In 1809 Möbius graduated from his College and he became a student at the University of Leipzig. His family had wanted him 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 him 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 1807-09 and the Mollweide map projection which preserves areas.
In 1813 Möbius travelled 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. Möbius wrote
This is the most horrible idea I have heard of, and anyone who shall venture, dare, hazard, make bold and have the audacity to propose it will not be safe from my dagger.He avoided the army and completed his Habilitation thesis on Trigonometrical equations. 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 which came early in his career.
However Möbius did not receive 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.
He 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 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 1820 he married and he was to have one daughter and two sons. 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. His contributions to mathematics are described by his biographer Richard Baltzer in  as follows:-
The inspirations for his research he found mostly in the rich well of his own original mind. His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way. He worked without hurrying, quietly on his own. His work remained almost locked away until everything had been put into its proper place. Without rushing, without pomposity and without arrogance, he waited until the fruits of his mind matured. Only after such a wait did he publish his perfected works...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.
Before the question on the four colouring 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. See THIS LINK. This discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Académie.
Although we know this 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.
Article by: J J O'Connor and E F Robertson