About the cover: Euler and Königsberg's Bridges: A historical view

Abstract

Graph theory almost certainly began when, in 1735, Leonhard Euler solved a popular puzzle about bridges. East Prussian city of Konigsberg (now Kalin- ingrad) occupies both banks of the River Pregel and an island, Kneiphof, which lies in the river at a point where it branches into two parts. There were seven bridges that spanned the various sections of the river, and the problem posed was this: could a person devise a path through Konigsberg so that one could cross each of the seven bridges only once and return home? Long thought to be impossible, the first mathematical demonstration of was presented by Euler to the members of the Petersburg Academy on August 26, 1735, and written up the following year under the title Solutio Problematis ad Geometriam Situs Pertinentis (The solution to a problem relating to the of position) (2) in the proceedings of the Petersburg Academy (the Commentarii). title page of volume appears on the cover of issue of the Bulletin. This story is well-known, and the illustrations in Euler's paper are often repro- duced in popular books on mathematics and in textbooks. Sandifer in (6) claims flatly that The Konigsberg Bridge Problem is Euler's most famous work, though scholars in other specialties (differential equations, complex analysis, calculus of variations, combinatorics, number theory, physics, naval architecture, music, . . . ) might disagree. N. L. Biggs, E. K. Lloyd, and R. J. Wilson in their history of graph theory (1) clearly view paper of Euler's as seminal and remark: The origins of graph theory are humble, even frivolous. Whereas many branches of mathematics were motivated by fundamental problems of calculation, motion, and measurement, the problems which led to the development of graph theory were often little more than puzzles, designed to test the ingenuity rather than to stimulate the imagina- tion. But despite the apparent triviality of such puzzles, they captured the interest of mathematicians, with the result that graph theory has become a subject rich in theoretical results of a surprising variety and depth. Euler provides only a neces- sary condition, not a sufficient condition, for solving the problem. But he does treat more than the original problem by beginning a generalization to two islands and four rivers, as is illustrated in the plate accompanying the original paper (Figure 1). In renowned paper Euler does not get around to stating the problem until the second page. On the first he states the reason for being interested in the problem— it was an example of a class of problems he attributes to Leibniz as belonging to something Leibniz called geometry of position. Euler says that this branch is concerned only with the determination of position and its properties; it does not