Ben Fisher's Möbius Report
Möbius's Work
In 1844, the mathematician Grassmann visited Möbius. Grassmann wanted him to review his major work Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, which was very similar to Möbius's work. However, Möbius didn't understand the significance of the work, and so he didn't review it. He persuaded Grassman to submit it for a prize, and when he won in 1847, Möbius did write a review of the winning entry.
By studying astronomy for such a long time, Möbius was able to 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.
Almost all of Möbius's mathematical work was published in Crelle's Journal (which was the first journal devoted just to publishing mathematics). Der barycentrische Calkul, which Möbius's wrote in1827 on analytical geometry, became a classic. It included many of his results on projective and affine geometry. He introduced homogeneous coordinates and discussed geometric transformations. Möbius introduced what is now called a Möbius net.
Möbius's name is attached to other important mathematical formulas such as the Möbius function, which he introduced in the 1831 paper Über eine besondere Art von Umkehrung der Reihen. He also invented 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.
Möbius posed the following 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 will can't be satisfied. This problem wasn't too hard, but his illustrates Möbius's interest in topological ideas, which I will discuss in the next page. Möbius's biographer said:
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.
