**Hieronymus Georg Zeuthen**'s father was the minister in Grimstrup where his son was born and began his education. However Zeuthen's father moved in 1849 from a church in Grimstrup to a church in Soro and at this time Zeuthen began his secondary education. After completing his schooling in Soro in 1857 he entered the University of Copenhagen to study mathematics.

At Copenhagen Zeuthen studied a broad mathematics course attending courses on topics in both pure and applied mathematics. In 1862 he graduated with a Master's degree and was awarded a scholarship to enable him to study abroad. He decided to visit Paris and there he studied geometry with Chasles. This was extremely important for Zeuthen since his research areas of mathematics were firmly shaped by Chasles during this period. The first topic on which Zeuthen undertook research was enumerative geometry.

In 1865 he submitted his doctoral dissertation on a new method to determine the characteristics of conic systems to the University of Copenhagen. Haas describes the thesis in [1]:-

In this work Zeuthen adhered closely to Chasles's theory of the characteristics of conic systems but also presented new points of view: for the elementary systems under consideration, he first ascertained the numbers for point or line conics in order to employ them to determine the characteristics.

Up until 1875 Zeuthen worked almost exclusively on enumerative geometry. He was appointed as an extraordinary professor at the University of Copenhagen in 1871 and, in the same year, he became an editor of *Matematisk Tidsskrift,* a position he held for 18 years. He was the secretary of the Royal Danish Academy of Sciences for 39 years, during which time he was also a lecturer at the Polytechnic Institute. He continued teaching at the University of Copenhagen where he was promoted to ordinary professor in 1886. He was twice Rector of the University.

After 1875 Zeuthen's contributions to mathematics became more varied. He began to write on mechanics and he also made significant contributions to algebraic geometry, particularly the theory of algebraic surfaces. As we mentioned above, he developed the enumerative calculus, proposed by Chasles, for counting the number of curves touching a given set of curves. The move towards rigour in geometry led to this theory being neglected for many years but recently some of the remarkable numerical results produced by it have been confirmed.

He was also an expert on the history of medieval mathematics and produced important studies of Greek mathematics. He wrote 40 papers and books on the history of mathematics, some of which have become classics. Unlike many historians of science Zeuthen explained the ancient texts in the manner of a colleague of the ancient mathematicians.

His historical studies covered many topics and several periods. In a major work in 1885 he looked in detail at the work of Apollonius on conic sections and showed that Apollonius used oblique coordinates. Caveing, in [3], looks at Zeuthen's ideas on the discovery of irrational numbers. Zeuthen argued that Pythagoras himself discovered that 2 was irrational when computing the diagonal of a square. The passage from Plato's *Theaetetus* where it states that Theodorus proved the irrationality of 3, 5, ... 17 was also carefully studied by Zeuthen. He suggested that the end of Theodorus's proof somehow involved the continued fractions for 17 and 19, a conjecture which is very much in line with modern ideas about Greek mathematics.

Zeuthen's largest historical work was published in 1896. It looked in detail at the work of Descartes, Viète, Barrow, Newton and Leibniz as he traced the development of algebra, analytic geometry and analysis.

In [7] Lützen and Purkert compare the historical approaches of Moritz Cantor and Zeuthen. They write:-

Moritz Cantor and Hieronymus Georg Zeuthen were probably the two most outstanding historians of mathematics at the end of the19th century. However, their methods of work differed strikingly. Moritz Cantor was an encyclopaedist who ... followed the development of mathematics in a survey of an almost innumerable collection of original and secondary sources from antiquity to the end of the18th century. ... Zeuthen's papers and books, on the other hand, present deep mathematical analyses of the methods found in classical works mostly from antiquity and from the16th and17th centuries in an attempt to capture their fundamental ideas.

Kleiman gives an interesting biography of Zeuthen in [6]. This indicates that:-

[

Zeuthen]was in many ways a leading light of the burgeoning intellectual life of his country, and with a selfless and generous disposition he seems happily to deserve Coolidge's epithet of the "ever kindly".

Finally we quote Hass's comments in [1] on Zeuthen's style. He writes:-

Zeuthen saw things intuitively: he constantly strove to attain an overall conception that would embrace the details of the subject under investigation and afford a way of seizing their significance. This approach characterised his historical research equally with his work on enumerative methods in geometry.

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

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