substituting an alternative for Playfair’s Axiom. Each of the three worked
with “Through each point not on a given line there exists more than
one parallel to the given line,” and each deduced much the same conclu-
sions—although history gives the lion’s share of the credit to Nikolai
Ivanovich Lobachevsky and János Bolyai.
Although Gauss was undoubtedly the first to reach the conclusion that a
consistent geometry was possible using the above alternative to Playfair’s
Axiom, Gauss lived in a different era—and played mathematics by a dif-
ferent set of rules than those commonly used today. Gauss’s unofficial
motto was Pauca, Sed Matura—which translates from the Latin as “Few,
but ripe,” and expresses his attitude toward publishing. Gauss did not
publish anything until he was convinced that doing so would add to his
prestige (which, considering his prestige, meant he would publish only
the crème de la crème), and also that the result had been polished to a
fare-thee-well. Of course, like any mathematician, he certainly did not
burn his papers, and he was willing to communicate his results privately.
One day, he received a visit from Carl Jacobi, at the time generally re-
garded as the second-best mathematician in Europe. Jacobi wanted to
discuss a result he had obtained, but Gauss extracted some papers from a
drawer to show Jacobi he had already obtained the result. A disgusted
Jacobi remarked, “It is a pity that you did not publish this result, since you
have published so many poorer papers.”^5
Newton had probably set the gold standard for recalcitrance when it
came to publishing. He stuck his work on gravity in a drawer—probably
as the result of a vicious academic dispute with Robert Hooke over the
nature of light. Some years later, the astronomer Edmond Halley (of Hal-
ley’s comet fame) came to visit Newton, and inquired of him what would
be the motion of a body under an inverse square law of gravitational at-
traction. Newton astounded Halley by telling him that he had calculated
it to be an ellipse, and Halley was so impressed that he underwrote the
cost of publishing Newton’s Principia—which Newton had difficulty
finding when Halley visited him, because he wasn’t sure where he had
hidden it. When Newton didn’t avoid publication, he published anony-
mously—but his solution to a problem posed by Johann Bernoulli was so
elegant that even though the solution was anonymous, Bernoulli knew it
was Newton’s, declaring that he knew the lion by his claw.
Publication is a very different matter nowadays. With rare exceptions
(such as when Andrew Wiles announced a solution to Fermat’s last theo-
rem), mathematicians generally publish, or try to publish, what they’ve
got—even if it isn’t a polished solution to a problem, or even a complete
one.
106 How Math Explains the World