‎"BERNOULLI, (JOHANN I) - THE TAUTOCRONE CURVE IN A MEDIA.‎
‎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.‎

Reference : 46599

‎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.].‎