Solution of a Problem Relating to the Geometry of Position
Can one walk cross all seven bridges of Königsberg exactly once? Euler's “no” founded graph theory.
Seven bridges, one city, and a simple dare: cross every bridge exactly once. The man who proved it couldn't be done quietly invented a whole new kind of mathematics.
The big idea
The old Prussian city of Königsberg sat on both banks of a river and around an island in it, all stitched together by seven bridges. A popular Sunday challenge was to walk through town crossing every bridge once and only once — never recrossing any. Nobody could manage it, but nobody could say why.
Leonhard Euler, the most prolific mathematician of his age, realised the layout of the streets didn't matter at all. What mattered was only this: how many bridges touch each piece of land. He showed that for the walk to be possible, almost every region has to have an even number of bridges — and Königsberg's regions all had an odd number. So the walk was impossible, and he could prove it without ever leaving his desk.
How it came about
The puzzle reached Euler around 1735 while he was at the St. Petersburg Academy. At first he thought it beneath serious mathematics — a riddle, not a theorem. But solving it forced him to invent a way of thinking about problems where, as he put it, position matters and distance does not: a “geometry of position” that Leibniz had named but no one had developed.
Instead of testing the thousands of possible routes, Euler found a single counting rule that settled the question in one stroke — and realised it would settle any town, with any number of bridges. He presented it to the Academy in 1735; it was printed in 1741. Mathematicians now date the birth of graph theory to this paper.
Why it mattered
Euler had shown that you can prove something impossible — cleanly, with certainty — and that the proof can throw away almost everything concrete about a problem and keep only its pattern of connections. That move, discarding distance and shape to study pure structure, opened two fields at once: graph theory, the mathematics of networks, and topology, the mathematics of shape without measurement.
A way to picture it
Think of a join-the-dots picture you want to trace in one continuous pen stroke, never lifting the pen and never going over a line twice. Try it and you'll feel the rule: every dot you pass through needs a line in and a line out — an even number of lines. Only the dot you start at and the dot you finish at may have an odd number. If three or more dots are “odd”, it can't be done in one stroke. Königsberg is four odd dots — hopeless.
Where it sits
Euler wrote at the dawn of a new mathematics of structure. The same instinct — keep the connections, drop the measurements — later grew into topology, and into his own formula linking the corners, edges and faces of any solid, V − E + F = 2. Today every map app finding a route, and every network you can name, are distant children of seven bridges in a Prussian town.
§1 · The geometry of position
The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the “geometry of position” (geometria situs).