Isaac Newton's ``of quadrature by ordinates'' (Q354159)

In his unfinished and unpublished short treatise \textit{Of quadrature by ordinates}, I. Newton proposed two methods for approximating the quadrature of curves whose ordinates at a finite number of equidistant abscissae are given. He seemed to think that both methods are equivalent and verified this for small degrees. In the present paper, these methods and their relation to well-known numerical integration formulae are explained. In particular, it is shown that Newton's second method leads to the so-called Newton-Cotes formulae while the first method agrees with them only in small degrees.