In 1873 Gram graduated with a Master's degree in mathematics. This degree was of a higher level than the present British/American Master's degree and more on a par with today's British/American Ph.D. Gram had published his first important mathematics paper before he had graduated. This was a work on modern algebra which appeared first in Tidsskrift for Mathematik but, in 1874, Gram published a fuller account of the same material in French. This fuller version appeared in Mathematische Annalen under the title Sur quelques théorèmes fondamentaux de l'algèbre moderne Ⓣ. This work provided a simple and natural framework for invariant theory.
In 1875 Gram was appointed as an assistant in the Hafnia Insurance Company. Around the same time he began working on a mathematical model of forest management. We give more details below of this research interest which Gram followed for many years. His career in the Hafnia Insurance Company progressed well and soon he was promoted to a more senior role in the company. However his work for the insurance company led him back into mathematical research. He began working on probability and numerical analysis, two topics whose practical applications in his day to day work in calculating insurance made their study important to him.
Gram's mathematical career was always a balance between pure mathematics and very practical applications of the subject. We have already mentioned the very practical applications to forestry which he continued to study at this time, and his work on probability and numerical analysis involved both the theory and its application to very practical situations. He published a paper on these topics On series expansions determined by the methods of least squares and for this work he was awarded the degree of Doctor of Science in 1879. This degree, equivalent to today's British D.Sc., is at a higher level than a doctorate. Gram later published this work in the Journal für Mathematik and it proved to be of fundamental importance in the development of the theory of integral equations.
The year 1879 was important for Gram in another context, for on 30 September of that year, he married Dorthe Marie Sørensen who was the daughter of a blacksmith. Also in 1879 Gram published the second of his four papers on forestry which all appeared in Danish Forestry Journals. The first had appeared in 1876, a year after he began this research, and it presented a mathematical model for maximising the profit in managing a forest. The paper was written in Danish and failed to be noticed outside his own country. In fact Gram never gained the international recognition for this work which he deserved but German researchers, unaware of Gram's contributions, published their own results on the same problems. There is little doubt that their work was far less satisfactory than that of Gram's yet because it was readily accessible it was the Germans and not Gram who gained international acclaim. Gram extended his work on forestry in later papers. After the 1879 paper he published two further papers in 1883 and 1889. These papers included developments of his forestry model which he was able to make in the light of experiments on trees which he carried out over a number of years. He work in this area was later widely used.
Gram's work on probability and numerical analysis led him in a natural way to study abstract problems in number theory. In 1884 he won the Gold Medal of the Royal Danish Academy of Sciences for his paper Investigations of the number of primes less than a given number which he published in the journal of the Society. Gram had corresponded with Meissel on this topic and in 1885 Meissel travelled to Denmark and met with Gram. It was 1885 that Meissel published his work on the number of primes less than 109 so the two had much to discuss on the topic. Gram also worked on the Riemann zeta function. Zeuthen writes in :-
His brilliance and scientific training together with his practical skills made his contributions to pure and applied mathematics very significant.Although he continued to work for the Hafnia Insurance Company in more and more senior roles, Gram founded his own insurance company, the Skjold Insurance Company, in 1884. He was the director of this company from its founding until 1910. From 1895 until 1910 Gram was also an executive of the Hafnia Insurance Company. On 9 April 1895 Gram's wife Dorthe died. Gram married again just over one year later on 15 May 1896, his second wife being Emma Birgitte Hansen. From 1910 until his death in 1916 Gram was Chairman of the Danish Insurance Council. It was through his work for the insurance companies that Gram became close to another Danish mathematician, Thorvald Thiele, who worked as an actuary.
Despite not teaching mathematics in a university and as a consequence never having any students, Gram still managed to influence the next generation of Danish mathematicians in a very positive way. He often lectured in the Danish Mathematical Society, he was an editor of Tidsskrift for Mathematik from 1883 to 1889, and he also reviewed papers written in Danish for the Jahrbuchs über die Fortschritte der Mathematik Ⓣ.
Gram received honours for his mathematical contributions despite being essentially an amateur mathematician. The Danish Academy had awarded him their Gold Medal in 1884 before he became a member, but in 1888 he was honoured with election to the Society. He frequently attended meetings of the Society and published in the Society's journals. For many years he was the Treasurer of the Danish Academy.
Gram is best remembered for the Gram-Schmidt orthogonalisation process which constructs an orthogonal set of vectors from an independent one. He was not however the first to use this method. The process seems to be a result of Laplace and it was essentially used by Cauchy in 1836.
Gram met his death in a rather strange and very sad way. He was on his way to a meeting of the Danish Academy when he as struck and killed by a bicycle. He was sixty-five years old when he met his death in this tragic accident.
Article by: J J O'Connor and E F Robertson