  In this paper I uncover and explain---using contour integrals and
residues---a connection between cubic splines and a popular compact finite
difference formula. The connection is that on a uniform mesh the simplest
Pad\'e scheme for generating fourth-order accurate compact finite differences
gives \textsl{exactly} the derivatives at the interior nodes needed to
guarantee twice-continuous differentiability for cubic splines. %I found this
connection surprising, because the two problems being solved are different. I
also introduce an apparently new spline-like interpolant that I call a compact
cubic interpolant; this is similar to one introduced in 1972 by Swartz and
Varga, but has higher order accuracy at the edges. I argue that for mildly
nonuniform meshes the compact cubic approach offers some potential advantages,
and even for uniform meshes offers a simple way to treat the edge conditions,
relieving the user of the burden of deciding to use one of the three standard
options: free (natural), complete (clamped), or "not-a-knot" conditions.
Finally, I establish that the matrices defining the compact cubic splines
(equivalently, the fourth-order compact finite difference formul\ae) are
positive definite, and in fact totally nonnegative, if all mesh widths are the
same sign.
