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Aksel Frederik Andersen (1891-1972) was born on 10 February 1891, the son of a farmer on Sealand (near Næstved

While still a student at the University, Andersen began his career as a teacher of mathematics at a Copenhagen gymnasium in 1912 and kept that position as a supplement to his other engagements until 1934. After his graduation from the University, he became an assistant at the Polyteknisk Læreanstalt (Polytechnical College) in Copenhagen in 1916 and was promoted to associate professor (lektor) in 1923. From 1926 to 1930, he lectured at the Kongelig Veterinær og Landbohøjskole (Royal Veterinary and Agricultural College), before returning to the Polytechnical College as professor in 1930. Until his retirement in 1960, Andersen lectured on mathematics for the students of engineering and was responsible for the preparatory course on mathematics for future applicants to the Polytechnical College.

Andersen was active in the Danish Mathematisk Forening (Mathematical Society) giving a number of lectures, some of which were published. He also served as the secretary of the society for a few years after 1917 (see RAMSKOV 1995, pp. 228-229 and DMF 1973). Furthermore, Andersen partook in the organisation of the Scandinavian congresses of mathematicians, and in 1932 he attended the International Congress of Mathematicians (ICM) in Zurich.

Andersen's research interests on Cesàro summability were in line with those of the mathematical group that was being established in the 1910s and 1920s in Copenhagen around Harald Bohr (1887-1951) and N. E. Nørlund (1885-1981). Infinite series, Cesàro summability, and, more broadly, function theory remained Andersen's main areas of mathematical research throughout his career, culminating in an article published in the proceedings of the London Mathematical Society (ANDERSEN 1958).

Andersen had a life-long interest in the secondary and tertiary mathematics education and became an experienced teacher and author of influential textbooks. Through these occupations, his interest in the theory of functions also came to exert a substantial influence on the teaching of mathematics in Denmark. When just 21 years old and still a student at the University, Andersen began reviewing textbooks for the Danish journal of mathematics Nyt Tidsskrift for Matematik (New Journal of Mathematics), which in 1918 was renamed Matematisk Tidsskrift (Journal of Mathematics). Focused on rigorous analysis, Andersen was concerned with presenting mathematics with the utmost precision:

"With his calm disposition, Andersen was an eminent teacher who was able to provide his students with a firm foundation for mathematics while distancing himself from extreme axiomatic tendencies" (BANG 1979).In a review of a new edition of one of the leading textbooks of mathematics for the gymnasium in 1935, Andersen elaborated on his views concerning axiomatics and the teaching in the gymnasium:

"When one notices how the axiomatic approach functions in this book, one is led directly to reflect upon whether it is at all appropriate to introduce axiomatics into the teaching in the gymnasium. According to my view, its inclusion cannot be dismissed right away. [. . . ] Of course, one can settle for a looser description without complete logical coherence [. . . ] that has been done previously and in other places [. . . ] and be satisfied with what one accomplishes. But I certainly understand if one wants something more. And I consider it both possible and useful to give a logically satisfactory foundation for the calculus in the gymnasium" (ANDERSEN 1935, p. 64).However, the axiomatics had to be presented in a way that connected it to the students' previously acquired understanding. Andersen expressed himself similarly on the role of geometrical intuition:

"I am well aware that in recent years a tendency has existed towards pure analytic geometry based on a completely arithmetic foundation and without appeal to concepts from intuition - but I doubt the appropriateness of such an approach. I think one is better served by, as is customary, connecting analytical geometry to the intuitive contents of the geometry that the students are already familiar with, even if something of the logical coherence is thereby lost" (ANDERSEN 1935, p. 65).In 1934, the year before writing the previous review, Andersen was dissatisfied with the teaching of the foundations of functions in the Danish gymnasium and wrote an article for the Journal of Mathematics addressed to the secondary schools (ANDERSEN 1934). A colleague, Johannes Mollerup (1872-1937), reviewed it for the Jahrbuch über die Fortschritte der Mathematik (JFM):

Da der Mathematikunterricht der dänischen Gymnasien sich stets sorgfältiger mit den feineren Teilen der Analysis beschäftigt, hat der Verf., der als Professor der Technischen Hochschule die Schüler der Gymnasien weiterführt, sich vorgenommen, die Sätze von der Exponentialfunction (e^{x}, a^{x}), von der Logarithmusfunktion (lx, logx ) und von der Potenzfunktion x^{n}so einfach wie nur möglich zu begründen" (MOLLERUP, JFM 60.0861.05).

(Because the instruction in mathematics in the Danish high schools is treating the finer parts of analysis with increasing care, the author - who as professor at the Polytechnical College is to bring the students of the gymnasium further - has taken it upon himself to present as simply as possible theorems concerning exponential function (e^{x}, a^{x}), the logarithmic function (lx, logx ), and the power function x^{n}.)

From 1916, Mollerup, together with Bohr, taught mathematical analysis at the Polytechnical College based on a new textbook that they wrote together during the years 1915-1918. This new textbook in four volumes was a considerable improvement over previous systems and continued to be used in subsequent editions at the universities in Copenhagen and Aarhus until 1959, and even as late as 1973 at the Technical University (formerly the Polytechnical College; see JØRSBOE 2000, p. 32). The first major reworking of the textbook was undertaken after Mollerup's death by the new professors and teachers at Polytechnical College, of which Andersen was one. Together with Bohr and Richard Petersen (1894-1968), Andersen revised the system into Lærebog i matematisk Analyse (ANDERSEN, BOHR and PETERSEN 1945-1949), bringing it up to date with changes in the curriculum while insisting on the precision and rigour that had marked those textbooks out from the beginning (see also RAMSKOV 1995, pp. 176-194 for a discussion of the textbook system and the changes it underwent at the hands of Andersen and Petersen). In an obituary of Bohr, Andersen commented upon the great impact of the textbook system throughout the Danish educational system in mathematics:

"For the Danish instruction in mathematics, the system "Bohr and Mollerup" was a milestone. Although the modern views were not unknown in this country, it was through this book that they managed to break through. The book did not only have a thorough impact on the teaching at the two institutions for which it was written [the Polytechnical College and the University of Copenhagen] but also sparked a renewal of the instruction in the gymnasium where the old traditions were fading away" (RAMSKOV 1995, p. 193).In the editing of the textbook, Andersen's insistence on precision and rigour seems to sometimes have been rather pedantic and almost counterproductive. In a letter from Bohr to Børge Jessen (1907-1993), Bohr describes the work of the editors (Petersen and ANDERSEN) as follows:

"I feel that Petersen - who has contributed 99% of the work +6% help from me -5% anti-help from Andersen - deserves some acknowledgement after his long struggle with this book" (Bohr to Jessen, 28 April 1949, quoted in RAMSKOV 1995, p. 189; see also JØRSBOE 2000, p. 35).Andersen's interest in and concern for the teaching of mathematics in the gymnasium led him to write a textbook for secondary education together with Poul Mogensen (born 1895). It appeared in its first edition in 1937-1940 and was revised as Lærebog i Matematik for Gymnasiets matematisk-naturvidenskabelige Linie (ANDERSEN and MOGENSEN 1942). As was to be expected of Andersen, the infinitesimal analysis was presented with some emphasis on rigour. In particular, the treatment of the elementary functions (powers, logarithms, and exponentials) closely followed his approach of 1934 mentioned above. This required the treatment of infinite series, which was not customary in the gymnasium, but according to a review, its clarity and comprehensible nature attracted the interest of the students (see DUERLUND 1942, pp. 73-74). The textbook was widely used until a reform of the curriculum in the gymnasium around 1960. Thus, by teaching in the gymnasium, teaching the introductory courses at the Polytechnical College and writing or editing influential textbooks for both the gymnasium and the Polytechnical College, Andersen was highly influential in forming the instruction in mathematics in Denmark in the period 1930-1960. Andersen died on 18 February 1972 at the age of 81.

A.F. ANDERSEN 1921, Studier over Cesàro's Summabilitetsmetode med særlig Anvendelse overfor Potensrækkernes Teori, København, Jul. Gjellerups Forlag

A.F. ANDERSEN 1934, Eksponential- og Logaritmefunktioner, Matematisk Tidsskrift, A, 38-64

A.F. ANDERSEN 1935, Review of Pihl, Kristensen and Rubinstein: Lærebog i Matematik for det matematisk-naturvidenskabelige Gymnasium I, Gyldendahl 1935, 220 pages, Matematisk Tidsskrift, A, 59-66

A.F. ANDERSEN 1958, On the extensions within the theory of Cesàro summability of a classical convergence theorem of Dedekind, Proceedings of the London Mathematical Society, series III, 8, 1-52

A.F. ANDERSEN, H. BOHR, R. PETERSEN 1945-1949, Lærebog i matematisk Analyse (4 vols.), København, Jul. Gjellerups Forlag

A.F. ANDERSEN, P. MOGENSEN 1942, Lærebog i Matematik for Gymnasiets matematisk-naturvidenskabelige Linie (4 vols.), København, Gyldendalske Boghandel - Nordisk Forlag, First edition 1937-1940

T. BANG 1979, Aksel Frederik Andersen (1891-1972), Dansk Biografisk Leksikon, 3rd Edn. (16 vols.), Gyldendahl, København, vol. 1, 152

DMF 1973, Dansk Matematisk Forening 1923-1973, København, Dansk Matematisk Forening

DUERLUND 1942, Review of A. F. Andersen and Poul Mogensen, Lærebog i Matematik for Gymnasiets matematisk-naturvidenskabelige Linie I-IV, København, Gyldendahl 1937-40, Matematisk Tidsskrift, A, 71-75

O.G. JØRSBOE 2000, Undervisningen i Matematik på DTU 1829-2000, Lyngby, Institut for Matematik

K. RAMSKOV 1995, Matematikeren Harald Bohr, Licentiatafhandling, Institut for de eksakte videnskabers historie Aarhus Universitet

Author

Henrik Kragh Sørensen

Department of Science Studies

University of Aarhus - Denmark

ivhhks@ivs.au.dk