Gauss quadrature amounts to integration of a polynomial interpolant
in Legendre points, and the periodic trapezoidal rule to integration
of a trigonometric interpolant in equispaced points. What about the
Euler-Maclaurin formula, and its fully-discrete cousin the Gregory
formula, for nonperiodic data in equispaced points? We show that
these formulas too can be interpreted as integrals of interpolants.
Then we consider more broadly the theory and practice of interpolation
of smooth nonperiodic functions in equispaced points.