Pierre René Jean Baptiste Henri Brocard (12 May 1845 – 16 January 1922) was a French

meteorologist A meteorologist is a scientist who studies and works in the field of meteorology aiming to understand or predict Earth's atmospheric phenomena including the weather. Those who study meteorological phenomena are meteorologists in research, while t ...

and

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 ...

, in particular a

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

. His best-known achievement is the invention and discovery of the properties of the

Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

, the

Brocard circle In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this s ...

, and the

Brocard triangle In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points construc ...

, all bearing his name. Contemporary mathematician Nathan Court wrote that he, along with Émile Lemoine 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 ...

, was one of the three co-founders of modern

triangle geometry A triangle is a polygon with three edges and three 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, any three points, when non-collin ...

. He is listed as an Emeritus at the International Academy of Science, was awarded the

Ordre des Palmes Académiques A suite, in Western classical music and jazz, is an ordered set of instrumental or orchestral/concert band pieces. It originated in the late 14th century as a pairing of dance tunes and grew in scope to comprise up to five dances, sometimes with ...

, and was an officer of the

Légion d'honneur The National Order of the Legion of Honour (french: Ordre national de la Légion d'honneur), formerly the Royal Order of the Legion of Honour ('), is the highest French order of merit, both military and civil. Established in 1802 by Napoleon ...

. He spent most of his life studying

meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...

as an officer in the

French Navy The French Navy (french: Marine nationale, lit=National Navy), informally , is the maritime arm of the French Armed Forces and one of the five military service branches of France. It is among the largest and most powerful naval forces in t ...

, but seems to have made no notable original contributions to the subject.

Biography

Early years

Pierre René Jean Baptiste Henri Brocard was born on 12 May 1845, in Vignot (a part of

Commercy Commercy () is a commune in the Meuse department in Grand Est in north-eastern France. The 18th-century Lorraine historian Nicolas Luton Durival (1713–1795) was born in Commercy. History Commercy dates back to the 9th century, and at that time ...

Meuse The Meuse ( , , , ; wa, Moûze ) or Maas ( , ; li, Maos or ) is a major European river, rising in France and flowing through Belgium and the Netherlands before draining into the North Sea from the Rhine–Meuse–Scheldt delta. It has a t ...

to Elizabeth Auguste Liouville and Jean Sebastien Brocard. He attended the

Lycée In France, secondary education is in two stages: * ''Collèges'' () cater for the first four years of secondary education from the ages of 11 to 15. * ''Lycées'' () provide a three-year course of further secondary education for children between ...

Marseilles Marseille ( , , ; also spelled in English as Marseilles; oc, Marselha ) is the prefecture of the French department of Bouches-du-Rhône and capital of the Provence-Alpes-Côte d'Azur region. Situated in the camargue region of southern Franc ...

as a young child, and then the

Strasbourg Strasbourg (, , ; german: Straßburg ; gsw, label=Bas Rhin Alsatian, Strossburi , gsw, label=Haut Rhin Alsatian, Strossburig ) is the prefecture and largest city of the Grand Est region of eastern France and the official seat of the Eu ...

. After graduating from the Lycée he entered the

Academy An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy ...

where he was prepared for the examination for entrance to the prestigious

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 ...

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 ...

, to which he was accepted in 1865.

École Polytechnique and military years

Brocard attended the École Polytechnique from 1865 to 1867. As was the norm at the time, he, after graduation, became a technical

officer An officer is a person who has a position of authority in a hierarchical organization. The term derives from Old French ''oficier'' "officer, official" (early 14c., Modern French ''officier''), from Medieval Latin ''officiarius'' "an officer," fro ...

in the

French military The French Armed Forces (french: Forces armées françaises) encompass the Army, the Navy, the Air and Space Force and the Gendarmerie of the French Republic. The President of France heads the armed forces as Chief of the Armed Forces. France ...

, which had been reorganized in 1866. He acted as a

in the

French navy The French Navy (french: Marine nationale, lit=National Navy), informally , is the maritime arm of the French Armed Forces and one of the five military service branches of France. It is among the largest and most powerful naval forces in t ...

, and general technician as well. Brocard soon saw active service, as

Napoleon III Napoleon III (Charles Louis Napoléon Bonaparte; 20 April 18089 January 1873) was the first President of France (as Louis-Napoléon Bonaparte) from 1848 to 1852 and the last monarch of France as Emperor of the French from 1852 to 1870. A nephew ...

declared

war War is an intense armed conflict between states, governments, societies, or paramilitary groups such as mercenaries, insurgents, and militias. It is generally characterized by extreme violence, destruction, and mortality, using regular o ...

upon

Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...

. Brocard was one of the 120,000 men under

Marshal MacMahon Marie Edme Patrice Maurice de MacMahon, marquis de MacMahon, duc de Magenta (; 13 June 1808 – 17 October 1893) was a French general and politician, with the distinction of Marshal of France. He served as Chief of State of France from 1873 to 1 ...

led to

Metz Metz ( , , lat, Divodurum Mediomatricorum, then ) is a city in northeast France located at the confluence of the Moselle and the Seille rivers. Metz is the prefecture of the Moselle department and the seat of the parliament of the Grand E ...

to free the

French army The French Army, officially known as the Land Army (french: Armée de Terre, ), is the land-based and largest component of the French Armed Forces. It is responsible to the Government of France, along with the other components of the Armed For ...

of the

Rhine ), Surselva, Graubünden, Switzerland , source1_coordinates= , source1_elevation = , source2 = Rein Posteriur/Hinterrhein , source2_location = Paradies Glacier, Graubünden, Switzerland , source2_coordinates= , so ...

. The French army, however, was defeated on 31 August at the

Battle of Sedan The Battle of Sedan was fought during the Franco-Prussian War from 1 to 2 September 1870. Resulting in the capture of Emperor Napoleon III and over a hundred thousand troops, it effectively decided the war in favour of Prussia and its allies, ...

, and was taken prisoner along with approximately 83,000 other combatants.

Middle years

After Brocard was freed, he returned to his military position and continued teaching, publishing his mathematical articles in the most popular mathematical journal of that time, (also called ''Nouvelles annales mathématiques''). He joined the

Société Mathématique de France Lactalis is a French multinational dairy products corporation, owned by the Besnier family and based in Laval, Mayenne, France. The company's former name was Besnier SA. Lactalis is the largest dairy products group in the world, and is the sec ...

in 1873, just a year after its founding. In 1875, he was inducted into the French Association for the Advancement of Science as well as the French Meteorological Society. He was shortly after sent to northern

Africa Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area ...

, where he served as a military technician for the French forces stationed in

Algiers Algiers ( ; ar, الجزائر, al-Jazāʾir; ber, Dzayer, script=Latn; french: Alger, ) is the capital and largest city of Algeria. The city's population at the 2008 Census was 2,988,145Census 14 April 2008: Office National des Statistiques ...

, the seat of French Africa. While in Algiers, Brocard founded the Meteorological Institute of Algiers. Brocard also visited

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 ...

while in northern Africa, which was occupied by the French in 1831.

Discovery of Brocard points

During a meeting of the French Association for the Advancement of Science, Brocard presented a self-written article entitled ''Etudes d'un nouveau cercle du plan du triangle'', his first paper on the

, the

, and the

, all of which today bear his name.

Later years

In 1884 Brocard returned to France. He served with the Meteorological Commission in

Montpellier Montpellier (, , ; oc, Montpelhièr ) is a city in southern France near the Mediterranean Sea. One of the largest urban centres in the region of Occitania (administrative region), Occitania, Montpellier is the prefecture of the Departments of ...

before moving to

Grenoble lat, Gratianopolis , commune status = Prefecture and commune , image = Panorama grenoble.png , image size = , caption = From upper left: Panorama of the city, Grenoble’s cable cars, place Saint- ...

and lastly

Bar-le-duc Bar-le-Duc (), formerly known as Bar, is a commune in the Meuse département, of which it is the capital. The department is in Grand Est in northeastern France. The lower, more modern and busier part of the town extends along a narrow valley, sh ...

. He honorably retired from the French military in 1910 as a

lieutenant colonel Lieutenant colonel ( , ) is a rank of commissioned officers in the armies, most marine forces and some air forces of the world, above a major and below a colonel. Several police forces in the United States use the rank of lieutenant colone ...

. His remaining two major publications were ''Notes de bibliographie des courbes géométriques'' (1897, 1899, published in two volumes) and the ''Courbes géométriques remarquables'' (1920, posthumous 1967, also published in two volumes) which was written in collaboration with T. Lemoyne. Brocard attended the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

at Zurich in 1897,

in 1900,

Heidelberg Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...

in 1904,

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

in 1908,

Cambridge, England Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge became ...

in 1912, and

in 1920. Brocard spent the last years of his life in

Bar-le-Duc Bar-le-Duc (), formerly known as Bar, is a commune in the Meuse département, of which it is the capital. The department is in Grand Est in northeastern France. The lower, more modern and busier part of the town extends along a narrow valley, sh ...

. He was offered the presidency of Bar-le-Duc's ''Letters, Sciences, and Arts Society'', of which he had been a longtime member and correspondent for several foreign academies of, but declined. He died on 16 January 1922 while on a trip to

Kensington Kensington is a district in the Royal Borough of Kensington and Chelsea in the West End of London, West of Central London. The district's commercial heart is Kensington High Street, running on an east–west axis. The north-east is taken up b ...

London London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a majo ...

England England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...

Contributions

Brocard triangle, Brocard circle, and Brocard points

Brocard's most well-known contributions to mathematics are the

Brocard point In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

s, the

, and the

. The positive Brocard point (sometimes known as the first Brocard point) of a Euclidean plane

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 ...

is the interior

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

of the triangle for which the three angles formed by two of the vertices and the point are equal. Their common value is the Brocard angle of the triangle. The Brocard circle of the triangle is a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

having a

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

of the

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

between the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

and symmedian. It contains the

. The Brocard triangle of a

is a triangle formed by the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

line 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 ...

from a vertex to its corresponding

and a line from another vertex to its corresponding Brocard point and the other two points constructed using different combinations of vertices and Brocard points. The Brocard triangle is inscribed in the

Other mathematical contributions

Brocard published various other papers on

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

during his time at

, none of which became as well known as ''Etudes d'un nouveau cercle du plan du triangle''. He also conjured up a famous unsolved problem thought to have no answers other than the three he provided. This problem is called

Brocard's Problem Brocard's problem is a problem in mathematics that asks to find integer values of n and m for which n!+1 = m^2, where n! is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa ...

. One other achievement of his is guessing at the meaning of the cryptic title of one of

Girard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are ...

' papers, ''DALG''. In his paper ''Analyse d'autographes et autres écrits de Girard Desargues'', he surmised that it stood for ''Des Argues, Lyonnais, Géometre'', which is the generally accepted title.

Meteorology

Though Brocard made no major notable original discoveries in

, he founded the Meteorological Institute in

and served as a meteorological technician during his time in the French military. He also published several notable papers on meteorology.

Notes

References

* Reprinted in * Guggenbuhl, Laura. (1970–90). "Brocard, Pierre René Jean-Baptiste Henri" in ''Dictionary of Scientific Biography'' (New York). *"Henri Brocard" in ''Encyclopaedia of Mathematics: An Updated and Annotated Translation''. * * {{DEFAULTSORT:Brocard, Henri 1845 births 1922 deaths People from Meuse (department) 19th-century French mathematicians 20th-century French mathematicians French geometers French meteorologists École Polytechnique alumni Recipients of the Ordre des Palmes Académiques

Biography

Early years

École Polytechnique and military years

Middle years

Discovery of Brocard points

Later years

Contributions

Brocard triangle, Brocard circle, and Brocard points

Other mathematical contributions

Meteorology

See also

Notes

References