  This paper discusses the extent to which one can determine the space-time
metric from a knowledge of a certain subset of the (unparametrised) geodesics
of its Levi-Civita connection, that is, from the experimental evidence of the
equivalence principle. It is shown that, if the space-time concerned is known
to be vacuum, then the Levi-Civita connection is uniquely determined and its
associated metric is uniquely determined up to a choice of units of
measurement, by the specification of these geodesics. It is further
demonstrated that if two space-times share the same unparametrised geodesics
and only one is assumed vacuum then their Levi-Civita connections are again
equal (and so the other metric is also a vacuum metric) and the first result
above is recovered.
