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history-page25 - Pappus Given four lines in a plane nd the...

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Pappus: Given four lines in a plane, find the locus of a point that moves so that the product of the distances from two fixed lines (along specified directions) is propor- tional to the square of the distance from the third line [three-line locus problem], or proportional to the product of the distances from the other lines [four-line locus problem]. Pappus stated without proof that the locus was a conic section. Descartes showed this algebraically. · Descartes: “[S]ince so many lines are confusing, I may simplify matters by consid- ering one of the given lines and one of those to be drawn . . . as the principal lines, to which I shall try to refer all the others.” Newton later showed it geometrically. Descartes “I have omitted nothing inadvertently but I have foreseen that certain persons who boast that they know everything would not miss the opportunity of saying that I have written nothing that they did not already know, were I to make myself sufficiently intelligible for them to understand me.”
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