constructed a model for a consistent geometry, but merely had convinced
himself that one was possible.
Gauss concluded his letter to Taurinus by saying, “In any case, consider
this a private communication, of which no public use or use leading to
publicity is to be made. Perhaps I shall myself, if I have at some future
time more leisure than in my present circumstances, make public my
investigations.” Several years later, in a letter to the astronomer Heinrich
Olbers, he reiterated both his results and his desire not to go public with
them.
Nonetheless, he was sufficiently impressed with the possibility that the
geometry of the real world might not be Euclidean that he conceived of an
experiment to resolve the matter. Saccheri and Gauss had both deduced
that if the parallel postulate did not hold, the sum of the angles in a trian-
gle would total less than 180 degrees. Gauss laid out a triangle using
mountains around his home in Göttingen; the sides of the triangle were
approximately 40 miles long. He measured the angles of the triangle and
computed their sum; had the result been significantly less than 180 de-
grees, he would have been able to reach an earthshaking conclusion. It
was not to be: the sum of the angles differed by less than 2 seconds
(1/1,800 of a degree), a difference that could certainly have been the result
of experimental error.
Wolfgang and János Bolyai
Wolfgang Bolyai (a.k.a. Farkas Bolyai) was a friend of Gauss from their
student days at Gottingen. As students, they had discussed what they re-
ferred to as the problem of parallels, and over the years they maintained
friendship by correspondence when Wolfgang returned to Hungary.
However, Wolfgang’s son János was unquestionably the mathematical
star of the family. Wolfgang gave János instruction in mathematics, and
the son proved to be an extraordinarily quick learner. Wolfgang fell ill one
day when János was thirteen, but the father had no qualms about sending
in his son to pinch-hit for his lectures at college. I’m not sure how I would
have felt if a thirteen-year-old showed up in place of my usual professor.
When János was sixteen, Wofgang wrote to Gauss, asking him to take
János into Gauss’s household as an apprentice to facilitate the advance-
ment of his career. Possibly the letter went astray, but Gauss did not an-
swer, and so János entered the Imperial Engineering Academy, planning
on a career in the army. In addition to being an extremely talented math-
ematician, János was a superb duelist and an enthusiastic violinist. He
once accepted a challenge in which he fought thirteen consecutive duels
108 How Math Explains the World