Carl Ferdinand Degen

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Carl Ferdinand Degen (1 November 1766 – 8 April 1825) was a

Danish-Norwegian
school system.

He was born in

Kant's philosophy[2] and was elected to the Royal Danish Academy of Sciences and Letters in 1800.[1]

In 1802 Degen got his first academic position as head teacher in mathematics and physics at the Odense cathedral school. After a few years there he was appointed rector at the corresponding school in Viborg. There he remained until 1814 when he became professor in mathematics at the University of Copenhagen. Although his lectures were not so well organized, he was loved by his students and he infused the courses with new and more modern mathematics. At the same time he pursued his own research and published results in many different directions. All this made him the most esteemed mathematician in Scandinavia at that time.[2]

When

elliptic functions which Degen had encouraged. He is buried on the Assistens Kirkegård at Nørrebro
in Copenhagen.

Mathematical contributions

Degen worked in many branches of what was then modern mathematics. Most of his contributions had to do with problems within number theory, but he also wrote papers on geometry and mechanics.[1]

The Pell equation

In 1817 Degen got printed his large work on the fundamental solutions (x, y) of

Euler had earlier shown that these could be systematically calculated with the use of continued fractions. Degen used this method and presented integer solutions for all n < 1000.[3] The same calculations also gave approximate, but very accurate rational results for the square root of n. In addition, he also found solutions of the adjoint equation with −1 on the right hand side for the n-values when they existed. These tables of numerical results became in the following years a standard reference for the Pell equation.[4]

The eight-square identity

While his work on the Pell equation can be considered a continuation of previous contributions made by

Lagrange and Legendre to this problem, Degen's discovery of the eight-square identity
was his most important and original discovery. Most probably it resulted from his attempts to generalize the Pell equation.

The two-square identity

had been known from the times of

St. Petersburg where Euler had worked, his eight-square identity of exactly the same structure as the two previous identities.[5]
The following year he was elected as a «corresponding member» to the same academic society.

His work on the eight-square identity was first published in 1822.[6] Almost thirty years later his identity was rediscovered by John T. Graves and Arthur Cayley as obeyed by the norm of octonions. These were an extension of Hamilton's quaternions. In 1898 Adolf Hurwitz proved that such identities involving 2k squares can exist only for k = 0, 1, 2 and 3.

The encounter with Abel

In 1821

quintic equation. None of his teachers or professors at the University of Oslo could find anything wrong with his work. The astronomy professor Christopher Hansteen recommended then that the paper ought to be published by the Science Academy in Copenhagen. It thus came in the hands of Degen to be evaluated.[2]
He again could not pinpoint any mistakes, but asked that this new method should first be tried out on a practical example. In a letter to Hansteen he proposed the equation x5 − 2x4 + 3x2 − 4x + 5 = 0. He ended the letter with the wish that

.... the time and efforts that Mr. Abel in my eyes spends on this rather sterile subject ought to be invested in a problem whose development will have the greatest consequences for Mathematical Analysis and its applications to practical investigations. I refer to elliptic transcendentals. A serious investigator with suitable qualifications for research of this kind would by no means be restricted to the many strange and beautiful properties of these most remarkable functions, but could discover a Strait of Magellan leading into the wide expanses of a vast Analytic Ocean.

This would soon turn out to be a very prophetic piece of advice. Abel himself soon discovered a mistake in his investigations of the quintic equation, but continued to work on the existence of solutions. Two years later he could prove that they in general have no

algebraic solutions
.

Degen's recommendation to concentrate instead on the elliptic integral had most probably made some impression on the young student. In the summer of 1823 Abel was on a short visit to Copenhagen where he met Degen. In a letter to his friend and former teacher Bernt Michael Holmboe in Oslo he wrote that he had constructed elliptic functions by inverting the corresponding integrals. The following year in a letter to Degen he could report that these new functions had two periods.[7] Although this discovery marks the beginning of a new and very important branch of modern mathematics, Abel waited with the publication of his results. That happened first in 1827. Degen had in the meantime died and was therefore unaware of the beautiful discoveries Abel had made and which he had prophesied.

References

  1. ^
    Projekt Runeberg
    , digitalized 2. edition (1916).
  2. ^
    ISBN 3-540-66834-9
    .
  3. ^ C.F. Degen, Canon Pellianus Sive Tabula simplicissimam Aequationis Celebratissimae, Bonnier, København (1817). Electronic version from Göttinger Digitalisierungszentrum.
  4. ^ D.H. Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, Washington D.C. (1941).
  5. ^ A. Rice and E. Brown, Commutativity and collinearity: A historical case study of the interconnection of mathematical ideas. Part I Archived 2016-10-20 at the Wayback Machine, Journal of the British Society for the History of Mathematics 31 (1), 1–14 (2016).
  6. ^ C.F. Degen, Adumbratio Demonstrationis Theorematis Arithmetici Maxime Universalis, Mémoires de l’Académie Impériale des Sciences de St. Pétersbourg, pour les années 1817 et 1818, 8, 207–219 (1822).
  7. ISBN 978-0821846445
    .
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