We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.