**Julius Petersen**'s father was Jens Petersen, who worked as a dyer, and his mother was Anna Cathrina Petersen who was born in Wiuff. From his early childhood Julius was friends with Hieronymous Georg Zeuthen who lived four houses away on the same street. Julius first attended a private school in Soro then, in 1849, he entered Soro Academy School. Most of the pupils at the school were boarders but the school also admitted some day pupils like Julius who lived in the town. His interest in mathematics began when he was at this high school. He was passionate about problem solving and spent a great deal of time trying to solve the classical problem of trisecting an angle with ruler and compass. He later wrote [9]:-

Mathematics had, from the time I started to learn it, taken my complete interest, and most of my work consisted in solving problems of my own and my friends, and in seeking the trisection of the angle; a problem that has had a great influence on my whole development.

His family were poor and although they had tried their best to give Julius a good education, by 1854 he had to leave school since his parents could no longer afford to keep him there. Petersen had an uncle who worked in Kolding, Jutland, and he went there to work as a grocery apprentice to his uncle. After about a year his uncle died and left some money to Petersen. He returned to Soro, passed the school leaving examinations, and entered the College of Technology in Copenhagen in 1856. His first publication was two years later when he published an elementary text on logarithms. In 1860 he passed his civil engineering examinations and, in the same year, submitted an essay on the history and properties of the cycloid for a University prize. When the judges found that he was not a university student they ruled out his entry.

Although he had decided that he wanted to study mathematics at university long before graduating from the College of Technology, he had used up his inheritance by this time so had no way of funding himself through university. He therefore taught in a prestigious private school from 1859 to 1871 and also took other part-time teaching jobs to support himself.

In 1862 he passed the entrance examinations for the University of Copenhagen and began his mathematical studies there. In the same year he married Kirstine Bertelsen (who was two year older than he was); they had two sons and a daughter. Although he was now a student he still taught six to seven hours a day, six days a week, to support himself, his wife, and children. He was considered a good teacher of mathematics, but school teaching did present a great problem to Petersen for he seemed incapable of controlling the pupils. In 1866 he graduated from the university with the degree of Master of Mathematics but remained there to study for his doctorate. He was awarded the Gold Medal for his treatise on the equilibrium of floating bodies in 1867 but although the work was judged to be excellent, nevertheless he did fail to know of one of the most important works on the topic, namely that by Pierre Dupin in 1814.

Petersen was awarded his doctorate in 1871. The interest he had shown in ruler and compass constructions when he was at school had continued to influence his research topic and his doctoral thesis was entitled *On equations which can be solved by square roots, with application to the solution of problems by ruler and compass.* Petersen was unsure about a specific result which he used in his thesis, namely:

If the equation of degree 2

^{n}can be solved by square roots, one of the roots can be expressed bynsuch different square roots, where each can appear several times.

Petersen corresponded with Sylow during 1870-71, mostly about this result. Details of the correspondence appear in [8].

After the award of his doctorate in 1871 he was appointed as a dozent at the College of Technology in Copenhagen. In 1877 he was appointed as professor of mathematics at the University of Copenhagen and continued to hold this post throughout his career. His friend and colleague at the University of Copenhagen was Zeuthen who was almost exactly the same age as Petersen. The two had been friends since childhood as we noted near the beginning of this article. Petersen also taught at the Officers School of the Army from 1881 to 1887.

He wrote a series of school and undergraduate texts which achieved international acclaim despite being too difficult for all but the ablest pupils. For example he published *Plane and spherical trigonometry* in 1863. First published three years later, his *Methods and theories for the solution of problems of geometrical construction* appeared in various editions and languages. This work aimed at making geometric problem solving systematic but the geometrical constructions he considered were always those of ruler and compass.

His research was on a wide variety of topics from algebra and number theory to geometry, analysis, differential equations and mechanics. He published *The theory of algebraic equations* in 1877 which was written in a concise style, treating as many topics as possible without using Galois theory. He wrote a series of textbooks based on courses he had given at the College of Technology: one on plane geometry in 1877; one on statics in 1881; one on kinematics in 1884; and one on dynamics in 1887.

He also wrote on mathematical physics, mathematical economics and a pamphlet on cryptography. His work on economics began in 1871 with the publication of a pamphlet investigating how goods could be redistributed to favour workers. Another work studied aspects of social policy. Petersen was enthusiastic about such matters and joined a committee to support Georg Brandes but this was short lived. Brandes was a Danish critic and scholar who set himself the task of liberating Denmark from its cultural isolation and provincialism with progressive ideas aimed at reforming society. Petersen joined the National Economic Society and translated an economics text by Henry Fawcett into Danish. He also developed his own ideas on economics beginning in 1872, publishing these ideas in 1874. Quite suddenly he changed from studying economics to studying cryptography. He published a pamphlet on this topic in 1875 which he funded privately. This was written in French, unlike all his previous works which were in Danish [9]:-

Petersen never returned to cryptography; this seems to be another instance of a problem that must have occupied him intensely for a period, until he found a satisfactory solution and moved on to something else.

His most important work however was in geometry with innovative ideas on regular graphs. A paper which he wrote in 1891 *Die Theorie der regulären Graphs* marks the birth of graph theory. Sabidussi [13] writes that this is:-

... the first major graph-theoretical paper, not only in size but also in significance.

Watkins writes about Petersen's paper in [16]:-

In1891Julius Petersen published a paper that contained his now famous theorem: any bridgeless cubic graph has a1-factor. These days Petersen's theorem is always proven indirectly using major results such as Hall's theorem from1935and Tutte's theorem on1-factors from1947.

Although the Petersen graph is usually the first time an undergraduate will today come across his name, this does not appear in this 1891 masterpiece. In fact this ten-vertex graph appears in a later paper published in 1898. It is reasonable to ask why Petersen began working in graph theory. In fact his aim was to discover results in invariant theory and he corresponded with Sylvester on this topic.

In [9] it is stated that:-

... the work of Petersen and Zeuthen is regarded as being responsible for the emergence of Danish mathematics on the international scene towards the end of the19th century.

In the last part of his career, from 1888 to 1909, Petersen worked on function theory, latin squares, and number theory.

Petersen contributed to many areas other than mathematics, some of which we have mentioned above. In addition to these he was a member of the "kontrolkomité" of the insurance company *Hafnia.* Petersen was also a founder member of the Danish Mathematical Society, which came into existence in 1873, and he remained a very active member of the Society. In 1887 he was appointed a member of the Commission of Education for the Learned Schools which was an education inspectorate under the Ministry of Education.

His obituary states:-

Among Danish mathematicians he was the embodiment of the best sense of humour and the most vigorous joy of life.

Perhaps his own words sum up best his attitude to his life:-

When throughout life you have obtained honour and money for enjoying yourself, what more can you ask for!

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

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