## Ludwig Sylow's 1913 Address

Your Majesty!

Ladies and Gentlemen!

Abel again! That is what you may have exclaimed on seeing the subject of the Congress's first meeting -- has the subject of Abel not been exhausted yet here in Norway! Yes, we have published his works twice, we have celebrated the 100th anniversary of his birth, and his name was not forgotten at the University's 100th anniversary either. That is the situation.

But for Norwegian mathematicians, Abel is still of greater importance than he is for others. He was the first Norwegian to make a contribution of lasting significance to scientific development. No doubt there was an even earlier brilliant Norwegian-born mathematician, but Caspar Wessel's astonishing work has been lying unheeded for almost a hundred years, despite having been printed in the publications of the Royal Danish Academy of Sciences and Letters. It was only in 1894 that Prof Thiele drew attention to the fundamental significance of this, but by then the rediscovery had taken place long ago. Things were different with Abel. His fame is that he has been an example and a stimulus for all Norwegian mathematicians. I might be tempted to say that everything of any significance that has been generated in pure mathematics, here in this country, stems directly or indirectly from Abel. The subjects he studied have had a captivating attraction, and his exhaustive treatment of them and the emphasis he placed on rigour also acted as an example. In addition, the fundamental nature of his discoveries has fired the ambition of young mathematicians. Personally, I am able clearly to trace his influence on all of those, who before those here present have occupied a professorship at our university -- even his old teacher Holmboe. His admittedly rather heavy textbooks in elementary mathematics strove unmistakably for precision, which has certainly had an influence on the older generations. In this I see, at least partially, an influence of the pupil on the teacher. The definition that Holmboe gave in his lectures for the differential ratio is remarkable to me. He says: When a function's growth is developed in power series according to the increment in the absolute variable, that term in the expansion, which contains the first power, is called the differential of the function. This restriction to analytical functions is very reminiscent of Abel's critique - I sense conversations between Holmboe and Abel behind this attentiveness. Also on Holmboe's successors, Broch and Bjerknes, even on the geometer Lie, as different as the latter was from Abel with regard to mathematical of interests and working method, Abel's influence is unmistakable. -- I beg you to forgive this digression.

This is the background: new details have been added to Abel's mathematical history since the second edition of Abel's Collected Works. This meeting's outstanding Swedish member, Prof Mittag-Leffler, whose admiration for Abel and great interest in his memory are known to all mathematicians, has deservedly called attention to two unknown manuscripts for Abel from collections abroad. The first of these is a whole treatise, which Abel sent to his friend Crelle, the publisher of the 'Journal für die reine und angewandte Mathematik'; the second is a letter, also to Crelle. Prof Mittag-Leffler has had these manuscripts printed in the special volumes of 'Acta Mathematica' [Volume 26 (1902), volume 27 (1903) and volume 28 (1904) of 'Acta Mathematica' were all dedicated to the memory of Abel, and contained articles by the most famous mathematicians of the day (among them Poincaré and Hilbert), taking Abel's mathematical discoveries as inspiration and starting point] on the occasion of the Abel Centennial Anniversary, with the letter as a photographic facsimile. He also did me the honour of forwarding in advance a copy to me. However, due to lack of time, it was not possible for me to include them in my presentation of Abel's scientific progress in our University's Festschrift in 1902. Since, as far as I know, Prof Mittag-Leffler has not published anything further about them other than his concise indication of their content, with which he accompanied them in the 'Acta', I felt I ought to make them the subject of further examination.

JOC/EFR July 2014

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