 In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,
which has been described as the first formula for the DFT,
and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.