Émile Lemoine

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Émile Lemoine
Charles-Adolphe Wurtz

J. Kiœs

Émile Michel Hyacinthe Lemoine (French:

École Polytechnique
. Lemoine taught as a private tutor for a short period after his graduation from the latter school.

Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called Géométrographie and a method which related algebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work.

For most of his life, Lemoine was a professor of mathematics at the École Polytechnique. In later years, he worked as a civil engineer in

L'Intermédiaire des Mathématiciens
.

Biography

Early years (1840–1869)

Lemoine was born in

military Prytanée of La Flèche on a scholarship granted because his father had helped found the school. During this early period, he published a journal article in Nouvelles annales de mathématiques, discussing properties of the triangle.[1]

Lemoine was accepted into the

tutor for a period before accepting an appointment as a professor at the École Polytechnique.[5]

Middle years (1870–1887)

École Polytechnique

In 1870, a

Société Mathématique de France, the Journal de Physique, and the Société de Physique, all in 1871.[1]

As a founding member of the Association Française pour l'Avancement des Sciences, Lemoine presented what became his best-known paper, Note sur les propriétés du centre des médianes antiparallèles dans un triangle at the Association's 1874 meeting in Lille. The central focus of this paper concerned the point which bears his name today.[6] Most of the other results discussed in the paper pertained to various concyclic points that could be constructed from the Lemoine point.[2]

Lemoine served in the French military for a time in the years following the publishing of his best-known papers. Discharged during the Commune, he afterwards became a civil engineer in Paris.[1] In this career, he rose to the rank of chief inspector, a position he held until 1896. As the chief inspector, he was responsible for the gas supply of the city.[7]

Later years (1888–1912)

During his tenure as a civil engineer, Lemoine wrote a

Académie française. He published additional papers on the subject in Mathesis (1888), Journal des mathématiques élémentaires (1889), Nouvelles annales de mathématiques (1892), and the self-published La Géométrographie ou l'art des constructions géométriques, which was presented at the meeting of the Association Française in Pau (1892), and again at Besançon (1893) and Caen (1894).[1]

After this, Lemoine published another series of papers, including a series on what he called transformation continue (continuous transformation), which related mathematical

transformation. His papers on this subject included, Sur les transformations systématiques des formules relatives au triangle (1891), Étude sur une nouvelle transformation continue (1891), Une règle d'analogies dans le triangle et la spécification de certaines analogies à une transformation dite transformation continue (1893), and Applications au tétraèdre de la transformation continue (1894).[1]

In 1894, Lemoine co-founded another mathematical journal entitled, L'intermédiaire des mathématiciens along with

Charles Laisant, a friend whom he met at the École Polytechnique. Lemoine had been planning such a journal since early 1893, but thought that he would be too busy to create it. At a dinner with Laisant in March 1893, he suggested the idea of the journal. Laisant cajoled him to create the journal, and so they approached the publisher Gauthier-Villars, which published the first issue in January 1894. Lemoine served as the journal's first editor, and held the position for several years. The year after the journal's initial publication, he retired from mathematical research, but continued to support the subject.[6] Lemoine died on 21 February 1912 in his home city of Paris.[2]

Contributions

Lemoine's work has been said to contribute towards laying the foundation of modern

American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement [of modern triangle geometry] ..."[1] At the annual meeting of the Paris Academy of Sciences in 1902, Lemoine received the 1,000-franc Francœur prize,[11] which he held for several years.[12][13]

Lemoine point and circle

The Lemoine point; L. The black lines are medians, the dotted lines are angle bisectors and the red lines are the symmedians (the reflections of the black lines in the dotted lines).

In his 1874 paper, entitled Note sur les propriétés du centre des médianes antiparallèles dans un triangle, Lemoine proved the concurrency of the

squares
of the other two sides.

Lemoine also proved that if

Lemoine circle, or simply the Lemoine circle.[2][15]

Construction system

Lemoine's system of constructions, the Géométrographie, attempted to create a methodological system by which constructions could be judged. This system enabled a more direct process for simplifying existing constructions. In his description, he listed five main operations: placing a compass's end on a given point, placing it on a given line, drawing a circle with the compass placed upon the aforementioned point or line, placing a straightedge on a given line, and extending the line with the straightedge.[14][16]

The "simplicity" of a construction could be measured by the number of its operations. In his paper, he discussed as an example the

Joseph Diaz Gergonne in 1816 with a construction of simplicity 400, but Lemoine's presented solution had simplicity 154.[2][17] Simpler solutions such as those by Frederick Soddy in 1936 and by David Eppstein in 2001 are now known to exist.[18]

Lemoine's conjecture and extensions

In 1894, Lemoine stated what is now known as

positive integers j and k such that m = 2p + q, 2 + pq = 2j + r and 2q + p = 2k + s. [...] the study has directed our attention to more subtle aspects of the additive theory of prime numbers. Our conjecture reflects this, dealing with interactions of sums involving primes whereas Goldbach's conjecture and Lemoine's conjecture deal with such sums only individually. This conjecture and the open questions about numbers at levels two and three are of interest in their own right because of the issues they raise within this fascinating and often baffling additive realm of the prime numbers."[20]

Role in modern triangle geometry

Lemoine has been described by

collinearity, concurrency, and concyclicity, as they do not involve the measures listed previously.[22]

Lemoine's work defined many of the noted traits of this movement. His Géométrographie and relation of equations to tetrahedrons and triangles, as well as his study of concurrencies and concyclities, contributed to the modern triangle geometry of the time. The definition of points of the triangle such as the Lemoine point was also a staple of the geometry, and other modern triangle geometers such as Brocard and Gaston Tarry wrote about similar points.[21]

List of selected works

  • Sur quelques propriétés d'un point remarquable du triangle (1873)
  • Note sur les propriétés du centre 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 spécification de certaines analogies à une transformation dite transformation continue (1893)
  • Applications au tétraèdre de la transformation continue (1894)
  • "Note on Mr. George Peirce's Approximate Construction for π". Bull. Amer. Math. Soc. 8 (4): 137–148. 1902.
    doi:10.1090/s0002-9904-1902-00864-1
    .

See also

Notes

  1. ^
    JSTOR 2968278
    .
  2. ^ a b c d e O'Connor, J.J.; Robertson, E.F. "Émile Michel Hyacinthe Lemoine". MacTutor. Retrieved 2008-02-26.
  3. ^ "École Polytechnique - 208 years of history". École Polytechnique. Archived from the original on April 5, 2008. Retrieved 2008-03-21.
  4. ^ Charles Lenepveu. Letter to Émile Lemoine. February 1890. The Morrison Foundation for Musical Research. Retrieved on 2008-05-19
  5. ^ Kimberling, Clark. "Émile Michel Hyacinthe Lemoine (1840–1912), geometer". University of Evansville. Retrieved 2008-02-25.
  6. ^
    JSTOR 3028804
    .
  7. ^ Weisse, K.; Schreiber, P. (1989). "Zur Geschichte des Lemoineschen Punktes". Beiträge zur Geschichte, Philosophie und Methodologie der Mathematik (in German). 38 (4). Wiss. Z. Greifswald. Ernst-Moritz-Arndt-Univ. Math.-Natur. Reihe: 73–4.
  8. ^ Greitzer, S.L. (1970). Dictionary of Scientific Biography. New York: Charles Scribner's Sons.
  9. ISBN 0-486-49524-8
    .
  10. ^ Kimberling, Clark. "Triangle Geometers". University of Evansville. Archived from the original on 2008-02-16. Retrieved 2008-02-25.
  11. doi:10.1090/S0002-9904-1903-00993-8
    . Retrieved 2008-04-24.
  12. doi:10.1090/S0002-9904-1912-02239-5
    . Retrieved 2008-05-11.
  13. ^ "Séance du 18 décembre". Le Moniteur Scientifique du Docteur Quesneville: 154–155. February 1906. Archived from the original on January 21, 2021. Lemoine won the Prix Francœur in the years from 1902–1904 and 1906–1912, with the single interruption by Xavier Stouff's win in 1905.
  14. ^
    ISBN 0-486-45805-9
    .
  15. ISBN 978-1-4297-0050-4
    .
  16. ^ Lemoine, Émile. La Géométrographie ou l'art des constructions géométriques. (1903), Scientia, Paris (in French)
  17. ^ Eric W. Weisstein CRC Concise Encyclopedia of Mathematics (CRC Press, 1999), 733–4.
  18. ^ David Gisch; Jason M. Ribando (2004-02-29). "Apollonius' Problem: A Study of Solutions and Their Connections" (PDF). American Journal of Undergraduate Research. 3 (1). University of Northern Iowa. Archived from the original (PDF) on 2008-04-15. Retrieved 2008-04-16.
  19. ISBN 0-8284-0086-5
    .
  20. JSTOR 2689513
    .
  21. ^
    ISBN 978-1-4181-7845-1
    .
  22. ^ Steve Sigur (1999). The Modern Geometry of the Triangle (PDF). Paideiaschool.org. Retrieved on 2008-04-16.

External links

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