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,-w TUXbcd ~Newton' s Organic Construction:
Isaac Newton published the following construction for a general conic section, which he called his "organic" construction. Newton's original description is
almost incomprehensible. A somewhat more understandable, although still a bit dated, one is given by Salmon:
Two angles of constant magnitude move about fixed points P, Q; the intersection of two of their sides traverses a right line AA'; then the locus of V, the intersection of the other two sides will be a conic passing through P, Q.
This construction is illustrated in the sketch.
Start with a line AA' and two points (P and Q). For each point (M) on the line, draw the lines l1 = MA and l2 = MB. Rotate l1 through a fixed angle (50 degrees in this example, this is one of the constant angles described above) to get line l1'. Rotate l2 through another fixed angle (25 degrees in this example, this is the other constant angle) to get the line l2'. Look at the point V at the intersection of l1' and l2'. As M sweeps across AA', V sweeps out a conic that goes through P and Q.
You can experiment by moving P and Q around to see if you can generate other conics. With a little bit more work you can change the angles MPV and MQV.
Reference:
Coolidge, Julian Lowell, A history of the conic sections and quadric surfaces, The Clarendon Press, Oxford, 1945.
Salmon, George, A treatise on conic sections: containing an account of some of the more important modern algebraic and geometric methods, Hodges and Smith, Dublin 1850. (I don't think this is the first edition of this classic, amd it certainly isn't the last!!)
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