By Singer I.
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From C\ to D I • Recall that the distances BIB2 = B2B3 = ... , B\B2 = glAB, • I B ,B2 ' From this it follows that BIB2 BIEI F\E2 ~ABI FIE2 1= - - = - - . - - = - - . 29). + dy2. 7. 1) for light and in an elegant manner saw how to transfer this principle to his own problem-perhaps this is why he was so pleased with the name "brachystochrone" since this notion of least time was basic to his approach. He certainly had at least an inkling of the importance that would attach to a careful pursuit of the analogy between optics and mechanics.
Newton says that the figure of revolution produced by rotating DNGB about the axis CB will experience less resistance in moving through a "rare and elastic medium" from B to A than will any other figure produced by rotating a curve with "the same longitude BC and latitude 2CD," provided that BS NTqq BG 4NS X ST cub In this figure N is an arbitrary point on the minimizing curve, NS is drawn through N parallel to CB, and NT is the tangent to the curve at N. 34 [In modern terms, this says that BS / BG = NT 4 /(4NS X ST 3 ).
CPA ~ D Solution Through the given point A draw the horizontal line APCZ and on it first describe any cycloid AQP cutting the line AB (produced if need be) 43 Andrade (1958), p. 100. PT, pp. 384-389. The paper appeared anonymously, presumably as Newton desired. 6. Leibniz's Solution of the Brachystochrone Problem 35 in the point Q and second another cycloid ABC whose base and altitude are, respectively, as AB is to AQ. This last cycloid passes through point B and is the curve along which a heavy particle will descend most quickly from the point A to the point B.
Abstract convex analysis by Singer I.