  We show that the Higgs and gauge fields for a BPS monopole may be constructed
directly from the spectral curve without having to solve the gauge constraint
needed to obtain the Nahm data. The result is the analogue of the instanton
result: given ADHM data one can reconstruct the gauge fields algebraically
together with differentiation. Here, given the spectral curve, one can
similarly reconstruct the Higgs and gauge fields. This answers a problem that
has remained open since the discovery of monopoles.
