**Eugen Netto**'s father was Heinrich Netto (1795-1889), an official at the Franckesche Stiftung at Halle. The Stiftung was a Protestant religious institute which included a school for the poor, an orphanage, a medical centre, and publishing house. Eugen's mother was Sophie Louise Neumann (1816-89). We note that several biographies give 1848 as Netto's year of birth but Netto himself gives 1846 in the Vita at the end of his dissertation [12] and it seems highly unlikely that he would have made an error. Up to the age of eleven Eugen attended a school in Halle, but from that time he went to the Friedrich-Wilhelm Gymnasium in Berlin, an excellent school founded in 1797.

Karl Ferdinand Ranke (1802-1876), a classical philologist, was the rector of the Friedrich-Wilhelm-Gymnasium from 1842 to 1876, so was the rector when Netto began his studies there. Ranke introduced more English, French and mathematics into the curriculum at the expense of Greek. Netto was fortunate to have two outstanding teachers of mathematics at the Gymnasium in Karl Heinrich Schellbach and August Rudolph Luchterhandt. Schellbach (1805-1892) had studied mathematics, physics and philosophy at Halle before being awarded a doctorate by the University of Jena in 1834. He had been a teacher at the Friedrich-Wilhelm Gymnasium from 1841 where he had taught Gotthold Eisenstein. Luchterhandt was born 1810 in Marienwerder, and had studied at the University of Königsberg before becoming a teacher of mathematics at the Friedrich-Wilhelm-Gymnasium. Both Luchterhandt and Schellbach were excellent mathematicians and exceptional teachers but it was Schellbach who showed Netto the excitement of mathematics and from that time on mathematics was clearly the only topic that he considered. After graduating from the Gymnasium in 1866, Netto entered the University of Berlin to study mathematics. He mentions in his Vita in [12] that Alexander Carl Heinrich Braun (1805-1877), who was a botanist, was rector of the University of Berlin when he entered.

At the University of Berlin he again had some inspiring teachers in Leopold Kronecker, Karl Weierstrass and Ernst Eduard Kummer. However, there were others whose courses he took while studying there. These included: Heinrich Wilhelm Dove (1803-1879), a physicist and meteorologist who was appointed an ordinary professor at Berlin in 1845; Wilhelm Julius Foerster (1832-1921), an astronomer who became a professor at Berlin University in 1858 and worked at the Berlin Observatory becoming director in 1865; Friedrich Harms (1819-1880), a philosopher who, after being appointed as an ordinary professor at Kiel in 1858, was appointed to Berlin in 1867; Georg Hermann Quincke (1834-1924), a physicist who was a docent in Berlin from 1858 and a professor from 1865; Wilhelm Ludwig Thomé, a mathematician who had been advised by Weierstrass and awarded a Ph.D. in 1865; and Friedrich Adolf Trendelenburg (1802-1872), a philosopher who was an ordinary professor at Berlin from 1837. Netto says in [12] that he greatly benefited by the outstanding teaching of all these men but his greatest thanks went to Leopold Kronecker, Ernst Eduard Kummer and Karl Weierstrass.

Netto graduated from Berlin in 1870 having worked specifically under Weierstrass and Kummer. It was in fact Weierstrass who examined his final 20-page doctoral dissertation, written in Latin, *De transformatione aequationis **y*^{n} = *R*(*x*)*, designante R(x) functionem integram rationalem variabilis x in aequationem *

*η*

^{2}=

*R*

_{1}(

*xi*) Ⓣ. He defended his thesis in a public examination on 2 November 1870 with three opponents, Ernst Heinrich Bruns, Friedrich Wilhelm August Ludwig Kiepert, and Paul Simon. Bruns has a biography in this archive. He was awarded a Ph.D. in 1871 advised by Weierstrass and Kummer. Kiepert (1846-1934) was awarded a Ph.D. in 1870, also advised by Weierstrass and Kummer. Simon was a student at Berlin at the time of the examination and was awarded a Ph.D. from Halle-Wittenburg in 1876. Netto had to submit three further minor theses in order to obtain his doctorate. These, again written in Latin, were:

*Facilis in analysin valorum negativorum introductio nocuit cognitioni valorum complexorum.*

2.

*Generalitatem atque necessitatem signa esse judiciorum a priori, Kantius non satis confirmavit.*

3.

*Trigonometria ita est docenda, ut a circulo non a triangulo rectangulo considerando initium capiatur.*

*Zur Theorie der zusammengesetzten Gruppen*Ⓣ (1874). He published two papers in 1877, namely

*Beweise und Lehrsätze über transitive Gruppen*Ⓣ and

*Neuer Beweis für die Unauflösbarkeit der Gleichungen von höherem als dem vierten Grade*Ⓣ. In 1878 he published three papers, two on substitution groups and one on entire functions.

For a list of Netto's publications, see THIS LINK.

It was the French-German war of 1870-71, which ended with Alsace being annexed by the German empire, that had led to a German university being set up in Strasbourg. In 1872 the so-called Kaiser-Wilhelms-Universität was opened in Strasbourg. The Mathematics Seminar there was directed by Elwin Christoffel and Theodor Reye, and Netto took part in this seminar. His involvement is described in [15] where interesting background information about the working conditions and the number of students is given. In 1880, while he was Strasbourg, Netto married Hedwig Elfriede Freund (1860-1921); they had one son and two daughters.

After three years at the University of Strasbourg Weierstrass recommended that Netto be appointed an extraordinary professor at the University of Berlin and he took up the appointment in 1882. There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry. Netto held this post in Berlin until 1888 when he was appointed ordinary professor at the University of Giessen. He held this post for twenty-five years until his retirement in 1913. He gave a course of lectures entitled *Einleitung in die Algebra* Ⓣ during the summer semester of each year in the University of Giessen. This course became the book *Elementare Algebra* Ⓣ (1904).

In 1878 he attempted the second general proof of the invariance of 'dimension' but, like the first by Johannes Thomae, it was not completely satisfactory. Despite this, Netto's "proof" was widely accepted as providing a solution to the dimension problem until Jurgens' criticism in 1899 of Netto's proof. Jurgens similarly criticised a proof of the invariance of 'dimension' which had been given by Georg Cantor. These events are fully described in [10]. Cantor showed in 1878 that the unit interval *I* can be mapped bijectively onto the unit square *I*^{2}. In the following year Netto showed that such a mapping cannot be a continuous function. These results by Cantor and Netto are starting points for the investigations of space-filling curves which are an active research area today.

Netto made major steps towards abstract group theory when he combined permutation group results and groups in number theory. He did not however include matrix groups! He published this work in his book *Substitutionentheorie und ihre Anwendung auf die Algebra* Ⓣ in Berlin in 1882 described by Biermann in [1] as:-

The book was translated into English by Frank Nelson Cole and published under the title... a milestone in the development of abstract group theory.

*The theory of substitutions and its applications to algebra*(1892). Oskar Bolza wrote a review in which he wrote [5]:-

For a longer extract from Bolza's review, see THIS LINK.The great merit of Netto's book consists in the skilful and highly pedagogical presentation of the theory of substitutions, given in the first part of the book. The reader is gradually led from the most elementary considerations on symmetric and alternating functions to the general theory of unsymmetric functions of n independent elements, out of which the theory of substitutions is step by step evolved, the unsymmetric functions serving all the while as a concrete substratum for the abstract conclusions of the theory of substitutions. By this means an easy and attractive entrance into the theory of substitutions is gained, accessible even to the beginner, and it may fairly be said that Netto's book has largely contributed to spread the knowledge of this important branch of mathematics.

For Cole's translation of Netto's Preface to the 1882 German text, see THIS LINK.

This was only the first of a number of important books published by Netto. Others were: *Vorlesungen über Algebra*. Band 1 Ⓣ (1896); *Vorlesungen über Algebra. Erste Lieferung des 2. Bandes* Ⓣ (1900); *Über die Grundlagen und die Anwendungen der Mathematik* Ⓣ (1900); *Lehrbuch der Combinatorik* Ⓣ (1901);* Elementare Algebra. Akademische Vorlesungen für Studierende der ersten Semester* Ⓣ (1904); *Gruppen- und Substitutionentheorie* Ⓣ (1908); *Die Determinanten * Ⓣ (1910); and *Grundlehren der Mathematik. 1. Teil, 2. Band: Algebra* Ⓣ (1915).

For extracts of reviews of some of these books see THIS LINK.

He further contributed to the development of group theory in other papers. In particular, in 1877 Netto had given new proofs of the Sylow's theorems. His book *Gruppen- und Substitutionentheorie* Ⓣ (1908) is an important introduction to group theory. Harold Hilton writes [9]:-

Reviewing the same book, William Benjamin Fite write [7]:-The author has succeeded in producing a book which is elementary and is quite the sort of thing a beginner wants. The exposition is very clear, and the earlier chapters contain many useful illustrative examples.

For longer extracts from these reviews, see THIS LINK.Professor Netto aims to give in this book an introduction to the theory of groups of finite order. He has succeeded admirably in his purpose. ... In publishing so excellent a treatment of the subject Professor Netto has performed a service of value to the mathematical public.

He also contributed a number of articles to *Enzyklopädie der Mathematischen Wissenschaften* Ⓣ. For example: *Combinatorik* Ⓣ; *Rationale Functionen einer Veränderlichen, ihre Nullstellen* Ⓣ; and *Rationale Functionen mehrerer Veränderlichen* Ⓣ.

Beginning in 1910, Netto suffered from Parkinson's disease, which as the disease progressed forced him to withdraw increasingly from the public. He retired from his professorship at the University of Giessen in 1913 and spent the last six years of his life in increasing difficulty as his health deteriorated.

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