plane curve, the envelope of a family of lines determined by a support function. More intuitively, sufficiently well-behaved hedgehogs are plane curves with one tangent line in each oriented direction. A **projective hedgehog** is a restricted type of hedgehog, defined from an anti-symmetric support function, and forms a curve with one tangent line in each direction, regardless of orientation.

- 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), 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, doi:10.1090/surv/097, ISBN 978-0-8218-2948-6, MR 1930577

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