Background

August Ferdinand Möbius was born on November 17, 1790, in Schulpforta, Naumburg, Germany to the family of Johann Heinrich Möbius, a dancing teacher, who died when August was three years old and Johanne Katharine Christiane Keil. His mother was a descendant of Martin Luther.

Education

Möbius was educated at home until he was 13 years old when already showing an interest in mathematics, he went to the College in Schulpforta in 1803.

In 1809 Möbius graduated from his College and he became a student at the University of Leipzig. His family had wanted him to study law and indeed he started to study this topic. However, he soon discovered that it was not a subject that gave him satisfaction and in the middle of his first year of study he decided to follow his own preferences rather than those of his family. He, therefore, took up the study of mathematics, astronomy, and physics.

The teacher who influenced Möbius most during his time at Leipzig was his astronomy teacher Karl Mollweide. Although an astronomer, Mollweide is well known for a number of mathematical discoveries, in particular, the Mollweide trigonometric relations he discovered in 1807-1809 and the Mollweide map projection which preserves areas.

In 1813 Möbius traveled to Göttingen where he studied astronomy under Gauss. Gauss was the director of the Observatory in Göttingen but of course the greatest mathematician of his day, so again Möbius studied under an astronomer whose interests were mathematical. From Göttingen Möbius went to Halle where he studied under Johann Pfaff, Gauss's teacher. Under Pfaff, he studied mathematics rather than astronomy so by this stage Möbius was very firmly working in both fields.

In 1815 Möbius wrote his doctoral thesis on the occultation of fixed stars and began work on his Habilitation thesis. In fact, while he was writing this thesis there was an attempt to draft him into the Prussian army. He avoided the army and completed his Habilitation thesis on Trigonometrical equations.

Career

Mollweide's interest in mathematics was such that he had moved from astronomy to the chair of mathematics at Leipzig so Möbius had high hopes that he might be appointed to a professorship in astronomy at Leipzig. Indeed he was appointed to the chair of astronomy and higher mechanics at the University of Leipzig in 1816. His initial appointment was as Extraordinary Professor and it was an appointment that came early in his career.

However, Möbius did not receive a quick promotion to full professor. It would appear that he was not a particularly good lecturer and this made his life difficult since he did not attract fee-paying students to his lectures. He was forced to advertise his lecture courses as being free of charge before students thought his courses worth taking.

Möbius was offered a post as an astronomer in Greifswald in 1816 and then a post as a mathematician at Dorpat in 1819. He refused both, partly through his belief in the high quality of Leipzig University, partly through his loyalty to Saxony. In 1825 Mollweide died and Möbius hoped to transfer to his chair of mathematics taking the route Mollweide had taken earlier. However, it was not to be and another mathematician was preferred for the post.

By 1844, Möbius's reputation as a researcher-led to an invitation from the University of Jena and at this stage, the University of Leipzig gave him the Full Professorship in astronomy which he clearly deserved.

From the time of his first appointment at Leipzig Möbius had also held the post of Observer at the Observatory at Leipzig. He was involved in the rebuilding of the Observatory and, from 1818 until 1821, he supervised the project. He visited several other observatories in Germany before making his recommendations for the new Observatory. In 1848 he became director of the Observatory.

In 1844 Grassmann visited Möbius. He asked Möbius to review his major work Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) which contained many results similar to Möbius's work. However, Möbius did not understand the significance of Grassmann's work and did not review it. He did, however, persuade Grassmann to submit work for a prize and, after Grassmann won the prize, Möbius did write a review of his winning entry in 1847.

Although his most famous work is in mathematics, Möbius did publish important work on astronomy. He wrote De Computandis Occultationibus Fixarum per Planetas (1815) concerning occultations of the planets. He also wrote on the principles of astronomy, Die Hauptsätze der Astronomie (1836) and on celestial mechanics Die Elemente der Mechanik des Himmels (1843).

Möbius's mathematical publications, although not always original, were effective and clear presentations. Almost all Möbius's work was published in Crelle's Journal, the first journal devoted exclusively to publishing mathematics. Möbius's 1827 work Der barycentrische Calcul, on analytical geometry, became a classic and includes many of his results on projective and affine geometry. In it, he introduced homogeneous coordinates and also discussed geometric transformations, in particular projective transformations. He introduced a configuration now called a Möbius net, which was to play an important role in the development of projective geometry.

Möbius's name is attached to many important mathematical objects such as the Möbius function which he introduced in the 1831 paper Über eine besondere Art von Umkehrung der Reihen and the Möbius inversion formula.

In 1837 he published Lehrbuch der Statik which gives a geometric treatment of statics. It led to the study of systems of lines in space.

Achievements

• August Möbius is best remembered for his developments in topology, including the one-sided figure since known as the "Möbius strip." The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates into projective geometry. He is recognized for the introduction of the Barycentric coordinate system.
  
  Many mathematical concepts are named after him, including the Möbius plane, the Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

Works

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  • Gesammelte Werke, Vol. 1 1855
  • Gesammelte Werke, Vol. 2 1861
  • Gesammelte Werke, Vol. 3 1886
  • Lehrbuch der Statik, Vol. 1 1837
  • Lehrbuch der Statik, Vol. 2 1857
  • Der Barycentrische Calcul: Ein Neues Hülfsmittel Zur Analytischen Behandlung Der Geometrie 1827
  • Die Elemente Der Mechanik des Himmels 1843
  • Astronomie: Groesse, Bewegung und Entfernung der Himmelskoerper 1906
  • Die Hauptsätze der Astronomie 1868
  • Die Wahre Und Die Scheinbare Bahn Des Halley'schen Kometen Bei Seiner Weiderkunst Im Jahre 1835 1836
  • Beobachtungen Auf Der Königlichen Universitäts-sternwarte Zu Leipzig 1823
  • De Minima Variatione Azimuthi Stellarum Circulos Parallelos Uniformiter Describentium Commentatio 1816
  • De Computandis Occultationibus Fiscarum Per Planetas 1815
  • De peculiaribus quibusdam aequationum trigomometricarum affectionibus disquisitio analytica: Quam auctoritate amplissimi philosophorum lipsiensium ordinis die XIX april.a.MDCCCXV 1815

Religion

Although it is known that Möbius was a descendant of Martin Luther there is no direct information on him being Lutheran and no information on his religious views.

Politics

Möbius was not involved in politics.

Views

Before the question on the four colorings of maps had been asked by Francis Guthrie, Möbius had posed the following, rather easy, problem in 1840. "There was once a king with five sons. In his will, he stated that on his death his kingdom should be divided by his sons into five regions in such a way that each region should have a common boundary with the other four. Can the terms of the will be satisfied?" The answer, of course, is negative and easy to show. However, it does illustrate Möbius's interest in topological ideas, an area in which he is most remembered as a pioneer. In a memoir, presented to the Académie des Sciences and only discovered after his death, he discussed the properties of one-sided surfaces including the Möbius strip which he had discovered in 1858. This discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Académie.

Although it is known as a Möbius strip today it was not Möbius who first described this object, rather by any criterion, either publication date or date of first discovery, precedence goes to Listing.

A Möbius strip is a two-dimensional surface with only one side. It can be constructed in three dimensions as follows. Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180-degree twist. It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A.

Membership

August Ferdinand Möbius was a member of the Saxon Academy of Sciences, the Royal Prussian Academy of Sciences (now Berlin-Brandenburg Academy of Sciences and Humanities), and the French Academy of Sciences.

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  Saxon Academy of Sciences , Germany

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  Royal Prussian Academy of Sciences , Germany

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  French Academy of Sciences , France

Personality

Möbius was a quiet, thoughtful, reserved, meticulous person. He virtually spent his whole life in one city, and one building.

Connections

August Ferdinand Möbius married Dorothea Christiana Juliane Rothe, who was completely blind. She died nine years before him. They had four children: August Theodor Möbius, Emilie Auguste Möbius, Paul Heinrich August Möbius, and Carl Theodor Möbius.

Father:

Johann Heinrich Möbius

Mother:

Johanne Katharine Christiane Keil

ancestor:

Martin Luther

Wife:

Dorothea Christiana Juliane Rothe

Son:

August Theodor Möbius

Daughter:

Emilie Auguste Möbius

Son:

Paul Heinrich August Möbius

Son:

Carl Theodor Möbius

son-in-law:

Heinrich d'Arrest

Heinrich Louis d'Arrest married Emilie Auguste Möbius.

teacher:

Karl Gauss

teacher:

Johann Pfaff

mentor:

Karl Mollweide

References

• Dictionary of Scientific Biography: Volumes 9 & 10
  1981
• August Ferdinand Möbius: Entre polyèdres et corrélation élémentaire
  2013
• The Mobius Strip: Dr. August Mobius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
  2006
• Mobius and his Band: Mathematics and Astronomy in Nineteenth-Century Germany
  1993