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"D'ALEMBERT, JEAN LE ROND.
Sur les Tautochrones.
(Berlin, Haude et Spener, 1767). 4to. Without wrappers as issued in ""Memoires de l'Academie des Sciences et Belles Lettres"", tome XXI, pp. 381-413.
Reference : 39045
First edition, in the periodical form. In this paper D'Alembert comments on the previous solutions - by Huygens, Fontaine, Euler, Lagrange - to the tautochrone problem, the problem of finding the curve for an object sliding without friction in uniform gravity to its lowest point and in its shortest time.- Withbound a paper by M.J.A. Euler: Recherches des Forces dont les Corps célestes sont sollicités qu'ils ne sont pas spherique. Berlin, Acad. Royale, tome XXI, 1767.
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2 book(s) with the same title
Méthode pour trouver les Tautochrones, dans des Milieux résistants, comme le Quarré des Vitesses.
(Paris, L'Imprimerie Royale, 1732). 4to. Without wrappers. Extracted from ""Mémoires de l'Academie des Sciences. Année 1730"". Pp. 78-101.
First printing of Johann Bernoulli's importent paper in which he for the first time (Euler did it at the same time) solved the problem of finding the tautochrone in a medium that resists a body's motion directly as the square of the body's speed.After Huygens first discovered that the cylcoid was a tautochronous curve in vacuo according to the hypothesis of uniform gravity" Newton and Hermann have also given tautochrones following the hypothesis of non-uniform gravity acting, and pulling towards some fixed point as centre. Moreover, they have considered the motion to arise in a vacuum, with no resistance. Truly pertaining to resisting media, Newton has also shown that the cycloid is a tautochrone in a medium for which the resistance is proportional tothe speed moreover, as far as any other kinds resisting media are concerned, there has been no progress made either in roducing the curves themselves or in demonstrating possible tautochronism in them [The 3rd edition of the Principia that Euler refers to finally in 35 alters this view to include the type of resistance offered here. It may be of interest to the reader to observe that Johan. Bernoulli published a paper in the Memoire de l'Acad. Roy. des Sciences in 1730, also present in his Opera Omnia, T. III, p.173, with the title (in tra. from French): Method for Finding Tautochrones in Media Resisting as the Square of the Speed in which Euler does not get a mention.].
Sur les Courbes Tautochrones. (and ) Nouvelles Réflexions sur les Tautochrones. - [THE CALCULUS OF VARIATIONS]
(Berlin, Haude et Spener, 1767 and Berlin, Ch. Fr. Voss, 1774). 4to. Without wrappers as issued in ""Mémoires de l'Academie Royale des Sciences et Belles Lettres"" Tome XXI, pp. 364-380 and "" Nouveau Mémoires..."", pp. 97-122.
Both first edition in the journal form. Huygens proved Geometrically in 1659 that the tautochrone was a cycloid curve. This solution was later used to attack the problem of the Brachistochrone curve. Jacob Bernoulli solved the problem by using calculus in a paper from 1690, which for the first time used the term 'integral'. Both Lagrange and Euler loked for an analytical solution to the problem. Lagrange, in the papers offered here, developed a formal calculus based on the analogy between Newton's theorem and the successive differentiations of the product of two functions. He also communicated this to Eule in a letter written in Latin slightly before the Italian publication. In a letter to D'Alembert in 1769 Lagrange confirmed that this method of maxima and minima was the first fruit of his studies - he was only 19 when he divised it - and that he regarded it as his best work.A paper by Leonhard Euler:Éclaircissement plus détailles sur La generation et Propagation du Son et sur la Formation de L'Echo"" Berlin Academy Royale 1767"" in first edition withbound.
