**Émile Lemoine**'s father had been a military man, serving in Napoleon Bonaparte's army from 1807. The College at La Flèche had been closed in 1793 due to the French Revolution but in 1808 Napoleon transferred the Prytanée de Saint-Cyr to the College buildings in La Flèche, founding the military school there. Émile Lemoine's father was involved in the founding of the military Prytanée of La Flèche at this time and, because of this, Émile was awarded a scholarship to enable him to be educated there. While at this school, Lemoine published *Note sur une conique et son cercle directeur* in the *Nouvelles annales de mathématiques* in 1858. In this paper, Lemoine describes himself as a pupil in the Spéciales of the Prytanee impérial of La Flèche. Along with Charles Kessler, a fellow pupil at the Prytanée Militaire, he published a solution to a geometric question posed in the *Nouvelles annales de mathématiques* in 1859. Their paper appeared in this journal in 1860. Graduating from the Prytanée Militaire in 1860, Lemoine entered the École Polytechnique in Paris.

Lemoine was very musical and while at the École Polytechnique he founded an amateur musical group named "La Trompette" which, several years later, was good enough to have Saint-Saëns write music especially for it. Saint-Saëns composed *Septet in E flat major* Op 65, at Lemoine's request. The piece was written for trumpet, two violins, viola, cello, double bass and piano. In fact Saint-Saëns and other leading musicians of the day would perform with Lemoine's "La Trompette". In October 1907, Saint-Saëns wrote to Lemoine about his Op 65:-

When I think how much you pestered me to make me produce, against my better judgment, this piece that I did not want to write and which has become one of my great successes, I never understood why.

Saint-Saëns composed other works for Lemoine. For example he gave Lemoine the work *Préambule* as a present for Christmas 1879. It was played at a concert in January 1880 and Saint-Saëns was so pleased with the result that he promised Lemoine that he would write a complete work with *Préambule* as its first movement, which he had done by the end of 1880.

Let us return to Lemoine's university career which he completed when he graduated from the École Polytechnique in 1866. His life after this was unconventional although he did do some teaching at the École Polytechnique [3]:-

Instead of accepting any of the careers offered by the State to all graduates of the Polytechnic School, M Lemoine determined to make his own way. Indeed, for the next few years, although engaged in science teaching in Paris, he seems to have run the round of pleasure of which that city is the home par excellence. Of great versatility and exceptional conversational powers, with an originality that fascinated and a personality that impressed his large circle of friends, he lived the life of a dilettante in the best sense of the term, and drank at the fountains of pleasure, of politics, of the arts, and of the sciences.

However, Lemoine's life was not simply one of enjoying the social life of Paris. He continued his scholarly interests in a remarkably wide range of different areas. He did not give up studying for he attended classes at the École des Mines. He served as an assistant to the astronomer Pierre Jules César Janssen (1824-1907) who had been appointed as professor of physics at the École Speciale d'Architecture in Paris in 1865. Janssen invented a way to observe the sun without there being an eclipse, and in that way discovered the element helium in 1868. Lemoine also worked with the chemist Charles Adolphe Wurtz (1817-1884) [3]:-

... for whom he always had a great admiration and between whom and himself there was much affection.

He spent a year studying law, but "his republican principles and his liberal views on church matters" meant that he did not fit into this particular profession and he was forced to abandon it. His love for travelling was limited by the costs involved, but he managed to make a number of journeys in the role of tutor to the children of wealthy families. However, in 1870 he had an accident which caused a problem with his larynx. This put an end to his teaching and for a while he left Paris and rested in Grenoble. Around this time he began to undertake research in mathematics again and he published *Note sur l'expression de la distance entre quelques points remarquables d'un triangle ABC* (1870) and *Note sur une question d'arithmétique* (1870), both in the *Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale.*

The popularity of Napoleon III, the French emperor, was declining in France and he thought a war with Prussia might change his political fortunes since his advisers had told him that the French Army could defeat Prussia. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states. With both sides feeling that a war was to their advantage, the Franco-Prussian War became inevitable. On 14 July, Bismarck sent a telegram which infuriated the French government and on the 19 July France declared war on Prussia. Because of the war, Lemoine served for a while in the army. However, the war went badly for France and in August the German army trapped part of the French army in Metz. The French army surrendered on 1 September, and on 19 September the German army began to blockade Paris. The city fell in January 1871 and the treaty following the French defeat was humiliating. From March to May 1871, as a result of dissatisfaction with the government, there was an insurrection in Paris against the French government. After peace was restored, Lemoine returned to Paris in the summer of 1871 and, changing career, he became a civil engineer. However he continued to work as an amateur mathematician and musician.

His next mathematical work was *Sur une question de probabilités* published in the *Bulletin de la Société Mathématique de France* in 1872. Although at first sight this does not appear to be on his favourite topic of geometry, in fact it does solve a geometrical problem since the paper answers the following question:

A rod is broken into three pieces; what is the probability that, with these three pieces, one can form a triangle?

This is such a nice little question that the reader will almost certainly like to know what answer Lemoine came up with. The answer is ^{1}/_{4}.

As a civil engineer he rose to the rank of chief inspector and in that capacity he was responsible for the gas supply to Paris. He worked in the gas inspection service from 1886 until 1896.

Lemoine's contribution to mathematics was mainly on geometry and he published *Note sur un point remarquable du plan d'un triangle* in 1873. He founded a new study of properties of a triangle in his paper *Sur quelques propriétés d'un point remarquable de triangle* of 1873 and his 1874 paper *Note sur les propriétés du centre des médianes antiparalleles dans un triangle*. In these papers he studied the point of intersection of the symmedians of a triangle, a concept which we define below. He had been a founder member of the Association Française pour l'Avancement des Sciences and it was at a meeting of the Association in 1873 in Lyon that he presented the first of these papers on the symmedians. The second paper of 1874 he presented to the Congrès de Lille.

A *symmedian* of a triangle from vertex *A* is obtained by reflecting the median from *A* in the bisector of the angle *A*. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper is the result that the symmedian from the vertex *A* cuts the side *BC* of the triangle in the ratio of the squares of the sides *AC* and *AB*. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle.

You can see the Lemoine point configuration by clicking THIS LINK and the Lemoine circle configuration by clicking HERE.

These results are interesting but Lemoine's next venture failed to interest many mathematicians. He produced a classification of geometry according to five operations:-

- placing a compass end on a given point,

- placing a compass end on a given line,

- drawing a circle with the compass so placed,

- placing a straight edge on a given line,

- drawing a line with the straight edge so placed.

Lemoine then classified the "simplicity" of a construction according to how many times these five operations had to be used. As an example of the types of results that he obtained was his study of the problem of constructing a circle tangent to three given circles: the Apollonius problem. The usual construction required over 400 of Lemoine's operations but he was able to reduce the number to 199. He presented these results to the meeting of the Association Française pour l'Avancement des Sciences in 1888 at Oran in Algeria. One would have to say that these results were not thought to be particularly interesting by mathematicians at the meeting and there has been a similar lack of interest ever since.

It is perhaps worth asking what is interesting in mathematics. Why are these results of Lemoine not found interesting? All I [EFR] can add is that I agree with the mathematicians of the time who preferred a construction with a large number of easily understood steps to a shorter one with sophisticated, rather obscure, steps. Let me add that I do find Lemoine's results on symmedians of a triangle to be very interesting and beautiful! He deserves much credit for showing that there are many interesting properties of the triangle waiting to be discovered.

Lemoine gave up active mathematical research in 1895 but continued to support the subject. He had helped to found the mathematical journal* L'intermédiaire des mathématiciens* in 1894 and he became its first editor, a role he held for many years. Lemoine had been helped to found the journal by Charles-Ange Laisant (1841-1920), who had been a fellow student at the École Polytechnique and had then served in the military before entering politics. The journal *L'intermédiaire des mathématiciens* continued to be published until 1925.

Let us end by quoting David E Smith's personal comments about Lemoine and his music. We note that the American David Smith was a great lover of Paris and its culture and was a frequent visitor there where he met Lemoine [3]:-

The soirées of M and Mme Lemoine are justly celebrated, and each week of the winter sees an assemblage representing the 'anciens élèves' of the École Polytechnique, the École Normale, the Marine, and in general a good part of the scientific, literary, and artistic circles of Paris, to listen to a musical programme as original as the mathematical labours of the host. These soirées have exerted a great influence in a musical way, the type which they have fixed being adopted by many societies in and about Paris. One amusing feature of these meetings is the name which designates them. If the writer may be pardoned a personal allusion, he once attended an examination in the École Polytechnique by M Hermann Laurent. It was one of the most severe he had ever seen, - an exceptionally bright young man submitted to an oral examination that would certainly have floored most American professors, - the examiner, a dyspeptic looking man as cold and as keen as steel and apparently as unsympathetic as ice, though in reality one of the most genial of men. To this justly celebrated mathematician, M Laurent, is due the name of M Lemoine's soirées, "La Trompette". Long ago he one day remarked to M Lemoine in a jesting way, as the latter was excusing himself to attend one of his musical reunions, "Stay here with me, let the trumpet alone." Struck by the name, Lemoine adopted it, and La Trompette has ever since designated the delightful soirées with which the Paris cultured world is familiar. A final word concerning the modesty of M Lemoine. He estimates his position exactly. He says that he is not a mathematician. He has no claim to rank with Hermite, Poincaré, Picard, Painlevé, Appell, Jordan, Bertrand, Tannery, Darboux, or any of that famous circle which is making Paris such a centre of study in the fields of higher modern mathematics. But all mathematicians feel that he has done a noteworthy work in other lines, and for this his name will be known and prominently known in the history of mathematics.

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