studied by Newton beginning in 1664. One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of an ellipse by a pinned thread. If one stretches a thread from a pin at one focus to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci. However, Newton rejected such constructions as insufficiently rigorous. He defined the oval as the solution to a differential equation, constructed its subnormals, and again investigated its optical properties. The French mathematician Michel Chasles discovered in the 19th century that, if a Cartesian oval is defined by two points and , then there is in general a third point on the same line such that the same oval is also defined by any pair of these three points. James Clerk Maxwell rediscovered these curves, generalized them to curves defined by keeping constant the weighted sum of distances from three or more foci, and wrote a paper titled Observations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions. An account of his results, titled On the description of oval curves, and those having a plurality of foci, was written by J.D. Forbes and presented to the Royal Society of Edinburgh in 1846, when Maxwell was at the young age of 14 (almost 15). See also Cassini oval Two-center bipolar coordinates References External links Benjamin Williamson, An Elementary Treatise on the Differential Calculus, Containing the Theory of Plane Curves (1884) Algebraic curves