Émile Michel Hyacinthe Lemoine | |
---|---|

Born |
Quimper, France | 22 November 1840

Died | 21 February 1912
Paris, France | (aged 71)

Alma mater | École Polytechnique |

Known for | Lemoine point, other geometric work |

Scientific career | |

Fields | Mathematics, engineering |

Institutions | École Polytechnique |

Doctoral advisor |
Charles-Adolphe Wurtz J. Kiœs |

**Émile Michel Hyacinthe Lemoine** (French:
[emil ləmwan]; 22 November 1840 – 21 February 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 Altshiller Court's *College Geometry*. Additionally, he founded a mathematical
journal titled, *
L'Intermédiaire des Mathématiciens*.

Lemoine was born in
Quimper, Finistère, on 22 November 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.^{
[1]}

Lemoine was accepted into the
École Polytechnique in Paris at the age of twenty, the same year as his father's death.^{
[2]}^{
[3]} As a student there, Lemoine, a presumed
trumpet player,^{
[4]} helped to found an influential
chamber music society called
La Trompette, for which
Camille Saint-Saëns composed several pieces, including the
Septet for trumpet, string quintet and piano. 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.^{
[1]} 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.^{
[1]} 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.^{
[5]}

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.^{
[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]}

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.^{
[1]} Instead of the original idea, Lemoine proposed a simplification of the construction process to a number of basic operations with the compass and straightedge.^{
[8]} 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.^{
[9]} 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).^{
[1]}

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).^{
[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]}

Lemoine's work has been said to contribute towards laying the foundation of modern
triangle geometry.^{
[10]} 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] ..."^{
[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]}

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.^{
[14]} This circle is now known as the first
Lemoine circle, or simply the Lemoine circle.^{
[2]}^{
[15]}

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
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 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]}

In 1894, Lemoine stated what is now known as
Lemoine's conjecture: Every
odd number which is greater than three can be expressed in the form *2p + q* where *p* and *q* are
prime.^{
[19]} 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 = 2 ^{j} + r* and

Lemoine has been described by
Nathan Altshiller Court as a co-founder (along with
Henri Brocard and
Joseph Neuberg) of modern triangle geometry, a term used by William Gallatly, among others.^{
[14]} In this context, "modern" is used to refer to geometry developed from the late 18th century onward.^{
[21]} 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.^{
[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]}

*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.

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Smith, David Eugene (1896). "Biography of Émile-Michel-Hyacinthe Lemoine".*American Mathematical Monthly*.**3**(2): 29–33. doi: 10.2307/2968278. JSTOR 2968278. - ^
^{a}^{b}^{c}^{d}^{e}O'Connor, J.J.; Robertson, E.F. "Émile Michel Hyacinthe Lemoine". MacTutor. Retrieved 2008-02-26. **^**"École Polytechnique - 208 years of history". École Polytechnique. Archived from the original on April 5, 2008. Retrieved 2008-03-21.**^**Charles Lenepveu. Letter to Émile Lemoine. February 1890. The Morrison Foundation for Musical Research. Retrieved on 2008-05-19**^**Kimberling, Clark. "Émile Michel Hyacinthe Lemoine (1840–1912), geometer". University of Evansville. Retrieved 2008-02-25.- ^
^{a}^{b}Gentry, F.C. (December 1941). "Analytic Geometry of the Triangle".*National Mathematics Magazine*. Mathematical Association of America.**16**(3): 127–40. doi: 10.2307/3028804. JSTOR 3028804. **^**Weisse, K.; Schreiber, P. (1989). "Zur Geschichte des Lemoineschen Punktes".*Beiträge zur Geschichte, Philosophie und Methodologie der Mathematik*(in German). Wiss. Z. Greifswald. Ernst-Moritz-Arndt-Univ. Math.-Natur. Reihe.**38**(4): 73–4.**^**Greitzer, S.L. (1970).*Dictionary of Scientific Biography*. New York: Charles Scribner's Sons.**^**Coolidge, Julian L. (1980).*A History of Geometrical Methods*. Oxford: Dover Publications. p. 58. ISBN 0-486-49524-8.**^**Kimberling, Clark. "Triangle Geometers". University of Evansville. Archived from the original on 2008-02-16. Retrieved 2008-02-25.**^**"Disseminate".*Bulletin of the American Mathematical Society*. American Mathematical Society.**9**(5): 272–5. 1903. doi: 10.1090/S0002-9904-1903-00993-8. Retrieved 2008-04-24.**^**"Notes" (PDF).*Bulletin of the American Mathematical Society*. American Mathematical Society.**18**(8): 424. 1912. doi: 10.1090/S0002-9904-1912-02239-5. Retrieved 2008-05-11.**^**"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.- ^
^{a}^{b}^{c}Nathan Altshiller Court (1969).*College Geometry*(2 ed.). New York: Barnes and Noble. ISBN 0-486-45805-9. **^**Lachlan, Robert (1893-01-01).*An Elementary Treatise on Modern Pure Geometry*. Cornell University Library. ISBN 978-1-4297-0050-4.**^**Lemoine, Émile.*La Géométrographie ou l'art des constructions géométriques*. (1903), Scientia, Paris (in French)**^**Eric W. Weisstein*CRC Concise Encyclopedia of Mathematics*(CRC Press, 1999), 733–4.**^**David Gisch and Jason M. Ribando (2004-02-29). "Apollonius' Problem: A Study of Solutions and Their Connections" (PDF).*American Journal of Undergraduate Research*. University of Northern Iowa.**3**(1). Retrieved 2008-04-16.`{{ cite journal}}`

: CS1 maint: uses authors parameter ( link)**^**Dickson, Leonard E. (1971).*History of the Theory of Numbers*(4 volumes). Vol. 1. S.l.: Chelsea. p. 424. ISBN 0-8284-0086-5.**^**John Kiltinen and Peter Young (September 1984). "Goldbach, Lemoine, and a Know/Don't Know Problem".*Mathematics Magazine*. Mathematical Association of America.**58**(4): 195–203. doi: 10.2307/2689513. JSTOR 2689513.`{{ cite journal}}`

: CS1 maint: uses authors parameter ( link)- ^
^{a}^{b}Gallatly, William (December 2005).*The Modern Geometry of the Triangle*. Scholarly Publishing Office. p. 79. ISBN 978-1-4181-7845-1. **^**Steve Sigur (1999). The Modern Geometry of the Triangle (PDF). Paideiaschool.org. Retrieved on 2008-04-16.

- O'Connor, John J.;
Robertson, Edmund F.,
"Émile Lemoine",
*MacTutor History of Mathematics archive*, University of St Andrews - Works by or about Émile Lemoine at Internet Archive