**Christian Juel**'s father, Andreas Peter Theodor Juel (1822-1856) was a judge but Christian never knew him for he died before Christian was one year old. Andreas Juel was born on 15 July 1822 in Copenhagen, Denmark, and married Augusta Sofie Hanne Caroline Emilie Jürgensen in 1853. Augusta was born on 16 October 1829 in Godthaab, Greenland. Christian was the eldest of his parents' two children having the younger sister Theodora Sophie Juel who was born in 1856. When Christian was five years old his mother remarried Jakob Georg Christian Wulff (1827-1908). Jakob and Augusta had five children: Joachimine Frederikke Stredtmann Wulff (born 1861); Christian August Jürgensen Wulff (born 1862); Johanne A Wulff (born 1865); Hans Christian Carl Wulff (born 1866); and Wilhelmine Amalie Wulff (born 1870). The family lived in Svendborg, a coastal town in southern Funen Island, Denmark, and it was in this town that Christian was brought up. He was educated there, attending the Realschule in Svendborg. When he was fifteen years old he went to Copenhagen and, in the following year, he entered the Technical University of Copenhagen.

He studied at the Technical University from 1871 until 1875, then he decided that technical studies were not to his liking and that he would prefer to study pure science in general and pure mathematics in particular. One reason for this was the influence of Julius Petersen who taught at the Technical University from 1871 and at the University of Copenhagen from 1877. In January 1876 Juel took the entrance examinations for the University of Copenhagen. He studied there with Adolph Steen (1816-1886), the professor of mathematics, and Hieronymus Georg Zeuthen who was a docent. We should note that this was a time when mathematics at Copenhagen was reaching a high point marked by the founding of the Danish Mathematical Society in Copenhagen in 1873. The Society's first committee consisted of Thorvald Thiele, who taught astronomy at the university, Steen, and Zeuthen. Juel completed his Master's Degree at the University of Copenhagen in 1879. Continuing with his doctoral studies at the University of Copenhagen he received a gold medal for a geometrical treatise in 1881 and was awarded his doctorate in 1885 for a dissertation on geometry entitled *Contributions to the geometry of imaginary lines and imaginary planes* (Danish). Juel had spent sixteen years completing his university education and so, despite starting young, was 30 years old before he received his doctorate.

Juel worked for several years as a mathematics teacher and private tutor of mathematics before being appointed to the Polytechnic Institute in Copenhagen in 1894. He was promoted to a full professorship there three years later. He also sometimes lectured at the University of Copenhagen. From 1889 to 1915 he was editor of the *Matematisk Tidsskrift, *taking over from Zeuthen who had been the editor from 1871 to 1889. This was the first Danish mathematical journal and it had been founded in 1859. However, at first it was a journal aimed at school teachers of mathematics but in 1890, when Juel was editor, it was split into Series A containing school level mathematics, and Series B which was a scientific journal containing articles on advanced topics. Series B allowed articles in foreign languages and contributions from foreign mathematicians began to appear in the journal beginning at this time. On 23 July 1902, Juel married Laura Thiele (born 26 May 1873 in Copenhagen, died 21 March 1925) a daughter of the professor of astronomy in Copenhagen, Thorvald N Thiele, and his wife Marie M Trolle (1841-89). Christian and Laura Juel had four children: Aase (born 28 April 1903, married Vagn Vanggaard), Sven (born 17 July 1904), Ole (born 16 May 1906, died 21 February 1914), and Inger (born 17 September 1909). In 1907 Juel was appointed to a newly created chair of rational mechanics at the Polytechnic institute.

He made substantial contributions to projective geometry and wrote an important book on the topic. Some of his papers, written in German, are *Über einige Grundgebilde der projectiven Geometrie* Ⓣ (1890), *Über die Parameterbestimmung von Punkten auf Curven zweiter und dritter Ordnung* Ⓣ. *Eine geometrische Einleitung in die Theorie der logarithmischen und elliptischen Funktionen* Ⓣ (1896), *Über einen neuen Beweis der Kleinschen Relation zwischen den Singularitäten einer ebenen algebraischen Kurve* Ⓣ (1905), *Einige Sätze über ebene, ein-und mehrteilige Elementarkurven vierter Ordnung* Ⓣ (1915), *Einleitung in die Theorie der Elementarflächen dritter Ordnung* Ⓣ (1915), and *Über die Kongruenz zweiten Grades und die Kummersche Fläche* Ⓣ (1926). Juel's programme of research extended certain theorems on real manifolds of algebraic geometry to non-algebraic manifolds, even to non-analytic manifolds provided the concepts are generalised in a suitable way. His approach is similar to that of von Staudt but goes beyond von Staudt's in places. Although he was not alone in making these improvements, since Corrado Segre also proved similar results, but there is no doubt that Juel's results were obtained independently of Corrado Segre. The book we referred to at the beginning of this paragraph was written in German and had the title *Vorlesungen über projektive Geometrie mit besonderer Berücksichtigung der v. Staudtschen Imaginärtheorie* Ⓣ (1934). Patrick Du Val reviewed this text and we give the first part of his review [3]:-

Juel also worked on the theory of finite equal polyhedra and on oval surfaces. In 1899 he published a paper on what he called "graphical curves" and thereby inaugurated the study of a new family of geometric objects. These objects, known as "géométrie finie" in French and "geometrische Ordnungen" in German, are surveyed in [4] (see also [6]) and more details are given in the book by H Künneth and O Haupt,Professor Juel has in this book provided a worthy successor to the classical treatise of Reye, which supplies just those deficiencies which in Reye's ex-position were most remarkable. Many readers must have felt that if all that projective geometry could tell us of a problem involving a cubic equation was that it has at least one solution, and not more than three, then projective geometry had not by any means justified its claims to replace the ordinary algebraic kind. This reproach Professor Juel has set himself(with no less industry than ingenuity)to remove. The method, of course, is implicit in the previous works - it is von Staudt's theory of imaginary elements.(

About the strict axiomatic foundations of the real projective geometry with which he begins, he is content to be brief, and to refer the reader to the German version of Enriques' 'Lezioni sulla geometria projettiva'. In essence, however, he allows himself purely projective axioms and an axiom of continuity, which, as is familiar, will suffice to build up what is from the algebraic point of view the coordinate geometry in the field of the real numbers and infinity.

The extension of this to the complex number field is not essentially different from that of von Staudt; the elementary formsrow of points, plane pencil of lines, pencil of planes, conic locus, conic envelope, and quadratic regulus)are defined; also the linear congruence, as an aggregate of lines related in a certain way to an involution in a regulus, the relation being such that if the involution is hyperbolic the congruence consists just of all lines which meet both the double lines. Imaginary points, lines of the first kind, and planes are then defined as elliptic involutions in elementary forms combined with orientations, and lines of the second kind as elliptic linear congruences combined with orientations; the device used by some writers to avoid the somewhat artificial introduction of orientations, of using a cubic instead of a quadratic involution, is disregarded.

So far all is familiar, and the reader is inclined to wonder if there are not enough books about this sort of thing. But now mapping an imaginary line of the second kind on the congruence of real lines which defines it, the author proceeds to prove a mass of theorems about chains, points conjugate with respect to a chain, projectivities and "symmetralities"(i.e. anti-projectivities)in the form, of the kind which one could also prove by introducing a complex coordinate and mapping it on a sphere. He now returns to the classical matter and introduces coordinates by the algebra of "Würfe", explaining the meaning in algebraic terms of what he has done geometrically. But his objective clearly is to justify all the algebraically obvious results by purely geometrical reasoning; and though in the remainder of the work he continually uses algebraic terms and ideas, I think it is fair to say that all these could be omitted(at the cost of brevity and intelligibility)and the arguments would still stand.

The rest of the book follows the general lines of some portions of Reye - by no means the whole, since hardly any solid geometry is attempted - with due regard to the refinements introduced by the use of imaginary elements. The main topics covered are problems leading to cubic and quartic equations in one variable, reduced to finding the intersections of two conics, of which in the first case one intersection is known; the two-dimensional chain, and its relations; anti-collineations; the projective metric, Euclidean and non-Euclidean; the quadratic transformation, the rational plane cubic, and the general plane cubic.

*Geometrische Ordnungen*Ⓣ (1967). In 1914 Juel introduced the concept of an elementary curve [1]:-

Although these ideas are clever, Juel did not treat them with care and his writings on this topic, although interesting, are less than precise and leave something to be desired.... which is in the projective plane without straight-line segments and has the topological image of a circle and a tangent at every point. Outside these points a convex arc can be described on each side. Thus an elementary curve consists of an infinite number of convex arcs passing smoothly one into another.

Juel served mathematics in other important ways. We have mentioned above that he was an editor of *Matematisk Tidsskrift* for over 25 years. He also wrote textbooks for school level mathematics as well as for university level mathematics courses. These include: *Vorlesungen über Mathematik für Chemiker* Ⓣ (1890); *Elementar stereometri* Ⓣ (1896); *Analytisk stereometri* Ⓣ (1897); *Ren og anvendt aritmetik* Ⓣ (1902); and *Forlaesinger over rational mekanik* Ⓣ (1913; 2^{nd} enlarged edition 1920). The textbook on rational mechanics provided the foundation for those entering the teaching profession who studied at the Polytechnic institute.

Among the honours which he received was being elected to the Danish Academy of Sciences and Letters in 1899, being elected to honorary membership of the Mathematical Association in 1925, then receiving an honorary degree from the University of Oslo four years later. Juel attended the International Congress of Mathematicians held in Bologna in September 1928. He delivered the invited lecture 'Beispiele von Elementarkurven und Elementarflächen' which was published as a 20-page paper in the *Proceedings* of the Congress.

In his final years Juel's eyesight deteriorated and he was only able to write his classic text *Vorlesungen über projektive Geometrie mit besonderer Berücksichtigung der v. Staudtschen Imaginärtheorie* Ⓣ (1934) with the help of his student David Fog.

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

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