This is a Dover reprint of the 1917 edition of Huntington’s innovative introduction to the work of Dedekind, Cantor and Russell, which was the first such book in the English language. The first edition appeared in 1905, which was very close to the period when the foundations of the real number system had been properly laid. Consequently, mathematicians trained in the Victorian period could have used this book as an introduction to (what was then) the recently rigorized real number system and transfinite numbers.

Ideas in this book are presented in the rather archaic language of early set theory. For example, sets are referred to as classes, which makes the definition of ‘equivalence class’ very unwieldy. Also, an order relation on a set K is called a ‘serial relation within class K’, which explains the phrase ‘serial order’ in the book’s title. An ordered set, K, is called a ‘series’. In modern parlance a ‘continuum’ is a compact, connected Hausdorff space, but it has another meaning with respect to continuum mechanics. Cantor, however, referred to compactness in his definition of a linear continuum as a *linear continuous series which has both a first and last element. *Interestingly, GH Hardy in the 1908 edition of *A Course in Pure Mathematics* defined ‘*the* continuum’ to be the set of all real numbers, and that seems to have been the everlasting convention.

Aside from the quaintness of its terminology, this book provides an excellent account of the real number system in an historical setting. Indeed, it wouldn’t be out of place on the reading list for a present-day course in real analysis. For example, the relationship between density and countability is discussed in many interesting contexts, and compactness is eventually re-defined with respect to cluster points of a set. There are many historical footnotes and references to original sources.

As one of its founding fathers, Huntington became elected President of the Mathematical Association of America in 1916. An interesting account of his life is available at http://www-history.mcs.st-and.ac.uk/Biographies/Huntington.html, and this little gem is a fitting testimony to his contributions in the field of mathematics.

Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.