1981 Niels Erik Nørlund, 26 October 1885 - 4 July T. Bang 1983 , 481-493, published 1 November 29 1983 Biogr. Mems Fell. R. Soc. Email alerting service here corner of the article or click this article - sign up in the box at the top right-hand Receive free email alerts when new articles cite http://rsbm.royalsocietypublishing.org/subscriptions , go to: Biogr. Mems Fell. R. Soc. To subscribe to on August 21, 2018 http://rsbm.royalsocietypublishing.org/ Downloaded from on August 21, 2018 http://rsbm.royalsocietypublishing.org/ Downloaded from
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1981Niels Erik Nørlund, 26 October 1885 - 4 July
T. Bang
1983, 481-493, published 1 November291983 Biogr. Mems Fell. R. Soc.
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N ie l s E r ik N o r l u n d was born on 26 O ctober 1885, the son of A lfred N orlund , the licentiate of the pharm acy in Slagelse, a tow n in w estern Sealand. In th is ra ther w ell-to-do hom e he grew up w ith his younger sister M argrethe and younger b ro th e r Poul. T hey form ed a trio tha t later attained a p rom inen t position in the scientific and scholarly life of D enm ark; M argrethe m arried the physicist N iels Bohr in 1912 and becam e a b rillian t hostess for scientists from all over the w orld in B ohr’s honorary residence at C arlsberg, while Poul, later d irector of the N ationalm useum in C openhagen, becam e well know n for his excavations of N orse ru ins in G reen land and of the V iking fortress at T relleborg .
A fter attending the elem entary school in his native tow n, N orlund w ent to the old renow ned public school Soro A kadem i in the ne ighbouring tow n of Soro. H is favourite subjects were astronom y and m athem atics, and w hen in 1903 he had finished school and left for Copenhagen U niversity it was obvious th a t these should be his field of study. H ow ever, he had difficulty in deciding w hich of the subjects he preferred, and he therefore followed them bo th sim ultaneously.
At that tim e the two m athem atical professors at the university were Z euthen and Julius Petersen. Z euthen was the famous geom eter, b u t at that tim e his in terests had changed over to the history of m athem atics in classical tim es and N orlund felt tha t to be ra ther rem ote. Julius Petersen w rote on algebra and graph theory and w hat he taught in the theory of functions was felt to be old-fashioned by N orlund . A th ird m athem atician was the docent N iels N ielsen and from him N orlund learned the theory of functions in the direction initiated by W eierstrass, and N orlund w ould soon surpass him in cogency. But N o rlu n d ’s preferred teacher was T . N . T hiele who held the chair of astronom y. As a result of an eye disease he was unable to do any practical astronom ical observation by this tim e, b u t he was a m aster in treating the num erical values derived from observations and his lectures on system atic and accidental errors were
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a ttended not only by studen ts in astronom y, b u t also by m any w ho w anted positions in insurance com panies (at th a t tim e a chair in actuarial m athem atics did no t exist at C openhagen). Soon N o rlu n d took part in the w ork at the observatory and even a lthough at the sam e tim e he pu rsued deep m athem atical studies, his first pub lished paper was in astronom y. T h is was w hen he was 20 years old, and for a few years he com bined the study of the stars w ith his favourite, m athem atics.
N o r l u n d a s a n a s t r o n o m e r
H e investigated the well know n double star in U rsa M ajor, and th ro u g h fine m easurem ents found a sm all deviation from the K eplerian m ovem ent of the com ponents th a t allowed h im to deduce th a t in reality it is a trip le star w ith a th ird dark com ponent, a resu lt th a t has later been confirm ed by o ther astronom ers. T h is he pub lished in Astronomische Nachrichten in 1905, and as a resu lt he ob tained an appo in tm en t as assistant at the observatory , a position he held un til 1912. In 1908 he w rote a prize paper tha t answ ered questions about the personal errors occurring in astronom ical observations and w on the un iversity prize— a gold m edal— w ith an extrem ely laudatory aw ard (he already had ano ther gold m edal as described in the next section). E m ploym ent at the observatory was no t a sinecure; he determ ined about 7000 star positions, and also pub lished a catalogue of the p roper m otions of 140 stars calculated from old and new observations. H e left the observatory in 1912 and for a num ber of years thereafter was active solely as a m athem atician , b u t in his later career w here he tu rn ed to geodetics he was able to m ake use of his know ledge of star observations.
N 0R L U N D AS A MATHEMATICIAN
In 1907 he w rote a m athem atical prize essay at the university and tha t was also rew arded w ith a gold m edal. H is m otto was the Jacobi quotation ‘Le b u t unique de la science, c ’est l’honneu r de l’esprit h u m ain ’. T h e topic was representations of functions in form of continued fractions and in particu lar ‘reciprocal differences’, a concept in troduced by T hiele . It contained results th a t were found deserving of publication in the F rench Comptes Rendus de V Academie des Sciences, and in F rance he becam e recognized as one of the best experts in the theory of Fuchsian functions, such as created by H enri Poincare in his youth . Shortly afterw ards, while he was still a s tuden t, he was invited to w rite com m ents to a proposed edition of the collected w orks of Poincare. T h e first volum e (called n u m ber II), contain ing notes by N orlund , did not appear un til 1916 (after Poincare’s death) and the later ones were also d ilatory (the last volum e appeared in 1956); in the m eantim e N o rlund had o ther occupations, b u t th roughou t his whole life he preserved a deep veneration for the w ork of Poincare.
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In the sum m er of 1910 he was aw arded his m aste r’s degree in astronom y, and a few m onths later his d o c to r’s degree in m athem atics (a D anish thesis is an unassisted in d ependen t w ork of research). T h e next day he could celebrate the successful defence of the thesis sim ultaneously w ith his 25th b irthday , and he th u s lived to see his 70 year d o c to r’s jubilee .
T h e thesis is the beg inn ing of the penetra ting study of difference equations th a t he accom plished in the following 15 years. T h e prob lem in difference equations is to find general m ethods for de term in ing a function w hen the size of its increase on in tervals of a given length is know n. In its d irect na tu re th is seem s m ore e lem entary than differential equations w here an analogous p rob lem is considered com bined w ith a passage to the lim it in w hich the length of the in tervals ten d to zero, b u t really it is m uch m ore difficult ow ing to the fact th a t the solutions are no t uniquely determ ined , and the real p rob lem is to choose w hat should be called the ‘best so lu tions’. T h e m any earlier studies often arose from in terpo lation p rob lem s from astronom y or physics, and in his thesis N o rlund m ethod ically system atized and generalized the investigations and showed how it can be advantageous to p erm it the length of the intervals to be considered as variables in the com plex plane. T h is dem ands heavy use of differential equations and of in tegrations along ingeniously determ ined curves in the com plex dom ain. I f the values of a function are given for say the integers, then it is a central p rob lem to determ ine w hat should be called its ‘tru e ’ values for the in terjacen t real num bers or even for the su rround ing com plex num bers, and N o rlu n d ob tained in perspicacious ways m ethods to obtain such ‘m ain so lu tions’. Besides the m ore practical applications, natu ral to h im as an astronom er, he was fascinated by the fact tha t in this way it is possible to create functions exceeding w hat can be obtained by m ore usual operations such as differential equations, som ething tha t from form er tim es only was know n from m ore fragm entary exam ples such as the gam m a function ob tained from the sequence 1, T2 = 2, T2-3 = 6, —
In the thesis N o rlu n d used the new m ethods from the theory of analytic functions developed in the preceding years, and m ost of w hat at th a t tim e had been w ritten on difference equations and in terpolation was felt obsolete by h im as he th ough t tha t it d id not reach the core of the problem s. In his review soon after of the textbook of W allenberg & G uldberg on difference equations he expressed it in this way:
‘T h e authors do not possess any general m ethod for solving the equation F(x + 1) —F(x) = f(x), where f(x) is a given function, and therefore the book m ust be characterized as a very audacious experim ent. Perhaps m any will th ink tha t it saw the light too early. It was probably not the in ten tion of the authors to w rite a work of lasting influence, b u t ra ther a w ork w hich by its im perfection could encourage fu rther investigations. A nd, if it is understood in this way,
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th en I for m y p a rt will give it a w elcom e. T h e book trea ts a theory th a t is on the po in t of being b o rn .’
T h e w ork of N o rlu n d from the follow ing 15 years was devoted to the creation of th is theory .
T h e year 1910 was eventful for m athem atics in D enm ark . Early in the year H arald B ohr had defended his b rillian t thesis; N o rlu n d la ter to ld how he rem em bered th is occasion as a fanfare hera ld ing a new era for the D an ish study of m athem atics. N ow he gained his degree w ith sim ilar h o n o u r (they had helped each o th er w ith the read ing of p roof-sheets and shortly after becam e rela ted by the m arriage of N o rlu n d ’s sister M arg re the to N iels B ohr, the b ro th e r of H arald ). B ut it was also a year w ith changes in the occupants of the chairs of m athem atics at the university . T h e previous year Ju liu s P etersen had re tired and his position was filled by N iels N ielsen. N ow Z eu then cam e to his re tiring age and as successor the un iversity took the topologist Poul H eegaard; H arald B ohr got the docen t-position th a t N iels N ielsen had form erly held. In astronom y T h ie le had re tired 3 years earlier and Elis S trom gren had been appo in ted to the chair. H ence it was difficult for N o rlu n d to look forw ard to a p roper, su itable position , and for 2 years he stayed as assistant at the observatory; as m en tioned above he d id no t consider th is as a sinecure.
A t the un iversity in L u n d in Sw eden (not too far away from C o p en hagen) there had un til th en been only one m athem atician , T o rs ten B roden, b u t in 1912 a new chair was estab lished and N o rlu n d was appoin ted .
T h e sam e year he m arried A gnete Waever. I t was the beg inn ing of a long and happy life together and after her death in 1959 he was lonely, the m ore so as th e ir ch ild ren , two daughters, had m arried and settled in N orw ay.
In his 10 years in L u n d he w orked exclusively as a m athem atician , w riting a long series of papers developing the theory of difference equations. H e stud ied the factorial series and in terpo la tion series en tering in th e ir solutions, de term in ing the ir region of convergence and by analytic p ro longation and different sum m ation m ethods he extended them in the com plex plane, de term in ing the ir singularities and the ir behav iour at z — oo, also by use of th e ir relations to con tinued fractions and asym ptotic series. In particu la r the little paper ‘S u r une application des fonctions p e rm u tab les’ from 1919 should be m entioned: there he states som e universal results on the sum m ability of series based on a specific— b u t ra ther general— choice of the w eights given to the elem ents; the m ethod includes the b e tte r of the know n sum m ability m ethods, such as C esaro ’s m ethod, and is now know n un d er the s tandard designation of N o rlu n d -su m m atio n . I t is in th is connection th a t his nam e is m ost often m et w ith in the p resen t-day m athem atical lite ra tu re .
In 1916 N o rlu n d becam e a m em ber of the editorial staff of A cta
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m athem atica. T h is h ig h -ran k in g jo u rn a l was fo u n d ed in 1882 by M ittag - Leffler, w ho he ld a p ro m in en t p o sition in S w eden and had m any co n nec tions w ith th e in te rn a tio n a l m athem atica l w orld . N o rlu n d becam e h is close co llabo ra to r, and w hen M ittag -L effle r, sho rtly before h is death in 1927, re tired as m ain ed ito r he tran sfe rred th is po st to N o rlu n d , w ho re ta in ed it u n til h is death , b u t o f course in la te r years w ith the assistance o f y o u n g er p ro fic ien t S w edish m athem atic ians.
A t th e U n iv e rs ity o f C o p enhagen it soon becam e ev iden t th a t its tw o chairs in m ath em atics w ere insufficient to fulfil m o d ern dem ands (the th ird positio n , the d o c en tu r o f H a ra ld B ohr, h ad been tran sfe rred to the T ech n ica l U n iv ers ity ). As it also becam e know n th a t M ittag -L effler in ten d ed to leave his large fo rtu n e and his inc red ib ly rich m athem atica l lib ra ry for the fo u n d a tio n o f a m ath em atica l in s titu te — now the renow ned M ittag -L effle r In s ti tu t in D ju rsh o lm near S tockho lm — it was realized th a t if N o rlu n d was to be saved for D an ish science it was necessary to act quickly; in 1919 a new chair in m ath em atics was estab lished in C o p en hagen and N o rlu n d decla red h im se lf w illing to take it over. H ow ever, because o f th e difficult h o u sin g s itua tion , it was n o t activated u n til S ep tem b er 1922.
N ow one m ig h t have expected a q u ie t co n tin u a tio n in D enm ark of N o r lu n d ’s m athem atica l w ork, b u t even ts in 1923, to be m en tioned below , changed his life and for ab o u t 30 years he v irtua lly gave up his m athem atica l research . P erh ap s it was a c o n trib u to ry facto r th a t it can be said th a t h is m any investigations in the p reced ing years in a certa in way had exhausted the sub jec t o f d ifference equations in the com plex dom ain . H is capab ility and energy needed to be used for m ore adm in istra tive and o rgan izational tasks, b u t in a place th a t only som eone w ith his know ledge and position could fill a righ t. A lso, in 1916, he was elected a m em b er of the Royal D an ish A cadem y and his dom icile in C openhagen m ean t an increased invo lvem ent in its w ork, to be m en tio n ed later.
W h at he w ro te on m athem atics in the m id-1920s is ra th e r to be considered as a codification o f his earlier w ork. H e p u b lished tw o volum es in F rance and above all, he w ro te the im pressive Vorlesungen iiber D ifferenzenrechnung from 1924 th a t gave a com prehensive exposition o f the w hole theory ; w hen s tu d en ts from o th e r faculties w ere th ink ing th a t m athem atics was a sub jec t n o t d em and ing m uch reading , th en it was possib le to convert th em by show ing th em the b ib liog raphy of about 1500 item s in th is book. H e also b ro u g h t ou t a m ore p o p u la r w ork, Conversations in science, w ith ed itions in b o th D en m ark and Sw eden, before he d iscon tinued his m athem atical research for m any years.
H e held the chair of m athem atics in C openhagen un til he re tired in 1955, b u t because o f his o th e r assignm ent his lectu ring was reduced . H e lectu red m ainly on advanced topics from the theory of functions of a com plex variable, w hat he h im self characterized w ith the w ords ‘h igher analysis’, and it was often on difference equations and adjoining
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subjects. His lectures were given in a rather formal way and they were followed by only a few students, who nevertheless acquired an advanced supplem ent to the general education they acquired from the more attractive m athem atical professors, H arald Bohr and Hjelmslev.
W hen he retired in 1955 N orlund gave a lecture on the three sciences that had filled his life: astronom y, m athem atics and geodesy; he said that the creative art of m athem atics was nearest to his heart. In his retirem ent he returned to the subjects of his youth, and in articles in Acta mathematica and other periodicals he published fu rther valuable investigations of Bernoulli polynomials (in a general sense) and hypergeom etric functions.
N o r l u n d as D ir e c t o r o f t h e G e o d e t ic I n s t it u t e
T he superior triangulation survey of D enm ark was perform ed by the institution Den danske Gradm aaling, created in 1816 by the astronom er Schum acher. It was a small institution, as a m atter of fact consisting only of a director and a few appointed m ilitary assistants. But it was on a high scientific level: since its foundation the directors had been m em bers of the Danish Academy or M inisters, and at the same time highly skilled technicians who held this post together with their main occupations. In 1918 came a rather young prom ising officer and dr phil., Buchwaldt, but he died in 1923. T he institution was sem i-m ilitary and had its premises in the same building as the G eneral Staff, the m onum ental Proviantgaarden next to Christiansborg Castle.
After the Peace of Versailles the scientific unions were created and among them was the U nion Geodesique et Geophysique Internationale; it was now appropriate for D enm ark to be associated with this union. N orlund had already in 1919 taken part in the negotiations about international scientific cooperation as a delegate from the Danish Academy, and he was a m em ber of the astronomical union and took part in the attem pts to create a mathem atical union. It was therefore fortunate that in 1923 it was possible to persuade him to take the post as director of Gradm aalingen, and in this way affiliate to the union.
From now on N orlund used most of his energy for the furtherance of geodesy. He organized guidance for students and after a few years the university had its first graduates with the official title of m aster of geodesy.
He took up new topics—the first was seismology; some may say that it belongs to geophysics, but at that time the university had no chair in this subject either. W ith financial support from the Carlsberg Foundation he established seismographic stations in D enm ark and G reenland in 1925. T he station near Copenhagen was erected in a very favourable place in a heavy concrete casemate of the recently dism antled fortifications,
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and the seism ograph was solem nly started by the w om an M in ister of Education , N ina Bang.
N ext door to G radm aalingen and in the same build ing existed the far larger topographical division of the G eneral Staff tha t since 1842 had taken care of the ordnance survey of D enm ark (earlier this had been done by the Academ y). T h e two divisions were closely related, as G radm aalin gen provided some fundam ental data for use in the ordnance m aps, and now N o rlund though t to have them united . W ith the great w eight of his w ords— he was at tha t tim e also p residen t of the Academ y, as m entioned later— it was possible to obtain a m ajority in the D anish Parliam ent, and in 1928 the Geodaetisk In s titu t was founded as a civil in stitu tion under the M in istry of Defence, w ith N orlund as d irector. Civil servants succeeded the ‘gu ides’ of the G eneral Staff and N o rlund becam e the leader of a large technical establishm ent. T hus, fortu itously , an in stitu tion of continuously increasing im portance had been created, b u t it gave him a large burden of labour.
H ere he could wholly use his capacity for organizing work, exacting a com bination on a h igh level of m athem atics, astronom y, physics and m echanics. Besides the necessary updating of the ordnance m aps of D enm ark, he p u t into operation new m ore precise triangulations and m easurem ent of the m atching new baselines and new astronom ical determ inations. In addition to the coordinates for the ordinary m aps he in troduced a new system suitable for land registration. G ravity anomalies and seismic m easurem ents were taken for both geodetic and geophysical applications (even if at th a t tim e they did not have the significance that they later were to obtain). A new survey of the Faroe Islands was undertaken , and in Iceland he com pleted the survey begun by the G eneral Staff. G reenland becam e an im portan t new sphere of activity; here he used aerial photogram m etric survey as a substantial aid to terrestria l m easurem ent, so troublesom e in m ountain regions. T h is initiative was to be a valuable con tribu tion to the advantageous issue for the D anes of the lawsuit at the W orld C ourt in T h e Hague in 1933, after the N orw egians had proclaim ed occupation of part of Eastern G reenland.
An interesting experim ent was the hydrostatic levelling across the G reat Belt in 1938, when by m eans of a 20 kilom etre pipe filled w ith w ater and laid across the bottom of the Belt it becam e possible to com pare the indications of height in Funen and Sealand w ith an accuracy of less than a m illim etre. As a by-product one could observe the tidal m ovem ent of the firm (but weakly elastic) crust of the Earth.
D uring the G erm an occupation from 1940 to 1945 the institute had to reduce its activity, bu t instead N orlund published a series of atlases, containing a detailed history of the m apping of D enm ark, the Faroe Islands and Iceland. T hey are am ong the m ost stately books published in D enm ark, prin ted w ith the aid of all the techniques of the institute, and now extrem ely highly priced in second-hand bookshops.
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N o rlund m anaged the Geodaetisk In s titu t in such a way tha t it fulfilled the dem ands from society, and the dem ands were increasing. T h e prem ises were able to be enlarged w hen the tactical division of the G eneral Staff left Proviantgaarden and the institu te had the entire build ing at its disposal; nevertheless it was necessary to have annexes elsew here in C openhagen, besides the seismic station and an astronom ical observatory at B uddinge.
H ow ever, it m ust be em phasized tha t at the same tim e N o rlu n d created an institu tion that in its sp irit and its w ork is a scientific research institu te . Even if it does not belong to the university there is close collaboration in the education of scientists, and as m asters of geodesy usually obtain em ploym ent in the institu te it m eans continued research activity for m ost of them .
N orlund was m ore an adm in istra to r than a practical geodesist, and w hat he published in this long period are m ainly ra ther general reports on the activity of the institu te . Perhaps, w hen he reached the age lim it in 1955 and retired , he felt tha t he had achieved enough for geodesy and at last he looked forw ard to have m ore tim e for his beloved m athem atics.
In the university N orlund som etim es lectured on m athem atical topics for geodetical use, and about 1950 the university supported him by establishing a readership in geodesy. W hen he retired , and sim ultaneously w ith the appoin tm ent of a successor in the chair of m athem atics, a professorship in geodesy was created, and the new d irector of the institu te , E inar A ndersen, was appointed.
N 0R L U N D AS AN ACADEMIC ADMINISTRATOR
N orlund was elected a m em ber of the Royal D anish Academ y in 1916 and therefore obtained seniority as a m em ber for 65 years, a record in the h istory of the Academ y. H e soon becam e engaged in its w ork and, because of his early international contacts, becam e an active m em ber of the com m ittee for international scientific cooperation, which had m any tasks during the creation of the scientific unions in the years im m ediately after W orld W ar I.
He becam e a m em ber of its finance com m ittee in 1924 and in 1927 he was elected President. He was only 42 years old, the youngest president in the h istory of the Academy. It is necessary to outline the situation: the previous presidents had invariably been re-elected and retained the position for life, b u t now a debate had been opened on this principle, and also the question was posed as to w hether the task should alternate betw een the two classes of the Academy, for sciences and hum anities respectively, and N orlund had declared him self sym pathetic to this. As President he proposed that re-election should not be possible; the proposal was defeated b u t he took the consequences and resigned w hen his period term inated in 1933. M uch later certain lim itations on the
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p o ssib ility o f re -e lec tio n w ere in tro d u ced . N o rlu n d becam e a m em b er of th e finance co m m ittee once m ore , and he co n tin u ed as one u n til 1959.
H e w as a m em b er o f m any co m m ittees and as a delegate p a rtic ip a ted in very m any in te rn a tio n a l m eetings, p a rtly in academ ic m eetings (p a rticu la rly in th e co u n trie s w here h is a tta ch m en t was s tro n g est, F ran ce and Sw eden) and p a rtly in th e u n io n s , th e geodetic, the astronom ical and th e m a th em atica l (a fte r th is cam e in to ex istence in 1950); he was also p re s id en t o f I .C .S .U . (In te rn a tio n a l C ouncil o f Scientific U n io n s) from 1934 to 1937.
As d ire c to r o f th e G eodaetisk In s ti tu t he o f course took p a rt in m any m eetings, p a rticu la rly in th e B altic geodetic C om m ission , o f w hich he w as p re s id en t for tw o periods.
C o llab o ra tio n b e tw een D en m ark and o th e r co u n tries was fu r th e red by th e th e n ex isting R a s k -0 rs te d F o u n d a tio n , D e n m a rk ’s in te rn a tio n a l fu n d fo r science and le tte rs , and here he was ch a irm an from 1928 to 1964.
In the U n iv e rs ity o f C o p en h ag en he was R ec to r from 1933 to 1934 and in th is capacity he was able, in th e p resence o f the P rim e M in iste r, to in au g u ra te the m ath em atica l in s titu te endow ed by the C arlsberg F o u n d a tio n . T h e re a fte r an d u n til he re tired he was a m em b er o f the execu tive co m m ittee o f the g overn ing body of the un iversity .
P e r s o n a l i t y
N o b o d y w ho m et N o rlu n d cou ld d o u b t th a t he was a g rea t personality , a conclusion in keep ing also w ith h is tall a ris tocra tic s ta tu re . H e was a m an of few w ords, n o t im m ed ia te ly ob lig ing to s tran g ers, and even his c loser co llabo rato rs o ften felt h is ta c itu rn ity em barrassing . T h is coolness cou ld be felt as an aloofness; m aybe it was du e to som e so rt of shyness, b u t on th e o th e r h an d h is w ords th u s gained m ore im p o rtan ce and one cou ld feel how he exerted h im se lf to find the rig h t so lu tion to p rob lem s. S tu d e n ts o r em ployees w ho cam e closer to h im could rejoice at h is w a rm h earted in te rest, b u t as an ad m in is tra to r he could be severe. M aybe th a t was necessary in h is position ; la ter the post as d irec to r o f the G eodaetisk In s titu t was filled by a ju ris t.
As a lec tu rer he was form al and n o t very insp iring ; the p resen t scientific leader o f the In s titu te tells th a t the m ost p leasan t lesson he rem em b ers was w hen N o rlu n d had issued a trea tise on the old D an ish u n its of leng th and th en used the double h o u r to tell the s tu d en t (he was the only one p resen t) abou t th is top ic , en tire ly neg lecting the m a th e m atics o f d ifference equations as was scheduled . N o rlu n d was n o t devoid o f hu m o u r: w hen a young geodesist once rep o rted on an unsuccessful voyage to E astern G reen lan d and the ice cond itions th a t had m ade a land ing for the trian g u la tio n s im possib le, N o rlu n d th en asked: ‘was it really qu ite im possib le to land? S uppose you had taken a row ing boa t . . . ?’, and th en he m ade row ing gestu res w ith his arm s; o f course
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both of them knew that there w ould have been no point in doing this and that it would have been too dangerous.
N orlund appreciated tasteful surroundings and had a beautiful hom e enriched by m any books in fine bindings; a few years ago he disposed of his im pressive m athem atical library and now it is the principal part of the m athem atical library in the new U niversity of Odense.
N orlund achieved m uch as an astronom er, even m ore later as a m athem atician, bu t perhaps m ost of all as a geodetic director. I t seems a fitting token that his last resting-place is at the G arnison cem etery in Copenhagen, near the tom bs of his two predecessors as d irector of G radm aalingen, F .I . Buchw aldt and V. H. O. M adsen.
D is t in c t io n s
Grand Cross of the Order of the Dannebrog, , t r praeter pluraMember of
Kungliga Fysiografiska Sallskapet, Lund (1913)Det kongelige danske Videnskabernes Selskab (1916)Societe des Sciences, Strasbourg (1920)Accademia Pontaniana, Napoli (1925)Kungliga Vetenskapsakademien, Stockholm (1925)Societas scientiarum Fennica, Helsinki (1926)Academie des Sciences, Paris (1926)Accademia Nazionale dei Lincei, Roma (1927)Deutsche Akademie der Naturforscher, Halle (1927)Royal Astronomical Society, London (1935)Bureau des Longitudes, Paris (1937)Royal Society, London (1938)Akademiet for de tekniske Videnskaber, Kobenhavn (1939)Det norske Videnskaps-Akademi, Oslo (1946)Vetenskapsakademien, Helsinki (1946)Det kungliga vetenskapliga Sallskapet, Uppsala (1951)Societas scientiarum Islandica, Reykjavik (1959)The New York Academy of Sciences (1960)
Honorary member ofSydsvenska Geografiska Sallskapet (1947)Societe mathematique de France (1952)Ferdafelag Islands (1953)Dansk matematisk forening (1955)Royal Institution, London (1960)
Honorary Doctor atTechnische Hochschule, Darmstadt (1936)University of London (1937)Lunds Universitet (1941)L’universite de Dijon (1950)Universitetet, Oslo (1951)Veterinaer- og Landbohojskolen, Kobenhavn (1958)
Grand Prix des Sciences de l’Academie, Paris (1916)Kungliga Fysiografiska Sallskapets Guldmedaille (1916)Ole Romer-medaillen (1954)Vitus Bering-medaillen (1958)
T he photograph reproduced was taken by Julie Laurberg, Copenhagen, in 1917.
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1905 Determination de l’orbite de \ ursae majoris = 1523. Astr. Nachr. 170, 117-132.1908 Sur les differences reciproques. C. r. hebd. Seanc. Acad. Sci., Paris 147, 521-524.
Sur la convergence des fractions continues. C. r. hebd. Seanc. Acad. Sci., Paris 147, 585-587.1909 Note om opstigende Kaedebrok. N yt Tidsskr. Mat. B , pp. 25-29.
Untersuchung der personlichen Gleichung und der Helligkeitsgleichung bei Beobachtungen mit dem Repsoldschen selbstregistrierenden Mikrometer. Astr. Nachr. 183, 17-44.
Sur les equations aux differences finies. C. r. hebd. Seanc. Acad. Sc:., Paris 149, 841-843. Bestimmung der Rektaszensionen von 184 Sternen mit dem Passageinstrument der Kopen
Sur les fractions continues d’interpolation. Overs. K. danske Vidensk. Selsk.forh., pp. 57-68. Bidrag til de linecere differentialligningers Theori. Thesis. (71 pages.) Kobenhavn.Halleys Komet. Fys. Tidsskr. 9, 13-22.Astrofysik, Grundrids ved folkelig Universitetsundervisning 178, Kobenhavn.T. N. Thiele (Obituary). Fys. Tidsskr. 9, 1-7.
1911 Ober lineare Differenzengleichungen. K. danske Vidensk. Selsk. Skr. 6, 307-326. Beobachtungen am Meridiankreis der Kopenhagener Universitats-Sternwarte. Astr. Nachr.
189, 17-26.Beobachtungen am Meridiankreis der Kopenhagener Universitats-Sternwarte. Astr. Nachr.
189, 277-288.1912 Untersuchungen iiber die Eigenbewegungen fur 140 Sterne des IV Secchischens Typens. K.
danske Vidensk. Selsk. skr. 6, 329-371.Sur les equations aux differences lineaires a coefficients constants. Nyt Tidsskr. Mat. B , pp.
53-65.Sur les equations lineaires aux differences finies. C. r. hebd. Seanc. Acad. Sci., Paris 155,
1485-1487.1913 Sur les equations lineaires aux differences finies. C. r. hebd. Seanc. Acad. Sci., Paris 156,
51-54.Sur le probleme de Riemann dans la theorie des equations aux differences finies. C. r. hebd.
Seanc. Acad. Sci., Paris 156, 200-203.Sur Integration des equations lineaires aux differences finies par des series de facultes. Rc.
Circ. mat. Palermo 35, 177-216,Sur une classe d’integrales definies. J. Math, pures appl. 9, 77-88.Sur une classe de fonctions hypergeometriques. Overs. K. danske Vidensk. Selsk. Forh., pp,
135-153.Om Fakultetraekker og deres Anvendelser. Den tredie skandinaviske Matematikerkongres i
Kristiania 1913, pp. 77-82.1914 Sur les series de facultes. Acta math., Stockh. 37, 327-387.
Wallenberg & Guldberg: Theorie der linearen Differenzengleichungen (review). Bull. Sci.Math. 38, 23-24. (Also in Nyt Tidsskr. Mat. B , pp. 84-87.
Sur l’existence de solutions d’une equation lineaire aux differences finies. Annls scient. Ec. norm, sup., Paris 31, 205-221.
Sur les series de facultes. C. r. hebd. Seanc. Acad. Sci., Paris 158, 1252-1253.Sur les series de facultes et les methodes de sommation de Cesaro et de M. E. Borel. C. r. hebd.
Seanc. Acad. Sci., Paris 158, 1325-1327.1915 Sur les equations lineaires aux differences finies a coefficients rationels. Acta math., Stockh.
40, 191-249.1916 Notes. CEuvres de Henri Poincare, vol. 2, pp. 619-629. Paris: Gauthier-Villars.1918 P. J. Myrberg: Zur Theorie der Konvergenz der Poincareschen Reihen (review). Bull. Sci.
math. 42, 9-11.Tycho Brahe: Opera omnia I—111 (review). Bull. Sci. math. 42, 201-209.S. Stadler: Sur les systemes d’equations aux differences finies lineaires et homogenes (review).
Bull. Sci. math. 42, 223-224.Sur le calcul aux differences finies. Acta Univ. lund. (2) 14, no. 15.H. Bohr, Mollerup & Andersen: Nyere Undersogelser over Integralregning (review). Nyt
Tidsskr. Mat. A, pp. 64-68.Der Mond. In Vortrage fiir Kriegsgefangene herausgegeben von Damschen Roten Kreun, ser. A,
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1919 De BernoullFske Polynomier. Mat. Tidsskr. B , pp. 33-41.De Euler’ske Polynomier. Mat. Tidsskr. B , pp. 49-55.Sur les polynomes d’Euler. C. r. hebd. Seanc. Acad. Sci., Paris 169, 166—168.Sur les polynomes d’Euler. C. r. hebd. Seanc. Acad. Sci., Paris 169, 221-223.Sur les equations aux differences finies. C. r. hebd. Seanc. Acad. Sci., Paris 169, 372—375. Sur la solution principale d’une certaine equation aux differences finies. C. r. hebd. Seanc.
Acad. Sci., Paris 169, 462-465.Sur les polynomes de Bernoulli. C. r. hebd. Seanc. Acad. Sci., Paris 169, 521—524.Sur une extension des polynomes de Bernoulli. C. r. hebd. Seanc. Acad. Sci., Paris 169,
608-610.Sur le calcul aux differences finies. C. r. hebd. Seanc. Acad. Sci., Paris 169, 770—773.Sur le calcul aux differences finies. C. r. hebd. Seanc. Acad. Sci., Paris 169, 894—896.Sur une application des fonctions permutables. Acta Univ. lund (2) 16, no. 8.
1920 Memoire sur les polynomes de Bernoulli. Acta math., Stockh. 43, 121—196.Sur la convergence de certaines series. C. r. hebd. Seanc. Acad. Sci., Paris 170, 506—509. Sur un theoreme de Cauchy. C. r. hebd. Seanc. Acad. Sci., Paris 170, 715—718.Sur l’etat actuel de la theorie des equations aux differences finies. Bull. Sci. math. 4, 174-192;
200- 220.Sur les equations aux differences finies. In Comptes rendus du Congres international des
mathematiciens, Strasbourg, pp. 98—119.Sur les equations aux differences finies. Ens. math. 21, 202—203.
1922 Sur la formule d’interpolation de Stirling. C. r. hebd. Seanc. Acad. Sci., Paris 174, 919—921. Sur la formule d’interpolation de Newton. C. r. hebd. Seanc. Acad. Sci., Paris 174,
1108-1110.Sur les formules d’interpolation de Stirling et de Newton. Annls scient. Ec. norm, sup., Paris
39, 343-403.Nogle Bemaerkninger angaaende Interpolation med aekvidistante Argumenter. K. danske
Vidensk. Selsk. mat. fys. Medd. 4, 1-34.Memoire sur le calcul aux differences finies. Acta math., Stockh. 44, 71—212.
1923 Sur les formules d’interpolation de Stirling et de Newton. Annls scient. Ec. norm, sup., Paris40, 35-54.
Neuere Untersuchungen fiber Differenzengleichungen. Enzykl. math. Wiss. C7, 675—721. Summen af en given Funktion. Mat. Tidsskr. B , pp. 65—68.Sur certaines equations aux differences finies. Trans. Am. math. Soc. 25, 13—98.On certain difference equations. Bull. Am. math. Soc. 29, 212.Remarques diverses sur le calcul aux differences finies. J. Math, pures appl. 2, 193-214. Extrait d’un memoire inedit d’Henri Poincare sur les fonctions fuchsiennes. Acta math.,
Stockh. 39, 58-93.Videnskabelige causerier. (125 pages.) Kobenhavn.Bland siffror och tal [Swedish translation of the preceding]. (113 pages.) Stockholm.Sunto dei suoi lavori sul calcolo delle differenze finite. Boll. Un. mat. ital. 2, 182—186.
1924 Sunto dei suoi lavori sul calcolo delle differenze finite. Boll. Un. mat. ital. 3, 28-31.Vorlesungen iiber Differenzenrechnung. (551 pages.) Berlin: Springer-Verlag.Sur Finterpolation. Bull. Soc. math. Fr. 52, 114—132.Stirlings Interpolationsraekke. K. danske Vidensk. Selsk. Skr. 7, no. 2, 212-280.
1925 Indledningstale ved Aabningsmodet. In Beretning om VP skandinaviske Matematikerkongres iKobenhavn 1925, pp. 15-25.
Om Interpolationsraekker. Beretning om VP skandinaviske Matematikerkongres i Kobenhavn 1925, p. 251.
Stirlings Interpolationsraekke. Mat. Tidsskr B , pp. 20—24.1926 Lemons sur les series d’interpolation. (236 pages.) Paris: Gauthier-Villars.
J. L. W. V. Jensen (Obituary). Overs. K. danske Vidensk. Selsk. Fork., pp. 43—51.J. L. W. V. Jensen (Obituary). Mat. Tidsskr. B, pp. 1—7.
1927 Sur la ‘somme’ d’une fonction. Meml. Sci. math. 24, 1-54.G. Mittag-Leffler (Obituary). Acta math., Stockh. 50, I—XXIII.J. L. E. Dreyer (Obituary). Overs. K. danske Vidensk. Selsk. Fork., pp. 51-59.
1929 Lefons sur les equations lineaires aux differences finies. (152 pages.) Paris: Gauthier-Villars.1930 Rapport sur les travaux geodesiques executes de 1924 a 1930. In Meddr Danm. geod. Inst.,
Copenhague 1930.
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1933 Rapport sur les travaux geodesiques executes de 1930 a 1933. In Meddr Damn. geod. Inst.,Copenhague 1933.
1934 Observations de l’intensite de la pesanteur avec le nouveau modele de penduleHolweck-Lejay. Memoires de VInst. Geod. de Danemark 1, 1-163.
1935 Untersuchungen uber die Genauigkeit relativer Schweresmessungen mit dem Holweck-Lejay Pendel. In Verhandl. Tagung Baltischen geod. Kommission, Helsinki.
Uber die Wahl von Sternen bei Zeit- und Langenbestimmungen. In Verhandl. Tagung Baltischen geod. Kommission, Helsinki.
Some methods and procedure developed during recent expeditionary surveys in Greenland. GeogrlJ. 86, 317-329.
1937 Kongen og den videnskabelige Forskning. Christian den Tiende Konge af Danmark og Island,Aarene 1912-37, Kobenhavn, pp. 196-213.
Om Laengdebestemmelser. Mat. Tidsskr. B, pp. 72-74.Presidential Address. In Third general Assembly of International Council of Scientific Unions,
Cambridge.The Beacon lamp of the Danish Geodetic Institute. Meddr Danm. geod. Inst. 8, 1-12.
1938 Rapport general du Danemark. In 5iemecongres int. de photogrammetrie a Rome. Copenhague.1939 J. P. Koch (Obituary). Geogr. Tidsskr. 42, 3-6.
Vermessungsarbeiten in Gronland, Island und Danemark. Darmstadt 1939.Kortlaegningen af Gronland og Island. Geogr. Tidsskr. 42, 21-30.The survey work of the Danish Geodetic Institute in Greenland and Iceland. Polar Rec. no.
17, 38-45.Rapport sur les travaux geodesiques executes de 1933 a 1939. In Meddr Danm. geod. Inst.,
Copenhague 1939.1940 Ausgleichung nach der Methode der kleinsten Quadrate bei gruppenweiser Anordnung der
Beobachtungen. Acta math., Stockh. 72, 283-353.1942 Johannes Mejers Kort over det danske Rige / - / / / . (294 plates,) Kobenhavn: Munksgaard.
Danmarks Kortlcegning I. (77 pages, 105 plates.) Kobenhavn: Munksgaard.1943 Bestemmelse af Vcegtene paa de Ubekendte ved Elementudjoevning. Kobenhavn: Munksgaard.1944 De gamle danske Laengdeenheder. Memoir es de VInst. Geod. de Danemark 3, 1-80.
1945 Hydrostatisk Nivellement over Store Baelt. Memoires de VInst. Geod. de Danemark 6, 1-122.1946 Hydrostatisk Nivellement over Oresund. Memoires de VInst. Geod. de Danemark 7, 1-84.1950 Hypergeometriske Funktioner. M at. Tidsskr. B , pp. 18-21.1951 Harald Bohr (Obituary). Overs K . danske Vidensk. Selsk. Fork., pp. 61-67.1952 Series hypergeometriques. K . fysiogr. Sallsk. Lund Fork. 21, no. 15.
F. Losch & F. Schoblik: die Fakultat (Gammafunktion) und verwandte Funktionen (review). Mat. Tidsskr. B , p. 105.
1953 Opmaalingen af Danmarks Veje. Meddr Danm. geod. Inst. 26, 1—14 (also in Geografisk Tidsskr.52, 232-241).
Theorie des fonctions—sur les fonctions hypergeometriques. C. r. hebd. Seanc. Acad. Sci., Paris 237, 1371-1373; 1466-1468.
1954 Report on the geodetic works executed in the period 1939-1953. In Meddr Danm. Geod. Inst.,Copenhagen 1954.
Uber hypergeometrische Funktionen. Arch. Math. 5, 258—266.1955 Hypergeometric functions. Acta math., Stockh. 94, 289-349.1956 Tanker om tre Videnskaber. Nord. Mat. Tidsskr, pp. 1-19.
Sur les fonctions hypergeometriques d’ordre superieur. Mat. fys. Skr. K . danske Vidensk. Selsk. 1, no. 2, pp. 1-47.
1960 Sur les valeurs asymptotiques des nombres et des polynomes de Bernoulli. C. r. hebd. Seanc.Acad. Sci., Paris 251, 2269-2271.
Sur la convergence de certaines series de facultes. A tti Accad. Lincei 28, 532—538.1961 Sur les valeurs asymptotiques des nombres et des polynomes de Bernoulli. Rc. Circ. mat.
Palermo, 10, 1-18.1963 The logarithmic solutions of the hypergeometric equation. Mat. fys. Skr. K . danske Vidensk
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