# Edmond Nicolas Laguerre

### Born: 9 April 1834 in Bar-le-Duc, France

Died: 14 August 1886 in Bar-le-Duc, France

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**Edmond Laguerre**had poor health as a boy and this impeded his studies. His parents were forced to move him from one public school to another because of these health problems. However he was able to enter the École Polytechnique in Paris in 1852 but he suffered from tiredness every day. Despite showing a talent for modern languages and mathematics he was only ranked 46

^{th}in his class. This in no way reflected his ability, rather it showed that he was badly affected by health problems. An indication that he was already a talented mathematician is given by the fact that he published his first work during this time.

*On the theory of foci*appeared in 1853 and it is one of his most important papers, investigating the angle between lines in the complex projective plane. Laguerre graduated from the École Polytechnique in 1854 and decided on a military career.

He was commissioned as an artillery officer working on the manufacture of armaments at Mutzig, near Strasbourg, from 1854 to 1864. However during this period he continued with his mathematical studies and in 1864 he resigned his commission and returned to the École Polytechnique as a tutor. He remained there for the rest of his life but, after 1874, he was an examiner at the École. Bertrand, who was a great admirer of his work, supported him for election to the Academy of Sciences, and also supported his appointment to an additional post, namely that of professor of mathematical physics at the Collège de France. He was appointed to this chair in 1883 but his health, which had always been poor, broke down completely in February 1886. He returned to Bar-le-Duc where he died six months later.

Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations. This work came out of his paper published in 1879 which examined

∫ exp (-

where the integral is from 0 to infinity. He found a divergent series, the first few terms of which gave a good approximation to the integral. He also found a continued fraction expansion for the integral, the convergents of which involved the Laguerre polynomials. He went on to investigate properties of the polynomials, proving orthogonality relations and also showing that an arbitrary function could be expanded in a 'Fourier type' series in Laguerre polynomials. Bernkopf writes in [1]:-*x*) /*x**dx*Other than mathematics, it was only his family which played a large role in Laguerre's life. He was married with two daughters and he devoted much time and energy to the education of the two girls. Bernkopf writes in [1]:-This memoir of Laguerre is significant not only because of the discovery of the Laguerre equations and polynomials and their properties, but also because it contains one of the earliest infinite continued fractions which was known to be convergent. That it was developed from a divergent series is especially remarkable.

His most important work was in the areas of analysis and geometry. His work in geometry was important at the time but has been overtaken by Lie group theory, Cayley's work and Klein's work. Laguerre wrote 140 memoirs which he published in the leading journals of his time so it is reasonable to ask why he is only known for the results mentioned specifically above. Bernkopf examines this question in [1]:-Laguerre was pictured by his contemporaries as a quiet, gentle man who was passionately devoted to his research, his teaching, and the education of his two daughters.

Despite this assessment (which must be considered as rather harsh), there is still interest in Laguerre's work as is seen for example in [7] where the following is discussed:-What, then, can be said to evaluate Laguerre's work? That he was brilliant and innovative is beyond question. In his short working life, actually less than twenty-two years, he produced a quantity of first-class papers. Why, then, is his name so little known and his work so seldom cited? Because as brilliant as Laguerre was, he worked only on details - significant details, yet nevertheless details. Not once did he step back to draw together various pieces and put them into a single theory. The result is that his work has mostly come down as various interesting special cases of more general theories discovered by others.

The complete works of Laguerre were published in two volumes; Volume 1 in 1898 and Volume 2 in 1905. Hermite, Poincaré and Rouché edited both volumes. These were considered interesting enough nearly 100 years later to be reprinted in 1972. In 1986 a reprint ofDeep relations between elliptic functions and Cartesian ovals were also established in1867, with the geometrical proofs of the addition theorem of elliptic functions given by Darboux and Laguerre. When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem and that they constitute the algebraic form of the integral solution. Laguerre, on the other hand, proved the addition theorem with the help of anallagmatic curves using Poncelet's theorem on inscribed and circumscribed polygons to two conics.

*Recherches sur la géométrie de direction*Ⓣ appeared which Laguerre had first published in 1885. Again, producing a reprint shows that there is still considerable interest in his results. The work contains six of Laguerre's papers originally published in the

*Nouvelles Annales de Mathématiques*:

*Sur le règle des signes en géométrie*Ⓣ (1870);

*Transformations par sémi-droites réciproques*Ⓣ (1882);

*Sur les anticaustiques par réflexion de la parabole, les rayons incidents étant parallèles*Ⓣ (1883);

*Sur quelques propriétés des cycles*Ⓣ (1883);

*Sur les courbures de direction de la troisième classe*Ⓣ (1883); and

*Sur les anticaustiques par réfraction de la parabole, les rayons incidents étant perpendiculaires à l'axe*Ⓣ (1885).

Let us end our biography by quoting Bonnet's assessment of Laguerre:-

He was one of the most penetrating geometers of our age: his discoveries in geometry place him in the first rank among the successors of Chasles and Poncelet.

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

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