**Christian Kramp**'s first name is often written as Chrétien. His father, Jean-Michel Kramp, was a teacher in the Gymnasium at Strasbourg. The city of Strasbourg has, over a long period, been on the border of the French and German speaking regions and, as a consequence, has been fought over on several occasions. It has been a free city, a part of France and a part of Germany but in 1760, when Kramp was born there, it had been part of France for over 60 years and had enjoyed a long period of peace and prosperity. With both French and German spoken in the city, it was natural that Kramp was brought up to speak and write in both languages. His interests were broad while he studied at the Gymnasium in Strasbourg and he entered the University in that city, again taking a broad range of courses, but specialising in medicine.

After graduating, he practised medicine in the Strasbourg region around where he lived, travelling to patients in a fairly wide area. The breadth of his interests, however, is seen since he published a work on aerostatics [1]:-

In1783, the year the Montgolfier brothers made the first balloon ascension, Kramp published in Strasbourg an account of aerostatics in which he treated the subject historically, physically, and mathematically. He wrote a supplement to this work in1786.

The works referred to in this quote are the 2-volume *Geschichte der Aerostatik, historisch, physisch and mathematisch ausgefuehrt* (1783) and *Anhang zu der Geschichte der Aerostatik* (1786). In 1788, when he was twenty-eight years old, he went to Paris where he practised as a doctor. In 1794 he moved to become a doctor and lecturer in obstetrics in Meisenheim and then in Zweibrücken. Two years later in 1796, he became a medial doctor in Speyer. He did not give up his academic interests over this period, however, and he had submitted his thesis *De vi vitalistic arteriarum addita nova de febrium indole generalized conjectura* to the University of Strasbourg in 1786. He also published a book on obstetrics and the article *Versuch, die Sterblichkeitstafeln durch einfache Gleichungen zu bestimmen* (1787). It is clear from this work on mortality tables that Kramp was intent on applying mathematical techniques in his medical research. However his interests certainly ranged outside medicine for, in addition to a number of medical publications and his books on aerostatics, he published in 1793 a work on crystallography *Krystallographie des Mineralreiches *written in collaboration with Karl Bekkerhinn.

The stability of French rule which we noted at the beginning of this biography, came to an end in 1792 when France went to war with Prussia and Austria. They suffered defeats and soon the allies invaded French territory, advancing towards Paris. However, by 1794 the position was totally reversed with the French winning victories and France began to annex lands as their armies pushed east. In 1795 France annexed the Rhineland region in which Kramp was carrying out his medical work and after this he decided to give up his work as a medical doctor and was appointed as a teacher in Cologne at the École Centrale of the Département of the Ruhr. He maintained his broad interests, however, teaching mathematics, chemistry and experimental physics. The following years saw Kramp embrace mathematics more and more seriously. This was a period of French dominance under Napoleon and Kramp responded to the new political order by publishing books in French instead of German. Before he had published in German, for example his work on fevers *Fieberlehre, nach mechanischen Grundsätzen* (Heidelberg, 1794), and his critique of practical medicine *Kritik der praktischen Arzneikunde, mit Rücksicht auf die Geschichte derselben und ihre neuern Lehrgebäude* (Leipzig, 1795).

It was around the turn of the century that he changed languages and published in French his most important astronomical work *Analyse des réfractions astronomiques et terrestres* (Strasbourg-Leipzig, 1799). The influence of the French Revolution is seen on the title page since the author is given as "citizen Kramp" and it was "printed in the 7^{th} year of the Republic". In this work he tried to solve the problem of refraction by making the simplifying assumption that the elasticity of air is proportional to its density. He followed this with the mathematical textbooks É*léments d'arithmétique* (Cologne-Paris, 1801), where again on the title page the author is given as "citizen Kramp" and the date is given as the 9^{th} year of the Republic, and É*léments de géométrie* (Cologne, 1806), where now the author is C Kramp and there is no dating by year of the Republic. His treatise É*léments d'arithmétique universelle* (Cologne, 1808) attempted to fuse ideas in the calculus of variations introduced by Louis Arbogast with basic combinatorial techniques [1]:-

He thus strove to create an intimate union of differential calculus and ordinary algebra, as had Lagrange in his last works.

Kramp was appointed professor of mathematics at Strasbourg, the town of his birth, in 1809. He published his memoir on double refraction *Sur la double refraction de la chaux carbonatée* in 1811 and, in 1820, he published *Equations des nombres* which contains a new approximate method to solve numerical equations.

Kramp was friendly with, and learnt much mathematics from, Carl Friedrich Hindenburg. This was important both in influencing Kramp to undertake some important work in combinatorics, the topic for which Hindenburg is most famed, and also for giving Kramp important outlets for his mathematical contributions in the various journals that Hindenburg published. As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial or Γ function which applied to non-integers. His work on factorials is similar to, but independent of, that of James Stirling and Alexandre Vandermonde. Kramp sent his work on factorials of non-integers to Bessel who was influenced by it. He did achieve one "first" in that he was the first to use the notation *n*! although he seems not to be remembered today for this widely used mathematical notation.

In fact Kramp did not use the term 'factorial' in his early work on this topic, which was around 1798, but rather the term 'faculty'. However, in the preface to É*léments d'arithmétique universelle* (1808) he wrote:-

I have given it the name 'faculty'. Arbogast has substituted the name 'factorial' which is clearer and more French. In adopting his idea I congratulate myself on paying homage to the memory of my friend.

Further on in the same work Kramp writes:-

I use the very simple notation n!to designate the product of numbers decreasing from n to unity, i.e.n(n- 1)(n- 2) ... 3 . 2 . 1. The constant use in combinatorial analysis, in most of my proofs, that I make of this idea, has made this notation necessary.

It is a notation that has become standard in use today, but Augustus De Morgan was highly critical of it in the *Penny Cyclopaedia* (1842). In the article on 'symbols' De Morgan wrote:-

Among the worst of barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common language. Writers have borrowed from the Germans the abbreviation n! to signify1 . 2 . 3 ... (n- 1) .n, which gives their pages the appearance of expressing surprise and admiration that2, 3, 4, etc. should be found in mathematical results.

Kramp went further than simple factorials, however, in his article *Mémoire sur les facultés numériques* published in Gergonne's *Annales de Mathématiques* in 1812. Kramp writes:-

I give the name of 'Faculties' to products whose factors are an arithmetic progression, such as

a(a+r)(a+ 2r)... [a+ (m- 1)r]

and to designate such a product, I suggested the notationa_{m/r}

The faculties form a class of very elementary functions, when their exponent is an integer, either positive or negative; but, in all other cases, these functions become absolutely transcendent.

I note that any numerical faculty whatever is always reduced to the very simple form1

_{m/1}= 1.2.3 ...m

or that other simpler form m!if one wishes to adopt the notation that I used in my 'Éléments d'arithmétique universelle'.

Kramp was elected to the geometry section of the Académie des Sciences in 1817. On the title page of *Éléments de géométrie* (1806) he is listed as belonging to the Academy of Erfurt, the Academy of Rovérédo, the Literary Society of Mainz and the Mineralogical Society of Jena.

Many letters written by Kramp survive and these show that he was a man with a great deal of confidence in his own abilities. If one wishes to be a little unkind this self-confidence could reasonably be described as boasting. However, he worked with tireless energy on a wide range of topics, making many worthwhile contributions.

**Article by:** *J J O'Connor* and *E F Robertson*

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