This image is adapted from a figure in Newton's Principia (1687). It illustrates his treatment of integration, and his proof that for a monotonic function f defined on an interval [A,E] the difference between the upper and lower sums with n equal subdivisions is equal in absolute value to (f(E)-f(A))(E-A)/n (and therefore goes to 0 as n goes to infinity). Here n=4.

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Anthony Phillips
Math Dept SUNY Stony Brook
email: tony at
February 5, 2008