The subject of Graph Theory in mathematics, via the Seven Bridges of Königsberg and involving Euler of all people!
"The theory of graphs is one of the few fields of mathematics with a
definite birth date." -Oystein Ore, graph theorist & number theorist
Namely, that birthdate is in 1736 when Leonard Euler refers to "geometry of position" that we've come to know and love as Graph Theory.
Quotes below are from Chapter 3 of Graphs & Digraphs, 5th Edition, by Chartrand, Lesniak, and Zhang:
Early in the 18th century, the East Prussian city of Konigsberg (now called
Kaliningrad and located in Russia) occupied both banks of the River Pregel
and the island of Kneiphof, lying in the river at a point where it branches into
two parts. There were seven bridges that spanned various sections of the river.
A popular puzzle, called the Konigsberg Bridge Problem, asked whether
there was a route that crossed each of these bridges exactly once. Although such
a route was long thought to be impossible, the first mathematical verification of
this was presented by the famed mathematician Leonhard Euler (1707–1783) at
the Petersburg Academy on 26 August 1735. Euler’s proof was contained in a paper that would turn out to be the beginning of graph theory. This paper
appeared in the 1736 volume of the proceedings of the Petersburg Academy.
Euler’s paper, written in Latin, started as follows (translated into English):
"In addition to that branch of geometry which is concerned with magnitudes, and which has always received the greatest attention, there
is another branch, previously almost unknown, which Leibniz first
mentioned, calling it the geometry of position. This branch is concerned only with the determination of position and its properties; it
does not involve measurements, nor calculations made with them.
It has not yet been satisfactorily determined what kind of problems
are relevant to this geometry of position, or what methods should
be used in solving them. Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did
not require the measurement of distances, nor did calculation help
at all, I had no doubt that it was concerned with the geometry of
position – especially as its solution involved only position, and no
calculation was of any use. I have therefore decided to give here the
method which I have found for solving this kind of problem, as an
example of the geometry of position.
Euler goes on to say (see an excerpt from this amazing lecture on Euler's genius):
"...this solution bears little relationship to mathematics, and I do not understand why to expect a mathematician to produce it, rather than anyone else, for the solution is based on logic alone."
I guess I find this story so serendipitous, because you had to have so many things fall into place:
- the rivers naturally flowing in a unique way
- people to settle and
build just the right bridges in just the right places to make the
underlying problem unsolvable
- an arbitrary human constraint to want
to cross bridges exactly once on a Sunday afternoon walk
one of the most prolific and insightful minds of mathematics for all
- ...who proceeded to solve it despite the fact that he
didn't think it had any applications in math, nor did he think a
mathematician was needed to solve this problem!
I find it hard to believe Graph Theory would not have been developed eventually by someone else for another of its many applications, but to think that this was the fashion that it was introduced makes me consider it a serendipitous outcome.