Caspar Wessel (8 June 1745, Vestby – 25 March 1818,

Copenhagen Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan a ...

) was a Danish– Norwegian

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 ...

and

cartographer Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an ...

. In 1799, Wessel was the first person to describe the

geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

interpretation of complex numbers as points in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

and vectors.

Biography

Wessel was born in Jonsrud, Vestby,

Akershus Akershus () is a traditional region and current electoral district in Norway, with Oslo as its main city and traditional capital. It is named after the Akershus Fortress in Oslo. From the middle ages to 1919, Akershus was a fief and main cou ...

in Norway to pastor Jonas Wessel (1707 - 85) and Helene Marie Schumacher (1715 - 89). Helene came from

Drammen Drammen () is a city and municipality in Viken, Norway. The port and river city of Drammen is centrally located in the south-eastern and most populated part of Norway. Drammen municipality also includes smaller towns and villages such as Konne ...

and was of Griffenfeldts heritage. Growing up in Wesselstuen in Vestby he was one of thirteen children in a family. His brothers include

Johan Herman Wessel Johan Herman Wessel (6 October 1742 – 29 December 1785) was an 18th-century Danish-Norwegian poet, satirist and playwright. His written work was characterized by the use of parody and satiric wit. Biography Wessel was born and raised at Ves ...

and

Ole Christopher Wessel Ole Christopher Wessel (12 January 1744 – 26 December 1794) was a Norwegian jurist, civil servant and landowner. He was born in Vestby in Akershus as the son of parish priest Jonas Wessel (1707-1785) and Helene Marie Schumacher (1715-1789 ...

. Wessel is the grandnephew of the nobleman Peter Tordenskjold, brother in law of Maren Juel and the great-great-great-grandfather to the famous radio engineer Jan Wessel. In 1763, having completed secondary school at Oslo Cathedral School, he went to Denmark for further studies. He attended the

University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in ...

to study law, but due to financial pressures, could do so for only a year. To survive, he became an assistant

land surveyor Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ...

to his brother and they worked on the

Royal Danish Academy of Sciences and Letters {{Infobox organization , name = The Royal Danish Academy of Sciences and Letters , full_name = , native_name = Det Kongelige Danske Videnskabernes Selskab , native_name_lang = , logo = Royal ...

' topographical survey of Denmark. This was not enough, however, and he took on extra work as a

. He worked as a surveyor for the rest of his life, stopping only for a sabbatical year in 1778 to finish his

law degree A law degree is an academic degree conferred for studies in law. Such degrees are generally preparation for legal careers. But while their curricula may be reviewed by legal authority, they do not confer a license themselves. A legal license is gra ...

. By 1798 Wessel had risen to the supervisory role of Royal Inspector of Surveying. It was the mathematical aspect of surveying that led him to exploring the geometrical significance of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s. His fundamental paper, ''Om directionens analytiske betegning,'' was presented in 1797 to the

. Since it was in Danish and published in a journal rarely read outside of Denmark, it went unnoticed for nearly a century. The same results were independently rediscovered by Jean-Robert Argand in 1806 and

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

in 1831.It was first when a Danish student wrote a doctoral dissertation on "Danish mathematics in the 18th century" that the universal significance of Wessel was brought up. While the student didn't realize the contribution by himself, the supervisors did. The dissertation created a great uproar and resulted in a front-page article in

Dagbladet ''Dagbladet'' (lit.: ''The Daily Magazine'') is one of Norway's largest newspapers and is published in the tabloid format. It has 1,400,000 daily readers on mobile, web and paper. Traditionally ''Dagbladet'' is considered the main liberal newsp ...

and

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius S ...

taking action to raise awareness of Wessel. In fact, Lie wrote in the article that "It is now Norway's cause to do what should be done, that this strange Man's Memory may be brought forth by Oblivion and his Name find its proper place before the History of Mathematics." One of the more prominent ideas presented in " On the Analytical Representation of Direction" was that of vectors. Even though this was not Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added together, when one joins them together so that one begins where the other one ends, and next one draws a straight line from the first to the last point of the joined lines, and takes this to be their sum". This is the same idea as used today in summing vectors. Gedenkplakette für Caspar Wessel

Wessel's priority to the idea of a complex number as a point in the

is today universally recognised. His paper was re-issued in French translation in 1897, and in English in 1999 as ''On the analytic representation of direction'' (eds. Branner & Lützen). From 1764 to 1805, Wessel participated in the survey of Denmark,

Schleswig The Duchy of Schleswig ( da, Hertugdømmet Slesvig; german: Herzogtum Schleswig; nds, Hartogdom Sleswig; frr, Härtochduum Slaswik) was a duchy in Southern Jutland () covering the area between about 60 km (35 miles) north and 70 km ...

Holstein Holstein (; nds, label=Northern Low Saxon, Holsteen; da, Holsten; Latin and historical en, Holsatia, italic=yes) is the region between the rivers Elbe and Eider. It is the southern half of Schleswig-Holstein, the northernmost state of German ...

and the

Duchy of Oldenburg The Duchy of Oldenburg (german: Herzogtum Oldenburg)—named after its capital, the town of Oldenburg—was a state in the north-west of present-day Germany. The counts of Oldenburg died out in 1667, after which it became a duchy until 1810, w ...

. His map of Schleswig-Holstein was requested by

Napoleon Bonaparte Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader wh ...

Personal relations

Wessel had a modest and quiet personality, he was described by his brother Johan Herman as "He draws maps and reads the law. He's as diligent as I am lazy". No portraits of Wessel seem to have survived. In 1780 he married Cathrine Elisabeth Brinck f. Müller in

. On 1 July 1815, Wessel was made a knight of the

Order of the Dannebrog The Order of the Dannebrog ( da, Dannebrogordenen) is a Danish order of chivalry instituted in 1671 by Christian V. Until 1808, membership in the order was limited to fifty members of noble or royal rank, who formed a single class known ...

for his contributions to surveying; the cross of the Order of the Dannebrog was presented to him at the Royal Palace of Amalienborg on 4 August 1815.

Sources

A Source Book in Mathematics
by David Eugene Smith, has "essential" passages of ''"Om directionens analytiske betegning"'' translated into English (see page 55).

Biography

Personal relations

Sources

References

Further reading