Daniel Bernoulli (1733; 1734); Euler (1736)

Abstract

Daniel Bernoulli and Euler discussed the problem of the small oscilla- tions of a hanging chain or its discrete analogue, the linked pendulum, in St. Petersburg before Bernoulli’s departure in 1733. Bernoulli submitted his results¹ to the St. Petersburg Academy before leaving and communicated his proofs² the following year. After receiving Bernoulli’s proofs, Euler submitted his own version to the Academy.³ These papers finally appeared in 1738, 1740, and 1741, respectively. Essentially, these works treat the hanging chain and the linked pendulum as Johann Bernoulli had treated the vibrating string and the loaded string; but where Taylor and Johann Bernoulli had found the shape of the vibrating string to be sinusoidal, these works find the corresponding shape to be given by a Bessel function or, in the discrete case, by Laguerre polynomials, thus introducing these functions and the problem of finding their zeros. Perhaps it is because these functions are defined only analytically and not geometrically (as was the sine function) that the higher zeros are not overlooked and higher modes are discovered mathematically.