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In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth
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 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.
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
oval is not an algebraic function In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
of the secant.
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
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
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, ...
, then there are counterexamples, such as triangles or one of the lobes of Huygens lemniscate ''y''2 = ''x''2 − ''x''4, 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.
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
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
* * * Alternative translation of earlier (2nd) edition of Newton's ''Principia''. * * * {{DEFAULTSORT:Newton's Theorem About Ovals Theorems about curves Isaac Newton Theorems in plane geometry