The author considers the one-parameter lattice models of classical statistical mechanics from a simple algebraic viewpoint. The way in which the limiting locus of partition function zeros emerges through a sequence of semi-infinite (m* infinity ) lattice sections is considered. Convergence to the limiting locus C_infinity is obtained through two sequences of algebraic curves. For all arbitrarily large but finite m the partition function per site is a branch of an algebraic function Lambda ₁ defined by an irreducible polynomial and possesses only a finite number of algebraic singular points. In the limit of m to infinity infinitely many branch points accumulate on C_infinity; an algebraic basis to the universality hypothesis is presented in terms of the accumulation of branch points at the critical point. It is argued that for many two-dimensional lattice models the critical exponents alpha and nu will be of the form alpha =2/s and nu =(s-1)/s, where s is an integer >or=3.