Émile Lemoine


Émile Lemoine

Infobox_Scientist
name = Émile Michel Hyacinthe Lemoine


image_width = 250px
caption =
birth_date = birth date|1840|11|22|fd=y
birth_place = Quimper, France
residence = Paris, France
nationality = French
death_date = death date and age|1912|2|21|1840|22|11|fd=y
death_place = Paris, France
field = Mathematics, engineering
work_institutions = Professor at the École Polytechnique
alma_mater = École Polytechnique
doctoral_advisor =Charles-Adolphe Wurtz, J. Kioes
doctoral_students =Uwe Jannsen
known_for = Lemoine point, other geometric work
prizes = Francœur prize, held for several years

Émile Michel Hyacinthe Lemoine (IPA2|emil ləmwan; November 22 1840February 21 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the 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 Paris, and he also took an amateur's interest in music. During his tenure at the École Polytechnique and as a civil engineer, Lemoine published several papers on mathematics, most of which are included in a fourteen-page section in Nathan Court's "College Geometry". Additionally, he founded a mathematical journal titled, "L'intermédiaire des mathématiciens".

Biography

Early years (1840–1869)

Lemoine was born in Quimper, France, on November 22 1840, the son of a retired military captain who had participated in the campaigns of the First French Empire occurring after 1807. As a child, he attended the 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.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America]

Lemoine was accepted into the École Polytechnique in Paris at the age of twenty, the same year as his father's death.cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lemoine.html|title=Émile Michel Hyacinthe Lemoine|accessdate=2008-02-26|author=O'Connor, J.J.; Robertson, E.F |publisher=MacTutor] [cite web|url=http://www.polytechnique.edu/page.php?MID=28|title=École Polytechnique - 208 years of history|accessdate=2008-03-21|publisher=École Polytechnique] As a student there, Lemoine, a presumed trumpet player, [Charles Lenepveu. Letter to Émile Lemoine. February 1890. The Morrison Foundation for Musical Research. Retrieved on 2008-05-19] helped to found an amateur musical group called La Trompette, for which Camille Saint-Saëns composed several pieces. After graduation in 1866, he considered a career in law, but was discouraged by the fact that his advocacy for republican ideology and liberal religious views clashed with the ideals of the incumbent government, the Second French Empire.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America] Instead, he studied and taught at various institutions during this period, studying under J. Kiœs at the École d'Architecture and the École des Mines, teaching Uwe Jannsen at the same schools, and studying under Charles-Adolphe Wurtz at the École des Beaux Arts and the École de Médecine.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America] Lemoine also lectured at various scientific institutions in Paris and taught as a private tutor for a period before accepting an appointment as a professor at the École Polytechnique.cite web|url=http://faculty.evansville.edu/ck6/bstud/lemoine.html|title=Émile Michel Hyacinthe Lemoine (1840–1912), geometer|accessdate=2008-02-25|author=Kimberling, Clark|publisher=University of Evansville]

Middle years (1870–1887)

In 1870, a laryngeal disease forced him to discontinue his teaching. He took a brief vacation in Grenoble and, when he returned to Paris, he published some of his remaining mathematical research. He also participated and founded several scientific societies and journals, such as the "Société Mathématique de France", the "Journal de Physique", and the "Société de Physique", all in 1871.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America]

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.cite journal| last = Gentry| first = F.C.| title = Analytic Geometry of the Triangle| year = 1941| volume = 16| issue = 3| magazine = National Mathematics Magazine| month = December| pages = 127–40|publisher=Mathematical Association of America] Most of the other results discussed in the paper pertained to various concyclic points that could be constructed from the Lemoine point.cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lemoine.html|title=Émile Michel Hyacinthe Lemoine|accessdate=2008-02-26|author=O'Connor, J.J.; Robertson, E.F |publisher=MacTutor]

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.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America] 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. [cite journal| last = Weisse| first = K.| coauthors = P. Schreiber | title = Zur Geschichte des Lemoineschen Punktes| magazine = Beiträge zur Geschichte, Philosophie und Methodologie der Mathematik| volume = 38| publisher = Wiss. Z. Greifswald. Ernst-Moritz-Arndt-Univ. Math.-Natur. Reihe| pages = 73–4| year = 1989| issue = 4| language = German]

Later years (1888–1912)

During his tenure as a civil engineer, Lemoine wrote a treatise concerning compass and straightedge constructions entitled, "La Géométrographie ou l'art des constructions géométriques", which he considered his greatest work, despite the fact that it was not well-received critically. The original title was "De la mesure de la simplicité dans les sciences mathématiques", and the original idea for the text would have discussed the concepts Lemoine devised as concerning the entirety of mathematics. Time constraints, however, limited the scope of the paper.cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America] Instead of the original idea, Lemoine proposed a simplification of the construction process to a number of basic operations with the compass and straightedge. [cite book| last = Greitzer| first = S.L.| title = Dictionary of Scientific Biography| city = New York| year = 1970|publisher=Charles Scribner's Sons] He presented this paper at a meeting of the "Association Française" in Oran, Algeria in 1888. The paper, however, did not garner much enthusiasm or interest among the mathematicians gathered there. [cite book|last = Coolidge| first = Julian L.| title = A History of Geometrical Methods| city = Oxford| year = 1980| page = 58|publisher=Dover Publications] Lemoine published several other papers on his construction system that same year, including "Sur la mesure de la simplicité dans les constructions géométriques" in the "Comptes rendus" of the 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).cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America]

After this, Lemoine published another series of papers, including a series on what he called "transformation continue" (continuous transformation), which related mathematical equations to geometrical objects. This meaning stood separately from the modern definition of 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).cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America]

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.cite journal| last = Gentry| first = F.C.| title = Analytic Geometry of the Triangle| year = 1941| volume = 16| issue = 3| magazine = National Mathematics Magazine| month = December| pages = 127–40|publisher=Mathematical Association of America] Lemoine died on February 21, 1912, in his home city of Paris.cite web |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lemoine.html |title=Émile Michel Hyacinthe Lemoine |accessdate=2008-02-26|author=J.J. O'Connor and E.F. Robertson|publisher=MacTutor]

Contributions

Lemoine's work has been said to contribute towards laying the foundation of modern triangle geometry.cite web|url=http://faculty.evansville.edu/ck6/bstud/tg.html|title=Triangle Geometers|accessdate=2008-02-25|author=Kimberling, Clark |publisher=University of Evansville] The "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] ..."cite journal| last = Smith| first = David Eugene| title = Biography of Émile-Michel-Hyacinthe Lemoine| magazine = American Mathematics Monthly| year = 1896| volume = 3| pages = 29–33|publisher=Mathematical Association of America] At the annual meeting of the Paris Academy of Sciences in 1902, Lemoine received the 1,000-franc Francœur prize, [cite journal|title=Disseminate|page=273|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183417334&view=body&content-type=pdf_1|publisher=American Mathematical Society|journal=Bulletin of the American Mathematical Society|volume=9|issue=5|year=1903|pages=272–5|accessdate=2008-04-24] which he held for several years. [cite journal|title=Notes|url=http://www.ams.org/bull/1912-18-08/S0002-9904-1912-02239-5/S0002-9904-1912-02239-5.pdf|publisher=American Mathematical Society|journal=Bulletin of the American Mathematical Society|volume=18|issue=8|year=1912|accessdate=2008-05-11|pages=424]

Lemoine point and circle

In his 1874 paper, entitled "Note sur les propriétés du centre des médianes antiparallèles dans un triangle", Lemoine proved the concurrency of the symmedians of a triangle; the reflections of the medians of the triangle over the angle bisectors. Other results in the paper included the idea that the symmedian from a vertex of the triangle divides the opposite side into segments whose ratio is equal to the ratio of the squares of the other two sides.

Lemoine also proved that if lines are drawn through the Lemoine point parallel to the sides of the triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, or that they lie on a circle.cite book|edition=2|title=College Geometry|author=Nathan Altshiller Court|publisher=Barnes and Noble|city=New York|year=1969] This circle is now known as the first Lemoine circle, or simply the Lemoine circle.cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lemoine.html|title=Émile Michel Hyacinthe Lemoine|accessdate=2008-02-26|author=J.J. O'Connor and E.F. Robertson|publisher=MacTutor] [cite book|title=An Elementary Treatise on Modern Pure Geometry|last=Lachlan|first=Robert|publisher=Cornell University Library|date=1893-01-01|isbn=978-1429700504]

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. [Lemoine, Émile. "La Géométrographie ou l'art des constructions géométriques". (1903), Scientia, Paris (in French)]

The "simplicity" of a construction could be measured by the number of its operations. In his paper, he discussed as an example the Apollonius problem originally posed by Apollonius of Perga during the Hellenistic period; the method of constructing a circle tangent to three given circles. The problem had already been solved by Joseph Diaz Gergonne in 1816 with a construction of simplicity 400, but Lemoine's presented solution had simplicity 199.cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lemoine.html|title=Émile Michel Hyacinthe Lemoine|accessdate=2008-02-26|author=J.J. O'Connor and E.F. Robertson|publisher=MacTutor] [Eric W. Weisstein "CRC Concise Encyclopedia of Mathematics" (CRC Press, 1999), 733–4.] Simpler solutions such as those by Frederick Soddy in 1936 and by David Eppstein in 2001 are now known to exist. [cite journal|url=http://www.ajur.uni.edu/v3n1/Gisch%20and%20Ribando.pdf|date=2004-02-29|accessdate=2008-04-16|coauthors=David Gisch and Jason M. Ribando|publisher=University of Northern Iowa|title=Apollonius’ Problem: A Study of Solutions and Their Connections|journal=American Journal of Undergraduate Research|volume=3|number=1]

Lemoine's conjecture and extensions

In 1894, Lemoine stated empirically what is now known as Lemoine's conjecture on primes: Every odd number which is greater than three can be expressed in the form "2p + q" where "p" and "q" are prime. [cite book|authorlink=Leonard Dickson|first=Leonard E.|last=Dickson|title=History of the Theory of Numbers|volumes=4|volume=I|page=424] In 1985, John Kiltinen and Peter Young conjectured an extension of the conjecture which they called the "refined Lemoine conjecture". They published the conjecture in a journal of the Mathematical Association of America: "For any odd number "m" which is at least 9, there are odd prime numbers "p", "q", "r" and "s" and 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." [cite journal|journal=Mathematics Magazine|year=1984|coauthors=John Kiltinen and Peter Young|month=September|title=Goldbach, Lemoine, and a Know/Don't Know Problem|volume=48|number=4|pages=195–203|publisher=Mathematical Association of America]

Role in modern triangle geometry

Lemoine has been described by Nathan Court as a co-founder (along with Henri Brocard and Joseph Neuberg) of modern triangle geometry, a term used by William Gallatly, among others. In this context, "modern" is used to refer to geometry developed from the late 18th century onward. Such geometry relies on the abstraction of figures in the plane rather than analytic methods used earlier involving specific angle measures and distances. The geometry focuses on topics such as collinearity, concurrency, and concyclicity, as they do not involve the measures listed previously. [Steve Sigur (1999). [http://www.paideiaschool.org/Teacherpages/Steve_Sigur/resources/modern%20geo%20of%20triangle.pdf The Modern Geometry of the Triangle] (PDF). Paideiaschool.org. Retrieved on 2008-04-16.]

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.cite book|title=The Modern Geometry of the Triangle|first=William|last=Gallatly|pages=79|publisher=Scholarly Publishing Office|year=2005|month=December |isbn=978-1418178451]

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)

ee also


*Brocard circle
*Brocard points
*Nagel point
*Tarry point

Notes

External links

*
* [http://ens.math.univ-montp2.fr/SPIP/-Emile-Lemoine- Lemoine at the University of Montpellier] Persondata
NAME = Lemoine, Émile Michel Hyacinthe
ALTERNATIVE NAMES =
SHORT DESCRIPTION = French mathematician (geometer in particular) and civil engineer.
DATE OF BIRTH = November 12 1840
PLACE OF BIRTH = Quimper, France
DATE OF DEATH = February 21 1912
PLACE OF DEATH = Paris, France


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