Download e-book for iPad: A History of the Calculus of Variations from the 17th by Herman. H. Goldstine

By Herman. H. Goldstine

ISBN-10: 1461381061

ISBN-13: 9781461381068

ISBN-10: 1461381088

ISBN-13: 9781461381082

The calculus of adaptations is a topic whose starting may be accurately dated. it'd be stated to start in the intervening time that Euler coined the identify calculus of diversifications yet this can be, in fact, no longer the genuine second of inception of the topic. it's going to no longer were unreasonable if I had long past again to the set of isoperimetric difficulties thought of through Greek mathemati­ cians corresponding to Zenodorus (c. 2 hundred B. C. ) and preserved through Pappus (c. three hundred A. D. ). i haven't performed this considering that those difficulties have been solved by way of geometric capacity. as an alternative i've got arbitrarily selected to start with Fermat's stylish precept of least time. He used this precept in 1662 to teach how a mild ray used to be refracted on the interface among optical media of alternative densities. This research of Fermat turns out to me specifically applicable as a kick off point: He used the tools of the calculus to reduce the time of passage cif a gentle ray during the media, and his strategy was once tailored by means of John Bernoulli to resolve the brachystochrone challenge. there were numerous different histories of the topic, yet they're now hopelessly archaic. One by means of Robert Woodhouse seemed in 1810 and one other by way of Isaac Todhunter in 1861.

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Thus it follows that (AP . GK)I/2 = CM + LO = arc GL, as Bernoulli asserted. ,,(a) cuts the family of cycloids through the point A : (Xl' Yl) at the points for which the time of fall is a constant T, let us suppose that the points of intersection are given by c:P = c:p(a). / d~ = (a1J/aa)/(a~/aa). It is easy to see that Cc:p - sinc:p) + aCI - cosC:P)C:Pa d~= (l-cosc:p)+asinc:p·c:pa fir! This must be the negative reciprocal of dy / dx, the slope of the cycloid. This implies directly that S2Curiously, Stroik says that Bernoulli noted that his synchrone is a cycloid.

33) becomes for = _ _x__ Jl - x 2 so that the curve is 2 + (y = 1, a circle as Bernoulli stated. In the last two paragraphs of his paper Bernoulli reintroduced a concept he had examined in connection with Huygens's theory of light in 1693. In effect, what he did was to find a curve orthogonal to each member of a sheaf or family of curves emanating from a given point. ) Thus in effect he found wave-fronts. 20 the plane of the paper is vertical, and the curves AB represent the one-parameter family of cycloids passing through the point A.

In a letter to a M. Basnage, doctor of law and author of the History of Works of SCientists,41 John Bernoulli discussed a number of aspects of the problem and interestingly notes that "I proposed the problem of swiftest descent in the Leipsic Acts, as being completely novel, not knowing that it had been attempted previously by Galileo" (p. 194). Again (p. 199) he says "M. Leibniz noted two remarkable things about Galileo: it is that this man who was, without contradiction, the most clairvoyant person of his times ...

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A History of the Calculus of Variations from the 17th through the 19th Century by Herman. H. Goldstine


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