# Newton's Ellipse

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Newton showed this construction in Book 1, Section 4, Lemma 15, of *Principia*. On March 13, 1964, Feynman resurrected the construction and used it in a lecture, "The Motion of Planets Around the Sun". The lecture is detailed in a book with audio CD, *Feynman's Lost Lecture*, by David and Judith Goodstein. In the lecture, Feynman used the diagram and differential geometry to prove the planetary laws of motion.

Contributed by: Bob Rimmer (March 2011)

Open content licensed under CC BY-NC-SA

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The construction is done in the complex plane, using the vector properties of complex numbers to construct the lines to the points. The equation of the circle is , where is the semi-major axis of the ellipse; the equation of the ellipse is , where is the radius of the ellipse from the primary focus at that point. The construction can be redone so that the secondary focus is outside the circle, creating a hyperbola, or on the circle, where it should create a parabola. A simple geometric proof can be found in chapter 9 section 5 of Jeremy Tatum's online text Celestial Mechanics.

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