Two millennia of mathematics : from Archimedes to Gauss / George M. Phillips

As stated in the preface, “This book is intended for those who love mathematics, including undergraduate students of mathematics, more experienced students, and the vast number of amateurs, in the literal sense of those who do something for the love of it”.

There is much here for all of these, as well as useful material for teachers of mathematics. The “two millenia” in the title refers to the period from Archimedes to Gauss. The topics treated reflect the author’s special areas of interest, number theory, approximation theory and numerical analysis.

The first chapter, “From Archimedes to Gauss,” opens with Archimedes’ method for approximating π (using inscribed and circumscribed regular polygons), then shows how to accelerate the convergence by repeated “extrapolation to the limit”. There is some history of the use of the inverse tangent to approximate π, and the double mean processes for accelerating convergence. It concludes with a section on Gauss and the AGM for approximating elliptic and related integrals.

Chapter 2, “Logarithms,” opens with general discussion of exponential and logarithmic functions, then discusses the contributions of Napier and Briggs, and the logarithm as an area.

The remaining three chapters are “Interpolation,” “Continued Fractions” and “More Number Theory”.

There are many proved theorems and a goodly number of exercises, sometimes substantial, with hints or outlines of a solution. Serious students as well as amateurs will be challenged, delighted, and not discouraged. The book deserves a place in the library of every undergraduate institution as well as in the private library of lovers of mathematics. (zbMath)