## ‘Ueber die Bestimmung des Inhaltes eines Polyëders’, pp. 31-68 in Berichte der Königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physisiche Classe 17 (1865). [With:] ‘Theorie der elementaren Verwandtschaft’, pp. 18-57 in ibid. 15 (1863).

Leipzig: S. Hirzel, 1866 [- 1864].

First edition, journal issues in the original printed wrappers, of two of the most important papers in the early history of topology, including the introduction (and illustration) in the 1865 paper of the famous ‘Möbius band’ (or ‘Möbius strip’). “August Möbius was one of the nineteenth century’s most influential mathematicians and astronomers” (Fauvel et al.). “Möbius first described the ‘Möbius band’ in a paper presented to the Paris Academy in 1861 as an entry to a competition on the theme “Improve in some important point the geometric theory of polyhedra.” Möbius’ paper, written in bad French and containing many new ideas, was not understood by the jury, and like the other papers submitted to the competition, was not awarded the prize. The contents of the paper were published by Möbius in his articles ‘Theorie der elementaren Verwandtschaft’ (1863) and ‘Ueber die Bestimmung des Inhaltes eines Polyëders’ (1865)” (Kolmogorov & Yushkevich, p. 101). From an examination of Möbius’s notebooks it is known that he discovered the Möbius strip in 1858; it was discovered independently in the same year by Johann Listing (who had coined the term ‘topology’ in 1847). On his discovery of a one-sided surface, Ian Stewart writes (Fauvel et al, p. 159): “It was typical that Möbius should notice a simple fact that anyone could have seen in the previous two thousand years – and typical that nobody did”. Norman Biggs (*ibid*., p. 112) speculates that both Listing and Möbius may have been influenced in their discovery by the great Carl Friedrich Gauss (1777-1855). Gauss, Listing and Möbius all worked for many years at Göttingen; Möbius studied under Gauss and Listing freely acknowledges that he was trying to develop the topological ideas of Gauss, who himself never published anything on the subject.

The Möbius band is a surface obtained by taking a strip of paper, giving one of the two ends a half twist, and then gluing together the two ends. Unlike the cylinder, obtained by joining the ends of the strip without a twist, the Möbius band has only one side. This is sometimes illustrated by saying that an ant walking around a Möbius band will return to its starting position but will be on the opposite side. This is famously illustrated in M. C. Escher’s woodcut ‘Möbius Strip II (Red Ants).’ A further surprising property of the Möbius band is that, whereas a cylinder has two boundary curves, the Möbius band has only one: if you start at any point on the boundary and move along you will pass through every point on the boundary before returning back to your starting place. The 1865 paper contains the first published illustration of a Möbius band.

In his 1865 paper, “Möbius pointed out that ‘having only one side,’ while intuitively clear, is difficult to make precise, and proposed a related property that could be defined in complete rigour. This property was ‘orientability’. A surface is ‘orientable’ if you can cover it with a network of triangles, with arrows circulating round each triangle, so that whenever two triangles have a common edge the arrows point in opposite directions. If you draw a network on a plane, for example, this is what happens. On a Möbius band, no such network exists” (Stewart). By gluing the circular edge of a disc to the single edge of his band, Möbius constructed the first example of a non-orientable ‘closed’ surface (i.e., one without boundary curves); this is now called a ‘projective plane’ and is the simplest of an infinite number of such surfaces. Möbius also showed that there is no well-defined concept of volume (‘Inhalt’) for a non-orientable closed surface.

Möbius’s 1863 paper gave a classification of closed orientable surfaces. Such surfaces had been studied by Bernhard Riemann as part of his theory of complex functions (they are now called ‘Riemann surfaces’ in that context), and he had grasped intuitively that they are classified by the number of ‘holes’ (zero for a sphere, one for the surface of a doughnut, etc.), a concept that had been introduced by Simon L’Huilier in 1813, building on work of Leonhard Euler in 1752. Möbius gave the first rigorous proof of this important result. For this it was first necessary to agree when two such surfaces are to be considered ‘equivalent’. Möbius introduced the idea of *elementaren Verwandtschaften* (‘elementary relationships’) between two surfaces, in which ‘each point of one corresponds to a point of the other, in such a way that two infinitely neighbouring points always correspond to two infinitely neighbouring points’ (such transformations are now called ‘homeomorphisms’). Möbius showed that if two closed orientable surfaces have the same number of holes, there is an ‘elementary relationship’ from one to the other (i.e., they are ‘equivalent’). His proof was remarkably modern. “He classified singular points of a ‘height’ function into ‘elliptic’ and ‘hyperbolic’ points and developed what from a 20^{th} century point of view reads as a geometric presentation of the Morse theory of differentiable closed orientable surfaces” (James, p. 37). In his 1865 paper, Möbius showed that the classification theorem no longer holds for closed surfaces that are not necessarily orientable.

August Ferdinand Möbius (November 17, 1790 – September 26, 1868) was born in Schulpforta, Saxony-Inhalt, and was descended on his mother’s side from religious reformer Martin Luther. He studied mathematics under Carl Friedrich Gauss and Johann Pfaff. His early work was devoted to astronomy, culminating in his *Die Elemente der Mechanik des Himmels* (1843), which gives a through treatment of celestial mechanics without the use of higher mathematics. His best-known book, *Der barycentrysche Calcul* (1827) is celebrated for the introduction of homogeneous coordinates into projective geometry. Many mathematical concepts are named after him, including the Möbius transformations, important in projective geometry, and the Möbius function and the Möbius inversion formula in number theory.

“August Möbius influenced mathematics on many levels. Specific ideas – his famous one-sided surface, his inversion formula, his number-theoretic function, his transformations of the complex plane, his geometrical nets – bear his name. But, in addition, and perhaps more importantly, Möbius was aware of the big ideas, the general principles, the major areas of research … What is Möbius’s modern legacy? It is a large part of today’s mathematical mainstream. The concepts that attracted his attention, and the methods that he helped to develop, play a central role in modern mathematics” (Ian Stewart in Fauvel et al, p. 120).

Biggs, ‘The development of topology,’ pp. 106-119 in J. Fauvel, R. Flood & R. Wilson (eds.), *Möbius and his band* (1993); I. M. James (ed.), *History of Topology* (1999); A. N. Kolmogorov & A. A. Yushkevich, *Mathematics in the 19 ^{th} Century*, Vol. II (1996); I. Stewart,

*Visions of Infinity: The Great Mathematical Problems*(2011), Chapter 10.

Two complete journal issues, 8vo (226 x 140 mm). [1865:] pp. [iv], xii, 116, with one folding plate (coloured); [1863:] pp. 81. Original printed wrappers, unopened (front wrapper of 1865 issue slightly soiled, very minor wear to ends of spine). Fine copies.

Item #4826

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Price:
$3,000.00
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