Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) was a French

civil engineer A civil engineer is a person who practices civil engineering – the application of planning, designing, constructing, maintaining, and operating infrastructure while protecting the public and environmental health, as well as improving existing ...

and a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, a geometer in particular. He was educated at a variety of institutions, including the

Prytanée National Militaire The Prytanée national militaire is a French military school managed by the French military, offering regular secondary education as well as special preparatory classes, equivalent in level to the first years of university, for students who wish ...

and, most notably, the

École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...

. 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 In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. Ross Honsberger called its existence "one of the crown jewels of m ...

(or the symmedian point) of a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

. Other mathematical work includes a system he called ''Géométrographie'' and a method which related

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

ic 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 Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...

, and he also took an amateur's interest in

music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

. During his tenure at the École Polytechnique and as a civil engineer, Lemoine published several

papers Paper is a thin, flat material produced by the compression of fibres. Paper(s) or The Paper may also refer to: Publishing and academia * Newspaper, a periodical publication * ''Paper'' (magazine), an American monthly fashion and culture magazin ...

on mathematics, most of which are included in a fourteen-page section in

Nathan Altshiller Court Nathan Altshiller Court (1881-1968) was a Polish-American mathematician, a geometer in particular and author of the famous book ''College Geometry - An Introduction to the Modern Geometry of the Triangle and the Circle''. Biography Nathan Court w ...

's ''College Geometry''. Additionally, he founded a mathematical journal titled, '' L'Intermédiaire des Mathématiciens''.

Biography

Early years (1840–1869)

Lemoine was born in Quimper, Finistère, on 22 November 1840, the son of a retired

military A military, also known collectively as armed forces, is a heavily armed, highly organized force primarily intended for warfare. It is typically authorized and maintained by a sovereign state, with its members identifiable by their distinct ...

captain Captain is a title, an appellative for the commanding officer of a military unit; the supreme leader of a navy ship, merchant ship, aeroplane, spacecraft, or other vessel; or the commander of a port, fire or police department, election precinct, e ...

who had participated in the

campaigns Campaign or The Campaign may refer to: Types of campaigns * Campaign, in agriculture, the period during which sugar beets are harvested and processed *Advertising campaign, a series of advertisement messages that share a single idea and theme *Bl ...

of the

First French Empire The First French Empire, officially the French Republic, then the French Empire (; Latin: ) after 1809, also known as Napoleonic France, was the empire ruled by Napoleon Bonaparte, who established French hegemony over much of continental Eu ...

occurring after 1807. As a child, he attended the military Prytanée of

La Flèche La Flèche () is a town and commune in the French department of Sarthe, in the Pays de la Loire region in the Loire Valley. It is the sub-prefecture of the South-Sarthe, the chief district and the chief city of a canton, and the second most po ...

on a

scholarship A scholarship is a form of financial aid awarded to students for further education. Generally, scholarships are awarded based on a set of criteria such as academic merit, diversity and inclusion, athletic skill, and financial need. Scholarsh ...

granted because his father had helped found the school. During this early period, he published a journal

article Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: G ...

in ''

Nouvelles annales de mathématiques The ''Nouvelles Annales de Mathématiques'' (subtitled ''Journal des candidats aux écoles polytechnique et normale'') was a French scientific journal in mathematics. It was established in 1842 by Olry Terquem and Camille-Christophe Gerono, and con ...

'', discussing properties of the triangle. Lemoine was accepted into the

in Paris at the age of twenty, the same year as his father's death. As a student there, Lemoine, a presumed

trumpet The trumpet is a brass instrument commonly used in classical and jazz ensembles. The trumpet group ranges from the piccolo trumpet—with the highest register in the brass family—to the bass trumpet, pitched one octave below the standard ...

player, helped to found an influential

chamber music Chamber music is a form of classical music that is composed for a small group of instruments—traditionally a group that could fit in a palace chamber or a large room. Most broadly, it includes any art music that is performed by a small numb ...

society called La Trompette, for which

Camille Saint-Saëns Charles-Camille Saint-Saëns (; 9 October 183516 December 1921) was a French composer, organist, conductor and pianist of the Romantic music, Romantic era. His best-known works include Introduction and Rondo Capriccioso (1863), the Piano C ...

composed several pieces, including the

Septet A septet is a formation containing exactly seven members. It is commonly associated with musical groups but can be applied to any situation where seven similar or related objects are considered a single unit, such as a seven-line stanza of poetry. ...

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 The Second French Empire (; officially the French Empire, ), was the 18-year Empire, Imperial Bonapartist regime of Napoleon III from 14 January 1852 to 27 October 1870, between the French Second Republic, Second and the French Third Republic ...

. 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 École may refer to: * an elementary school in the French educational stages normally followed by secondary education Secondary education or post-primary education covers two phases on the International Standard Classification of Education scal ...

, teaching Uwe Jannsen at the same schools, and studying under

Charles-Adolphe Wurtz Charles Adolphe Wurtz (; 26 November 181710 May 1884) was an Alsatian French chemist. He is best remembered for his decades-long advocacy for the atomic theory and for ideas about the structures of chemical compounds, against the skeptical opinio ...

at the École des Beaux Arts and the École de Médecine. Lemoine also lectured at various scientific institutions in Paris and taught as a private

tutor TUTOR, also known as PLATO Author Language, is a programming language developed for use on the PLATO system at the University of Illinois at Urbana-Champaign beginning in roughly 1965. TUTOR was initially designed by Paul Tenczar for use in co ...

for a period before accepting an appointment as a professor at the École Polytechnique.

Middle years (1870–1887)

Ecole Polytechnique France seen from lake DSC03389

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. 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 Lille ( , ; nl, Rijsel ; pcd, Lile; vls, Rysel) is a city in the northern part of France, in French Flanders. On the river Deûle, near France's border with Belgium, it is the capital of the Hauts-de-France Regions of France, region, the Pref ...

. The central focus of this paper concerned the point which bears his name today. Most of the other results discussed in the paper pertained to various

concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line ar ...

that could be constructed from the Lemoine point. 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. In this career, he rose to the rank of chief

inspector Inspector, also police inspector or inspector of police, is a police rank. The rank or position varies in seniority depending on the organization that uses it. Australia In Australian police forces, the rank of inspector is generally the ne ...

, a position he held until 1896. As the chief inspector, he was responsible for the gas supply of the city.

Later years (1888–1912)

During his tenure as a civil engineer, Lemoine wrote a

treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject and its conclusions."Treat ...

concerning

compass and straightedge constructions In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

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. Instead of the original idea, Lemoine proposed a simplification of the construction process to a number of basic operations with the compass and straightedge. He presented this paper at a meeting of the ''Association Française'' in

Oran Oran ( ar, وَهران, Wahrān) is a major coastal city located in the north-west of Algeria. It is considered the second most important city of Algeria after the capital Algiers, due to its population and commercial, industrial, and cultural ...

Algeria ) , image_map = Algeria (centered orthographic projection).svg , map_caption = , image_map2 = , capital = Algiers , coordinates = , largest_city = capital , relig ...

in 1888. The paper, however, did not garner much enthusiasm or interest among the mathematicians gathered there. 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 An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, higher learning (and generally also research or honorary membershi ...

. He published additional papers on the subject in '' Mathesis'' (1888), ''Journal des mathématiques élémentaires'' (1889), ''

'' (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 Caen (, ; nrf, Kaem) is a commune in northwestern France. It is the prefecture of the department of Calvados. The city proper has 105,512 inhabitants (), while its functional urban area has 470,000, After this, Lemoine published another series of papers, including a series on what he called ''transformation continue'' (continuous transformation), which related mathematical

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

s to geometrical objects. This meaning stood separately from the modern definition of

transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...

. 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). 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. Lemoine died on 21 February 1912 in his home city of Paris.

Contributions

Lemoine's work has been said to contribute towards laying the foundation of modern triangle geometry. The ''

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...

'', in which much of Lemoine's work is published, declared that "To none of these eometersmore than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement f modern triangle geometrynbsp;..." At the annual meeting of the

Paris Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at th ...

in 1902, Lemoine received the 1,000- franc Francœur prize, which he held for several years.

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

symmedian In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corr ...

s of a triangle; the reflections of the

medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...

of the triangle over the angle bisectors. Other results in the paper included the idea that the symmedian from a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

of the triangle divides the opposite side into segments whose

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

is equal to the ratio of the

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

of the other two sides. Lemoine also proved that if

lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

are drawn through the Lemoine point

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

to the sides of the triangle, then the six points of intersection of the lines and the sides of the triangle are

concyclic In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line a ...

, or that they lie on a circle. This circle is now known as the first

Lemoine circle In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the cor ...

, or simply the Lemoine circle.

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. 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 In Euclidean geometry, Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his w ...

originally posed by

Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...

during the

Hellenistic period In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 3 ...

; the method of constructing a circle

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

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. Simpler solutions such as those by Frederick Soddy in 1936 and by David Eppstein in 2001 are now known to exist.

Lemoine's conjecture and extensions

In 1894, Lemoine stated what is now known as

Lemoine's conjecture In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. History ...

: Every

odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

which is greater than three can be expressed in the form ''2p + q'' where ''p'' and ''q'' are

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. 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 The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

: "For any odd number ''m'' which is at least 9, there are odd prime numbers ''p'', ''q'', ''r'' and ''s'' and

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

s ''j'' and ''k'' such that ''m = 2p + q'', ''2 + pq = 2^j + r'' and ''2q + p = 2^k + 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 Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...

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."

Role in modern triangle geometry

Lemoine has been described by

as a co-founder (along with Henri Brocard and

Joseph Neuberg Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926) was a Luxembourger mathematician who worked primarily in geometry. Biography Neuberg was born on 30 October 1840 in Luxembourg City, Luxembourg. He first studied at a local scho ...

) 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 Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

rather than analytic methods used earlier involving specific

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

measures Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Meas ...

and

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

s. The geometry focuses on topics such as collinearity,

concurrency Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

, and concyclicity, as they do not involve the measures listed previously.Steve Sigur (1999)
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

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

s 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 Gaston Tarry (27 September 1843 – 21 June 1913) was a French mathematician. Born in Villefranche de Rouergue, Aveyron, he studied mathematics at high school before joining the civil service in Algeria. He pursued mathematics as an amateur. In ...

wrote about similar points.

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) *

Notes

External links

* * {{DEFAULTSORT:Lemoine, Emile 1840 births 1912 deaths People from Quimper French civil engineers French geometers 19th-century French mathematicians 20th-century French mathematicians École Polytechnique alumni Members of the Ligue de la patrie française École Spéciale d'Architecture alumni