Carl Gustav Reuschle (1812–1875) is mostly forgotten today. Perhaps the only mathematicians who still remember him are those who make occasional use of his Tafeln complexer Primzahlen aus Wurzeln der Einheit gebildet, first published in 1875, which is a table of primes in cyclotomic number fields (of small degre) inspired by the work of Ernst Eduard Kummer.
During the period covered in this book, Kummer was a professor at the University of Berlin, at the apex of his career and reputation. A teacher by profession, Reuschle seems to have been something of a generalist, writing on a wide variety of subjects at several different levels. He published papers in Crelle's Journal, but he also wrote books for the general public and for children, textbooks, and encyclopedia articles. His publications in 1875 include not only the tables mentioned above, but also an atlas and two articles about Darwin's ideas. It is Reuschle's interest in Kummer's new theory of the arithmetic of "complex numbers built from roots of unity", however, that drove the correspondence contained in this book.
Reuschle and Kummer wrote to each other, mostly about Kummer's work on algebraic number theory, between 1856 and 1875. Most of Reuschle's letters have been lost, but clearly he valued and preserved what Kummer sent to him. This book collects the whole surviving correspondence (all of it in German), totalling 53 letters from Kummer to Reuschle and 8 letters from Reuschle to Kummer. To complete the package, the editors have included a letter from Kronecker to Reuschle, a note from Kronecker on Reuschle's tables, information on Reuschle's life and work, and an analysis of the number theory that appears in the letters.
Kummer started working on the arithmetic of cyclotomic fields in the early 1840s, so these letters will not serve to give us a glimpse of the development of those ideas. Neither do they give much insight into Reuschle's table-making. Rather, they give us a chance to see Kummer's mature take on his theory. Since Kummer also includes comments on other subjects, they will also be valuable to those interested in his life. Written from an eminent mathematician to a school teacher, these remarkable letters also reveal a bit of Kummer's personality and style. Their publication will be welcomed by all historians interested in Kummer, his work, and his times.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.