Some Methods of Approximate Integration and of Computing Areas

Abstract

IN NATURE of March 18 Mr. A. S. Percival gives an example (the quadrant of a circle) in which Simpson's rule (sometimes called his first rule) is more accurate than the “three-eighths” rule, and he remarks: “This result is curious, and shows that a small arc of a circle approaches more nearly to a small arc of a parabola than to a small are of any cubic curve.” Permit me to point out that this inference is not valid, and is based on the almost universal illusion that Simpson's rule is correct to the second order only, i.e. for the parabola y = a + bx + cx². It is easy to show by simple integration that Simpson's rule holds to the third order, i.e. for all cubics of the form y = a + bx + cx² + dx³, passing through the three chosen points. It is thus precisely accurate, not only for the parabola, but also for a singly infinite number of curves passing through the three points, even if an inflexion occurs.