  We study the indentation of ultrathin elastic sheets clamped to the edge of a
circular hole. This classical setup has received considerable attention lately,
being used by various experimental groups as a probe to measure the surface
properties and stretching modulus of thin solid films. Despite the apparent
simplicity of this method, the geometric nonlinearity inherent in the
mechanical response of thin solid objects renders the analysis of the resulting
data a nontrivial task. Importantly, the essence of this difficulty is in the
geometric coupling between in-plane stress and out-of-plane deformations, and
hence is present in the behaviour of Hookean solids even when the slope of the
deformed membrane remains small. Here we take a systematic approach to address
this problem, using the membrane limit of the F\"{o}ppl-von-K\'{a}rm\'{a}n
equations. This approach highlights some of the dangers in the use of
approximate formulae in the metrology of solid films, which can introduce large
errors; we suggest how such errors may be avoided in performing experiments and
analyzing the resulting data.
