Discusses Riemann postulates characterizing manifolds of constant curvature. This book also shows how to determine the conditions under which a divergent series may be manipulated as the analytic representative of an unknown function, to develop the properties of the function, and to formulate methods of deriving a function from the series.

The 1903 colloquium of the American Mathematical Society was held as part of the summer meeting that took place in Boston. Three sets of lectures were presented: Linear Systems of Curves on Algebraic Surfaces, by H. S. White, Forms of Non-Euclidean Space, by F. S. Woods, and Selected Topics in the Theory of Divergent Series and of Continued Fractions, by Edward B. Van Vleck. White's lectures are devoted to the theory of systems of curves on an algebraic surface, with particular reference to properties that are invariant under birational transformations and the kinds of surfaces that admit given systems. Woods' lectures deal with the problem of the classification of three-dimensional Riemannian spaces of constant curvature.The author presents and discusses Riemann postulates characterizing manifolds of constant curvature, and explains in detail the results of Clifford, Klein, and Killing devoted to the local and global classification problems. The subject of Van Vleck's lectures is the theory of divergent series. The author presents results of Poincare, Stieltjes, E. Borel, and others about the foundations of this theory. In particular, he shows 'how to determine the conditions under which a divergent series may be manipulated as the analytic representative of an unknown function, to develop the properties of the function, and to formulate methods of deriving a function uniquely from the series'. In the concluding portion of these lectures, some results about continuous fractions of algebraic functions are presented.