This book traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century work of Sergei Bernstein defining a new branch of function theory. The main focus is on the St. Petersburg Mathematical School and its interaction with mathematicians in Goettingen. The author presents a mathematical analysis of the subject together with a discussion of the philosophical underpinnings of the differences in approach of the various schools of thought. There is an exciting exposition in which history, philosophy, and mathematics are integrated. While demonstrating excellent technical control of the underlying mathematics, the work is focused on the essential results for the development of the theory.

Biographical data about some sixty eminent mathematicians and an excellent index round out the work. Historians as well as students interested in the history of mathematics will find this book useful.