Johann Bernoulli (1728)

Abstract

Johann Bernoulli announced his results on the vibrating string in letters of October and December of 1727 to his son Daniel in St. Petersburg. Excerpts of these letters were published by the St. Petersburg Academy in its volume for 1727 which appeared¹ in 1729. He presented the Academy with details in a paper that was published in the volume for 1728 which appeared² in 1732. To a considerable extent this paper can be regarded as a set of notes on Taylor’s analysis. But in addition, it introduces an analysis of the corresponding discrete problem of the weightless string loaded with equal and equally spaced point masses; it gives the pendulum condition for the continuous case in the form â d²y/ dz² = -y that was circumvented by Taylor; and finally it introduces the potential and kinetic energies as the sums or integrals of the respective energies of the masses or string elements. In solving the equation of the pendulum condition Bernoulli suppresses the harmonic constant a; he then finds this constant in two ways—either by substituting the solution back into the pendulum condition or by finding the maximum velocity of the mid-point from the equality of the maximum potential energy with the maximum kinetic energy.