against cavalry officers, but stipulated that he be allowed to play a violin
piece after every two duels. He won all thirteen duels; there are no re-
views of his violin performances.
While at the academy, János evinced interest in the parallel postulate.
Like everyone else, his initial efforts were devoted to trying to prove it.
His father, who had battled unsuccessfully with the problem, urged his
son to expend his effort elsewhere. “Do not waste one hour’s time on that
problem,” wrote Wolfgang. “It does not lead to any result, instead it will
come to poison all your life.... I believe that I myself have investigated
all conceivable ideas in this connection.”^9
János was not the first son to disregard his father’s advice, and in 1823
sent this communiqué to his father: “I am resolved to publish a work on
parallels as soon as I can complete and arrange the material, and the op-
portunity arises. At the moment I still do not clearly see my way through,
but the path which I have followed is almost certain to lead me to my goal,
provided it is at all possible.... All I can say at present is that out of noth-
ing I have created a strange new world.”^10
János had, indeed, created a strange new world. He developed a com-
plete system of geometry, constructing three distinct families of different
sets of postulates. The first system incorporated the five classic postu-
lates of Euclid—this is obviously Euclidean geometry. The second sys-
tem, now known as hyperbolic geometry, included the first four postulates
of Euclid and the negation of the parallel postulate. This was to be János’s
great contribution, a systematic development of non-Euclidean geometry.
Finally, his last system, absolute geometry, was based only on Euclid’s
first four postulates.
János’s work, the only thing he ever published, was included as a twenty-
four-page appendix to a textbook written by his father. His father sent the
work to Gauss, who wrote to a friend that he considered János Bolyai to be
a genius of the first order. However, his letter to Wolfgang was quite dif-
ferent. Gauss commented, “To praise it would amount to praising myself.
For the entire content of the work... coincides almost exactly with my
own meditations which have occupied my mind for the past thirty or
thirty-five years.”^11
Although this was not intended to be a put-down, it had a devastating ef-
fect on János, who was tremendously disturbed that Gauss had earlier tra-
versed the same path. János’s life deteriorated significantly thereafter. He
received a small pension when he was mustered out of the army and went
to live on a family estate. Isolated from the mathematical community, he
continued to develop some of his own ideas, and left twenty thousand
pages of notes on mathematics behind him. János was to become even
Never the Twain Shall Meet 109