In 1921 Juel received a rank of the Cross of Honour (Dannebrogordenens haederstegn) of the Order of Dannebrog.
University of Copenhagen, Copenhagen, Denmark
In January 1876 Juel took the examinations for admission to the University of Copenhagen. Completing his university studies in 1879 with the state examination, he received his doctor’s degree in 1885.
Technical University, Copenhagen, Denmark
Juel entered the Technical University in 1871.
Juel attended the Realschule in Svendborg. At the age of fifteen, he went to Copenhagen, where in 1871 he entered the Technical University. In January 1876, being more interested in pure science, he took the examinations for admission to the University of Copenhagen. Completing his university studies in 1879 with the state examination, he received his doctor’s degree in 1885.
From 1894 Juel was a lecturer at the Polytechnic Institute, where in 1907 he became a full professor. He occasionally lectured at the University of Copenhagen. From 1889 to 1915 he was editor of the Matematisk Tidsskrift.
Juel’s writings include schoolbooks, textbooks, and essays. He made substantial contributions to projective geometry for the cases of one and two complex dimensions, and to the theory of curves and surfaces. His book on projective geometry is very similar in approach to that of Staudt but is easier to understand; his treatment of autocollineations goes beyond Staudt’s. Segre arrived at similar results independently.
In 1914 Juel devised the concept of an elementary curve, 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, whose treatment of his subject was loose and incomplete, dealt mainly with fourth-order curves, developing the concept of the order of an elementary curve and setting up a correspondence principle and theory of inflection points. His third-order elementary curve is very close to a third-order algebraic curve but no longer has three points of inflection on one straight line.
Juel worked also on the theory of finite equal polyhedra, on cyclic curves, and on oval surfaces.
In 1922 Juel was made a member of the Finnish Academy of Sciences. In 1925 he was made a member of the Norwegian Academy of Sciences and an honorary member of the Danish Mathematical Association.
Physical Characteristics: Failing eyesight plagued Juel's life in later years.
Juel was married to a daughter of T. N. Thiele, professor of mathematics and astronomy.