**Paul Bachmann**'s father was Johann Friedrich Bachmann (1799-1876), a Lutheran minister, and his mother was Julie Lieder (1806-1885). Johann Bachmann had been born in Drossen, Prussia, the son of the Lutheran minister Christian Friedrich Bachmann. He had studied theology in Halle and was ordained before moving to Lisbon, Portugal, in 1825 as a preacher and teacher of religion. He returned to Berlin in the spring of 1828 to marry his fiancée Julie Lieder but was unable to go back to Portugal because of a civil war there. He became a preacher in the Luisenstadt district of Berlin. Johann and Julie's first two children, a son and a daughter, died as infants. They had another three sons and one daughter: Johann (1832-1888); Julie (1833-1900); Paul (1837-1920); and Gustav (1842-1890). Paul is the subject of this biography, but the other two sons also had notable careers, Johann as a professor of theology at Rostock and Gustav as an engineer, officer and teacher at the military academy in Kassel. Paul's father Johann became a pastor at the St Jacobi Kirche in the Luisenstadt district of Berlin soon after it was built in 1844-45.

Paul was brought up in a strict home where his modest nature was encouraged, and where he became an enthusiastic and highly talented musician. He began attending school at the age of seven. Later, he attended the Friedrich-Wilhelm-Gymnasium in Berlin at which he appears to have had some difficulty with mathematics. His mathematical talents, however, came to the fore under the excellent mathematics teacher Karl Heinrich Schellbach (1805-1892) who encouraged him to study mathematics at university. Schellbach, in addition to being professor of mathematics at the Friedrich-Wilhelm-Gymnasium from 1841, was also a professor at the Kriegsakademie in Berlin from 1843. An excellent research mathematician, Schellbach was particularly interested in 'the difficult art of teaching'. It was undoubtedly his skills which turned Bachmann into a top class mathematician. Bachmann graduated from the Gymnasium in March 1855 but his health at this stage was not good so, in order to recover from tuberculosis, he was advised to spend the summer in Switzerland which he did before beginning his university education.

Bachmann entered the University of Berlin (the Friedrich-Wilhelms-Universität) in the autumn of 1855 and studied mathematics there in the Faculty of Philosophy. In 1856 he went to the University of Göttingen so that he could continue to study courses by Lejeune Dirichlet who had just left Berlin to succeed to Gauss's chair in Göttingen. In Göttingen, Bachmann became close friends with Dedekind who had only a few years earlier been awarded his doctorate under Gauss's supervision. At the University of Göttingen Bachmann attended courses by Wilhelm Weber, Friedrich Woehler, Hermann Lotze, Bernhard Riemann and Richard Dedekind. Bachmann returned to Berlin where he received a doctorate in 1862 for a thesis on group theory under Eduard Kummer's supervision. He mentions, in a vita in the thesis, attending lectures by Kummer, Encke, Magnus, Dove, Rose, Trendelenburg, Weierstrass, Poggendorf, Borchardt and Arndt. His dissertation *De substitutionum theoria meditationes quaedam* Ⓣ was examined by E Fischer, F Bachmann and J Teichert on 24 March 1862.

From Berlin, Bachmann went to Breslau to study for his habilitation. There he worked on number theory and he was awarded his habilitation in 1864 for a thesis *De unitatum complexarum theoria* Ⓣ on complex units which had been a topic that he had been inspired to work on though the lectures by Dirichlet which he had attended. He taught at Breslau after the award of the habilitation, becoming an extraordinary professor there in 1867 after submitting *Theorie der komplexen Zahlen* Ⓣ. He published *Vorlesungen über die Lehre von der Kreisteilung* Ⓣ in 1872 and, in the following year, was appointed as an ordinary professor.

In 1875, he was appointed as a full professor at the University of Münster. He taught for many years at Münster frequently lecturing on topics from his favourite area of number theory. It gave him special satisfaction to see how number theory continued to grow with new friends and other researchers entering the area. However Bachamnn divorced his wife in 1890 and resigned his chair in Münster. Then, as Øystein Ore writes in [1]:-

With his second wife he settled in Weimar, where he combined his mathematical writing with composing, playing the piano, and serving as a music critic for several newspapers.

Freed from administrative tasks, Bachmann was able to carry out his plans for major publications that he had been thinking about for a long time. His most important work is a complete survey of number theory giving both the results and an evaluation of the methods of proof. *Zahlentheorie. Versuch einer Gesammtdarstellung dieser Wissenschaft in ihren Hauptteilen* Ⓣ, which he worked on after resigning his chair in Münster, was published in five volumes between 1892 and 1923. The first volume, subtitled *Die Elemente der Zahlentheorie* Ⓣ, was published in 1892. Jacob William Albert Young (1865-1948), associate professor of the pedagogy of mathematics at the University of Chicago, writes [8]:-

This work is the beginning of a homogeneous presentation of the subject in its present status, the object being rather to make a synopsis or an outline of this branch of mathematics than a compendium containing all that has ever been written on it. ... The work is a pleasing and welcome addition to the literature of the theory of numbers, on account of its contents and their manner of presentation, but still more as the forerunner of other volumes written in the same style and with the same aims, and dealing with the less elementary portions of the subject. ... It is to be hoped that[the author]will be able speedily to bring this difficult task to a satisfactory conclusion.

Bachmann followed this by: Volume 2, *Die analytische Zahlentheorie* Ⓣ (1894); Volume 3, *Die Lehre von der Kreisteilung und ihr Beziehungen zur Zahlentheorie* Ⓣ (1872); Volume 4, *Die Arithmetik der quadratischen Formen* Ⓣ (Part 1, 1898, Part 2, 1923); and Volume 5, *Allgemeine Arithmetik der Zahlkörper* Ⓣ (1905). His other major works include *Niedere Zahlentheorie* Ⓣ, published in two volumes in 1902 and 1910, *Das Fermat-Problem in seiner bisherigen Entwicklung* Ⓣ published in 1919, and *Grundlehren der neueren Zahlentheorie* Ⓣ (1907, Second Edition 1921, Third Edition edited by Robert Haussner 1931). Let us now look briefly at some of the comments made by reviewers of these works.

Jacob Young gives a very full review of *Die Elemente der Zahlentheorie* Ⓣ (1892) in [8]. Let us quote a short passage from the review which he gives before a detailed description of the work:-

The volume before us treats of the elements of the subject - congruences, quadratic residues, and forms, the last being restricted to binary forms. ... Regarding the work as a whole, the following points deserve especial remark:

Clearness of Presentation. - This is characteristic not only of the discussions, but also and especially of the formulations of the problems and the analyses of methods. This, together with the well-balanced mean preserved between excessive conciseness and too great diffuseness, makes the book unusually readable. It must be said, however, that the simpler presentation is not always preferred, but rather that which fits more consistently into the general plan of treatment....

Introduction of Recent Results. - Since the time of Gauss not much has been accomplished, relatively speaking, by elementary methods in the elementary field as outlined above, but still the author finds sufficient opportunity for presenting recent results to warrant the statement that he has given us, as proposed, not simply a sketch of the subject as Gauss left it, but of the elementary theory of numbers in its principal outlines as it stands to-day. ... Throughout the book well-chosen references direct the reader to original sources.

Historical Remarks. - Concise sketches of the salient features of the historical development of the subjects considered are made as occasion arises and add materially to the interest and also to the clearness of the treatment. The author is usually scrupulous in crediting even simple and commonly current results to their original publisher.

Leonard Dickson reviews *Niedere Zahlentheorie* (1902) in [3] and explains how Bachmann decided that a second introductory volume on number theory would be a useful addition to the literature:-

In view of the ambitious series of volumes by Bachmann; giving a comprehensive exposition of number theory, a series not yet completed, the appearance of a new volume on the elements of the subject, quite independent of the series mentioned, will doubtless cause some surprise. When the invitation came to the author to contribute to Teubner's 'Sammlung' a text upon the subject on which he is so eminent an authority, he hesitated long, fearing that a text on the elements of number theory ran the risk of conflicting with his 'Elementen'. The author has attempted to avoid this conflict in two directions: first by the addition of much important material; second, by employing a method of construction different at least in essential points. The author believes that the present book, both in contents and in foundation, may well be considered as a supplementary volume to his former series. ... The author has certainly succeeded admirably in the task he has undertaken. It will appeal especially to the student who wishes a complete account of the results and methods of the elementary parts of the theory of numbers.

Reviewing the second volume, published 8 years later, Dickson writes [4]:-

The first volume of this work appeared in1902... The present second volume has the subtitle 'Additive Zahlentheorie', a term suggested by Kronecker for the properties of numbers relating to their additive combinations. ... Bachmann has fully maintained his reputation as to clearness, thoroughness, and exhaustiveness.

Bachmann surveyed the attempts that had been made over nearly 300 years attempting to give a positive or a negative solution to Fermat's Last Theorem in *Das Fermatproblem in Seiner Bisherigen Entwicklung* (1919). Derrick N Lehmer, reviewing the book, writes:-

This little book, written in the terrible days of1918, is in honour of the fiftieth anniversary of the doctorate of Felix Klein. It is avowedly a glorification of German scientific effort in the theory of numbers, but the author is too great a scientist to allow any unworthy motive to colour the presentation of the subject. He treats the work of German and non-German with scholarly impartiality and thoroughness. The book gives in very convenient form the chief results of284years of struggle with the problem of proving the possibility or impossibility in integers of the equation x2.^{n}+ y^{n}= z^{n}for values of n greater than

E T Bell, reviewing the third edition of *Grundlehren der neueren Zahlentheorie* Ⓣ which was edited by Robert Haussner and published in 1931 (11 years after Bachmann's death) writes:-

The book can be warmly recommended to beginners with a good reading knowledge of German. The exposition is unusually detailed, and Bachmann spares no pains to make clear his reasons for attacking various arithmetical problems in the way he does.

Let us end this biography with the following assessment of Bachmann's contribution given by Robert Haussner in [5]:-

Common to all Bachman works is the full loving care and the great skill with which he knew how to extract from countless difficult individual works, an overview and a generous summary which he could achieve due to his mastery of this abstract domain and, even in his old age, admirable unparalleled knowledge of the literature. His constant endeavour was to achieve an artistically completed rounding of his works. His mathematical works show he had the mind of an artist, as he also showed as a musician. Many mathematicians have a love of music, and some of them they shall exercise in this or that instrument; and some of them they shall practice in this or that instrument; but it is rarely granted to them to be creative in music, as Bachmann's friends know that he was.

**Article by:** *J J O'Connor* and *E F Robertson*

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