About the cover: Isaac Newton and Roger Cotes

Isaac Newton and the astronomical refraction.

In a short interval toward the end of 1694, Isaac Newton developed two mathematical models for the theory of the astronomical refraction and calculated two refraction tables, but did not publish his theory. Much effort has been expended, starting with Biot in 1836, in the attempt to identify the methods and equations that Newton used. In contrast to previous work, a closed form solution is iden...

Error of the Newton-Cotes and Gauss-Legendre Quadrature Formulas

Abstract. It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex plane containing the interval of integration. In the present paper, a bound on the error of the Newton-Cotes quadrature formula for analytic functions is derived. Also the bounds on the Legendre polyno...

Isaac Newton as a Probabilist

In 1693, Isaac Newton answered a query from Samuel Pepys about a problem involving dice. Newton’s analysis is discussed and attention is drawn to an error he made. On November 22, 1693, Samuel Pepys wrote a letter to Isaac Newton posing a problem in probability. Newton responded with three letters, first answering the question briefly, and then offering more information as Pepys pressed for cla...

Isaac Newton on Mathematical Certainty and Method

Centroidal Mean Derivative - Based Closed Newton Cotes Quadrature

In this paper, a new scheme of the evaluation of numerical integration by using Centroidal mean derivative based closed Newton cotes quadrature rule (CMDCNC) is presented in which the centroidal mean is used for the computation of function derivative. The accuracy of these numerical formulas are higher than the existing closed Newton cotes quadrature (CNC) fromula. The error terms are also obta...