The geometric formalism devised by Gibbs in his development of capillary thermodynamics is replaced by an algebraic method in which no mention is made of mathematical dividing surfaces. The results are therefore free of the necessity of invariance theorems concerning dividing surfaces, and improve the clarity of Gibbs' presentation. Using algebraic methods, the various capillary excess quantities are interrelated by sets of linear equations whose coefficients involve bulk phase properties only. Some new results concerning highly curved interfaces are derived.