Émile Lemoine

Émile Lemoine

Émile Michel Hyacinthe Lemoine [ emil ləmwan ] (born November 22, 1840 in Quimper , France , † February 21, 1912 in Paris ) was a French mathematician and engineer. He was best known for his proof of the meeting of the symmedians at one point, the Lemoine point .

Life

Lemoine was born on November 22, 1840 in Quimper, Brittany. His father, a captain in the French army, helped found the Prytanée national militaire in La Flèche in 1808 . That's why Lemoine received a scholarship to attend this school. During his school days he published an article on geometric relations in triangles in the magazine Nouvelles annales de mathématiques . When he was twenty, the same year his father died, Lemoine was accepted into the École polytechnique . While he was studying there, he helped found a chamber ensemble called La Trompette , in which he probably played the trumpet. Camille Saint-Saëns composed several pieces for the ensemble.

After graduating in 1866, Lemoine initially wanted to become a lawyer. However, he abandoned this plan because his political and religious attitudes were in conflict with the ideals of the government of the time. Therefore he studied and taught in the next period at various institutions such as the École d'Architecture , École des Mines , École des Beaux-Arts and the École de Médecine . He also worked as a private tutor before accepting the appointment as professor at the École Polytechnique.

When Lemoine fell ill with the larynx in 1870, he stopped teaching and went to Grenoble for a short time . After his return to Paris, he published some results of his mathematical research. That year he became an engineering consultant at the Commercial Court in Paris. In the following time he founded several scientific associations and journals, including the Société Mathématique de France , the Société de Physique and the Journal de Physique .

At the meeting of the Association Française pour l'Avancement des Sciences in 1874, of which he was also a founding member, Lemoine presented his work Note sur les propriétés du center des médianes antiparallèles dans un triangle , which would later be one of his most famous works. In this writing he proved that the symmetries intersect at a point that was later named Lemoine point in his honor.

After a few years in the French military, he was responsible for the gas supply to Paris as an engineer until 1896. In these and the following years, Lemoine wrote most of his works, such as La Géométrographie ou l'art des constructions géométriques , which he presented in 1888 at the meeting of the Association Française in Oran , Algeria . In the work Lemoine describes a system with which the complexity of constructions can be specified.

Other works from this period were a series of writings on the relationship between equations and geometric objects, which he called transformation continue (continuous transformation). The theme of the works has nothing to do with today's concept of transformation .

In 1894 Lemoine realized a long-planned project and together with Charles-Ange Laisant , a friend of the École polytechnique, founded another mathematical journal called L'intermédiaire des mathématiciens . Lemoine was the editor-in-chief of the journal for several years and thus continued to support mathematics, although he had not done research since 1895.

Émile Lemoine died on February 21, 1912 in his hometown of Paris.

Services

Nathan Altshiller-Court says of Lemoine that, alongside Henri Brocard and Joseph Neuberg , he was one of the founders of modern triangular geometry (18th century or later).

At that time, triangular geometry was mainly concerned with investigations into whether certain points lie on a circle or a line, or whether three lines intersect at one point. With his works on triangular geometry, Lemoine fit in perfectly with the zeitgeist of the time, as he too examined the intersections of lines and circles in his works.

At the meeting of the Académie des Sciences in 1902, Lemoine received the Francœur Prize, endowed with 1000 francs , which he received for several years.

Lemoine point

Triangle with Lemoine point L.
bisector (green), bisector (blue),
symmetrians (red)

In his work Note sur les propriétés du center des médianes antiparallèles dans un triangle (1874), Lemoine proved that the symmetries of a triangle intersect at a point. Lemoine called the point center des médianes antiparallèles . He also cited properties of the point in the work. Since some mathematicians like Grebe or P. Hossard had already dealt with the point before Lemoine, Lemoine's achievement consisted only in the scientific summary of the results. These organizational merits led to the fact that from 1876 the point was mostly called Lemoine point or Lemoine point. According to other sources, Joseph Jean Baptiste Neuberg (1840-1926) proposed in 1884 that the point be named in honor of Lemoines Lemoine point.

The Lemoine hexagon and the first Lemoine circle.

E. Hain called the point in 1876 the point Grebeschen point, because he mistakenly thought that Ernst Wilhelm Grebe (1804–1874) had dealt with the point in 1847 first. As a result, the point was called Grebe point (or Grebescher point) for a while in Germany, but Lemoine point in France. Robert Tucker (1832-1905) suggested for the sake of consistency that the point should be called the symmetry point.

If you draw parallels to the three sides of the triangle through the Lemoine point and connect the points of intersection of the parallels with the sides of the triangle, you get a hexagon, the so-called Lemoine hexagon . The parallels are often called Lemoine parallels. The circumference of the hexagon is called the first Lemoine circle . The center of this circle lies in the middle between the Lemoine point and the intersection of the perpendiculars (i.e. the circumcenter ) of the triangle. If you draw the anti-parallels (also called Lemoine anti-parallels) through the Lemoine point of a triangle, they intersect with the sides of the triangle at six points. If you connect these, you get the cosine hexagon . The circumference of this hexagon is called the cosine circle or second Lemoine circle .

Construction system

Lemoine developed a system, which he called geométrography , with which the "simplicity" of geometric constructions could be assessed. He also recognized that this designation is actually wrong and should better be called "degree of complexity". The simplicity of a construction can be determined by the number of basic operations required. Lemoine calls the number of executions of operations 1, 2 and 4 the accuracy of the construction. The basic operations named by Lemoine are:

• Placing a compass on a given point,
• Placing a compass on a given line,
• Draw a circle with the compass on the point or line,
• Place a ruler on a line and
• Extend the line with the ruler.

Solving the Apollonian problem

This system also made it possible to simplify existing designs more easily. However, Lemoine did not have a sufficiently general algorithm with which he could prove whether a solution is optimal or whether there is a better one. Lemoine treated the system in the work La Géométrographie ou l'art des constructions géométriques , which he presented at the meetings of the Association Française in Pau (1892), Besançon (1893) and Caen (1894). He published other writings on this subject in Mathesis (1888), Journal des mathématiques élémentaires (1889) and Nouvelles annales de mathématiques (1892). As a result of the presentations and the introduction in some journals, the construction system received some attention in Germany and France, but was eventually forgotten because the mathematicians of the time preferred longer but simpler solutions to shorter and more complex ones. From today's perspective, it can be said that Lemoine was ahead of his time, and his geométrography represents a remarkable approach in measuring the complexity and optimization of algorithms.

In his work La Géométrographie ou l'art des constructions géométriques , Lemoine dealt with the Apollonian problem that had been posed by Apollonios von Perge : For three given circles, a fourth circle is to be constructed that affects the other circles. The problem had already been solved by Joseph Gergonne in 1816 with a simplicity of 400 (479 according to Coolidge: A history of Geometrical Methods ), but Lemoine presented a solution of simplicity 199. Today even simpler solutions are known, such as that of Frederick Soddy from the Year 1936 and that of David Eppstein from 2001.

Fonts

• Sur quelques propriétés d'un point remarquable du triangle (1873)
• Note sur les propriétés du center des médianes antiparallèles dans un triangle (1874)
• Sur la mesure de la simplicité dans les tracés géométriques (1889)
• Sur les transformations systématiques des formules relatives au triangle (1891)
• Étude sur une nouvelle transformation continue (1891)
• La Géométrographie ou l'art des constructions géométriques (1892)
• Une règle d'analogies dans le triangle et la specification de certaines analogies à une transformation dite transformation continue (1893)
• Applications au tétraèdre de la transformation continue (1894)

literature

• Nathan Altshiller-Court: College geometry . Barnes & Noble, inc., New York 1952
• Siegfried Gottwald (ed.): Lexicon of important mathematicians . Harri Deutsch, Thun 1990, ISBN 3-8171-1164-9 .
• Katrin Weiße, Peter Schreiber: On the history of the Lemoine point . In: Peter Richter (Ed.): Contributions to the history, philosophy and methodology of mathematics . Ernst Moritz Arndt University of Greifswald, Neubrandenburg 1988, ISSN  0138-2853 , pp. 73-74.

Web links

• John J. O'Connor, Edmund F. Robertson :  Émile Lemoine. In: MacTutor History of Mathematics archive .
• Biography Lemoine on the side of the University of Evansville (English)
• Digitized works by Lemoine on Numdam.org (English)

Individual evidence

1. ↑ http://www.morrisonfoundation.org/charles lenepveu.htm (link not available)
2. ↑ Clark Kimberling : Triangle Geometers . University of Evansville. Retrieved November 20, 2008.
3. ↑ Disseminate ( English , PDF) In: Bulletin of the American Mathematical Society . American Mathematical Society. Pp. 273, 1903. Retrieved November 21, 2008.
4. ↑ Notes ( English , PDF; 554 kB) In: Bulletin of the American Mathematical Society . American Mathematical Society. P. 424, 1912. Retrieved November 21, 2008.
5. ↑ Katrin Weisse, Peter Schreiber: On the history of the Lemoine point . In: Peter Richter (Ed.): Contributions to the history, philosophy and methodology of mathematics (II) . ISSN  0138-2853 , p. 73-74 .
6. ^ Carl D. Meyer: Earliest Known Uses of the Words of Mathematics ( English , PDF) p. 199. 2000. Accessed November 20, 2008.
7. ↑ Eric W. Weisstein: Lemoine Hexagon ( English ) Retrieved October 23, 2008.
8. ↑ Eric W. Weisstein: First Lemoine Circle ( English ) Retrieved October 23, 2008.
9. ↑ Eric W. Weisstein: Cosine Hexagon ( English ) Retrieved November 20, 2008.
10. ↑ Eric W. Weisstein: Cosine Circle ( English ) Retrieved November 20, 2008.
11. ↑ Lemoine, Émile. La Géométrographie ou l'art des constructions géométriques . (1903), Scientia, Paris (French)
12. ^ Julian Lowell Coolidge: A history of geometrical methods . Dover Publications, inc., New York 1963
13. ↑ David Gisch, Jason M. Ribando: Apollonius' Problem: A Study of Solutions and Their Connections ( English , PDF; 891 kB) In: American Journal of Undergraduate Research . University of Northern Iowa. February 29, 2004. Retrieved November 21, 2008.