Émile Lemoine

Emile 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 especially for his proof of meeting the Symmedianen in one point, the Lemoinepunkt, famous.

Life

Lemoine was born on November 22, 1840 in Quimper in Brittany. His father, a captain in the French army was in 1808 Militaire in La Flèche is taking part in the founding of Prytanée. Therefore Lemoine received a scholarship to attend this school. Even in his school days he published in the magazine Nouvelles de Mathématiques annales an article about geometric relationships in the triangle. When he was twenty, the same year when his father died, Lemoine was the École polytechnique added. While studying there, he was instrumental in the establishment of a chamber ensembles called La Trompette, where he probably played the trumpet. Camille Saint- Saëns composed several pieces for the ensemble.

After his graduation in 1866, Lemoine wanted to become legal scholars. He let go of this project, as his political and religious views were at odds with the ideals of the then government. Therefore, he studied and taught in the near future at various institutions such as the Ecole d'Architecture, École des Mines, École des Beaux -Arts and the École de Médecine. In addition, he worked as a private tutor before he accepted the appointment as professor at the École Polytechnique.

As Lemoine in 1870, fell ill in the larynx, he finished his teaching career and went for a short time to Grenoble. After his return to Paris, he published some results of his mathematical research. On the commercial court in Paris he was Ingenieurconsulent this year. In the following years 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' advancement of Sciences in 1874, of which he was also a founding member, Lemoine presented his work Note sur les propriétés du center of the median antiparallèles dans un triangle, which should include later his most famous works. In this paper he proved that the Symmendianen intersect at a point which was later named in his honor Lemoinepunkt.

After a few years the French military, he was responsible for the supply of gas to Paris as an engineer until 1896. In these and the following years wrote Lemoine most of his works, such as La Géométrographie ou l'art of 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 the structures 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 the concept of transformation of today.

1894 Lemoine realized a long -planned project and, together with Charles -Ange Laisant, a friend of the École polytechnique, another mathematical journal called L' intermédiaire of Mathématiciens. Lemoine was for several years the editor of the journal and so supported the further mathematics, although he no longer conducted research since 1895.

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

Services

Nathan Altshiller court says about Lemoine that he, alongside Henri Brocard and Joseph Neuberg, one of the founders of modern triangle geometry is (18th century or later).

At this time, the triangular geometry was mainly involved with investigations into whether certain points lie on a circle or a line, or whether three lines intersect at a point. Lemoine acquiesced with his works for the triangle geometry perfectly in the spirit of his times, since he also examined the intersections of lines and circles in his works.

At the meeting of the Académie des sciences in 1902, Lemoine was endowed with 1000 Francesco Francœur price he received for several years.

Lemoinepunkt

In his work Note sur les propriétés du center of the median antiparallèles dans un triangle (1874 ) Lemoine proved that intersect the Symmedianen a triangle at one point. Lemoine point called the center of the median antiparallèles. In addition, he led in the work characteristics of the point. As already had dealt with the point in front Lemoine some mathematicians such as Grebe or P. Hossard, consisted of Lemoine power only in the scientific summary of the results. These organizational achievements have meant that the point was in 1876 mostly called Lemoinepunkt or lemoinescher point. According to other sources suggested Joseph Jean Baptiste Neuberg (1840-1926) in 1884 before, glycolate the point in honor of Lemoine Lemoinepunkt.

E. Hain called the point 1876 point Grebe 's point because he mistakenly thought that Ernst Wilhelm Grebe ( 1804-1874 ) had the point taken as the first in 1847. Then, the point was called for a time in Germany Grebe - point (or Grebe shear point ), but in France Lemoinepunkt. Robert Tucker (1832-1905) suggested that for the sake of uniformity, to call the point Symmedianenpunkt.

If we take through the Lemoinepunkt parallels to the three sides of the triangle and one connects the intersections of the parallels with the triangle sides together, then a hexagon, the so-called lemoinesche hexagon. The parallels are often called Lemoine parallels. The perimeter of the hexagon is called first lemoinescher circle. The center of this circle is located in the middle between the Lemoinepunkt and the intersection of the bisectors (ie, the radius center point ) of the triangle. Considering the anti- parallel (also called anti - Lemoine parallels ) by the Lemoinepunkt of a triangle, as they intersect with the sides of the triangle in six points. By connecting those, one obtains the cosine hexagon. The perimeter of the hexagon is called cosine circle or second lemoinescher circle.

Construction system

Lemoine developed a system, called by him Géométrographie, with the "simplicity" of geometrical constructions could be evaluated. He also realized that this name is actually wrong and should better be called " degree of complexity ". The simplicity of a design can be determined by the number of required basic operations. The number of executions of the operations 1, 2 and 4 Lemoine called the accuracy of the construction. The basic operations mentioned by Lemoine are:

• Place a compass on a given point,
• Place a compass on a given line,
• Draw a circle with the compasses on the point or line,
• Creating a ruler on a line and
• Expand the line with a ruler.

This system allowed easier to simplify existing constructions. However, Lemoine had no sufficiently general algorithm by which he could prove whether a solution is optimal or whether there is a better one. Lemoinebehandelte the system in the work Géométrographie La ou l'art of constructions géométriques, which he presented at the meeting of the Association Française in Pau (1892 ), in Besançon ( 1893) and Caen ( 1894). He has published more writings on this subject in Mathesis (1888 ), Journal of Mathématiques élémentaires (1889 ) and Nouvelles de Mathématiques annales (1892 ). As a result of the presentations and the presentation in some journals was the construction system in Germany and France some attention, but was eventually forgotten, as the mathematicians of that time shorter longer but simpler solutions and complicated preferring. From today's perspective it can be said that Lemoine was ahead of his time, and his Géométrographie represents a significant approach in the measurement of complexity and optimization of algorithms.

In his work La Géométrographie ou l'art of constructions géométriques Lemoine treated the Apollonian problem that had been set up by Apollonius of Perga: For three given circles is to be constructed, a fourth circle that is tangent to the other circuits. The problem was already in 1816 by Joseph Gergonne with a simplicity of 400 ( 479 according to Coolidge: A history of Geometrical Methods ) have been solved, but Lemoine presented a solution to the simplicity 199 Today even simpler solutions known as the Frederick Soddy from the 1936 and by David Eppstein from the year 2001.

Writings

• Sur quelques propriétés d'un point Remarquable du triangle (1873 )
• Note sur les propriétés du center of the median antiparallèles dans un triangle (1874 )
• Sur la mesure de la simplicité dans les tracés géométriques (1889 )
• Sur les transformations systématiques of formules relative au triangle (1891 )
• Étude sur une nouvelle transformation continue (1891 )
• La Géométrographie ou l'art of constructions géométriques (1892 )
• Une regle d' analogies dans le triangle et la spécification de certaines analogies à une transformation dite continue transformation (1893 )
• Applications au Tetraedre de la transformation continue (1894 )