Sums of powers of integers – how Fermat may have found them

On 22 September 1636, Fermat wrote to Roberval, [I, FO.II.XIII, p.71], saying that he had found the quadrature of an infinite number of curves. He named in particular the ‘solid parabola’, y = x 3 , and said that his method was different from Archimedes‘ quadrature of the parabola. He invited Roberval to share his thoughts on this and other matters. On 11th October 1636, Roberval replied to Fermat [1. FO.II.XIV, p.75], indicating his method for determining the quadrature of the ‘solid parabola’ and extending it to the quadrature of y = x 4 and y = x 5 .