The self--duality transformation is applied to the Fisher zeroes near the critical point in the thermodynamic limit in the q>4 state Potts model in two dimensions. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic critical behaviour satisfy the latter requirement, the full locus of Fisher zeroes is shown to be a circle. This locus, together with the density of zeroes is shown to be sufficient to recover the singular form of all thermodynamic functions in the thermodynamic limit. (Contribution to LATT97.)