The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, R x+m/n dx, or under a higher hyperbola, R x-m/n dx— with the appropriate limits of integration in each case—, has a second part which was not...

A discrete method of accuracy O(hm) is constructed to solve the integral equation eigenvalue problem λϕ(x)=∫−11K(x,y)ϕ(y)dy, where K(x, y)=log|x−y| or K(x, y)=|x−y|−α, 0α1, ϕ(x) being the eigenfunction associated with the eigenvalue λ. The method is based first on improving the...