Mathematical Treasure: Works of Paul Bachmann

Author(s):

Frank J. Swetz (The Pennsylvania State University)

Paul Bachmann (1837-1920) was a German mathematician who studied under Dirichlet, Kummer, and Weierstrass. Eventually, Bachmann became a professor at the University of Münster, where he specialized in number theory; however, it was only after he resigned his professorship and moved to Weimar that he was able to focus on the writing of his greatest works. In 1892, he published Die Elemente der Zahlentheorie, the first book of what would be a five-volume series on number theory:

Volume 2, Die analytische Zahlentheorie (1894);

Volume 3, Die Lehre von der Kreistheilung unde ihre Beziehungen zur Zahlentheorie (1872);

Volume 4, Die Arithmetik der quadratischen Formen (Part 1, 1898, Part 2, 1923); and

Volume 5, Allgemeine Arithmetik der Zahlenkörper (1905).

The title page for Die Elemente der Zahlentheorie is shown here, courtesy of Wellesley College and the Internet Archive:

Title page of Die Elemente der Zahlentheorie by Paul Bachmann, 1892

The first three pages of the table of contents shown below provide an appreciation of the depth of Bachmann’s approach to number theory. The signature at the top of the first page belongs to Ellen Louisa Burrell, who taught mathematics at Wellesley College from 1886 to 1916, retiring as Professor of Pure Mathematics.

First page of table of contents for Die Elemente der Zahlentheorie by Paul Bachmann, 1892

Second page of table of contents for Die Elemente der Zahlentheorie by Paul Bachmann, 1892

Third page of table of contents for Die Elemente der Zahlentheorie by Paul Bachmann, 1892

The title page of Volume 2 of Paul Bachmann’s work, Die analytische Zahlentheorie, is shown below, courtesy of the University of Toronto and the Internet Archive:

Title page of Die analytische Zahlentheorie by Paul Bachmann, 1894

In his preface, Bachmann provides a brief summary of the history of the subject:

First page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

Second page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

Third page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

Fourth page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

Fifth page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

Sixth page of the preface to Die analytische Zahlentheorie by Paul Bachmann, 1894

A 1925 edition of the first part of the fourth volume is shown below, courtesy of the University of Michigan Historical Math Collection. This page bears the signature of Alexander Ziwet (1853-1928), a previous owner and fellow German mathematician. Ziwet spent his teaching career at the University of Michigan in the United States.

Title page of part one of Die Arithmetik der quadratischen Formen by Paul Bachmann, 1925