Lobachevsky’s results, having anticipated them, but nonetheless he was
intrigued by the methods he had used to derive them. Gauss even studied
Russian in his old age so that he could read Lobachevsky’s other papers!
Lobachevsky’s life differed significantly from János Bolyai’s. Lobachevsky
became the rector of Kazan University at age thirty-four and lived com-
fortably thereafter, yet he never ceased his attempts to have his efforts in
non-Euclidean geometry recognized. For the fiftieth anniversary of Ka-
zan University, he made one final attempt. Even though he had become
blind, he dictated “Pangeometry, or a Summary of the Geometric Foun-
dations of a General and Rigorous Theory of Parallels,” which was pub-
lished in the scientific journal of Kazan University.
Recognition for Lobachevsky would eventually follow, although not un-
til after his death. Just as Hilbert had saluted the efforts of Cantor, the
English mathematician William Clifford said of Lobachevsky, “What Ve-
salius was to Galen, what Copernicus was to Ptolemy, that was Lo-
bachevsky to Euclid.”^12 Today, all three of the major participants are
recognized as codiscoverers of non-Euclidean geometry, although the
bulk of the credit goes to Bolyai and Lobachevsky, who developed their
ideas independently—and published them. Sadly, when Bolyai learned of
Lobachevsky’s work, he initially believed that it was an attempt by Gauss
to rob him of his rightful place in the mathematical firmament, and that
Gauss had given Lobachevsky some of Bolyai’s ideas. Nonetheless, when
Bolyai examined Lobachevsky’s work, he retained enough integrity to
comment that some of Lobachevsky’s proofs were the work of a genius,
and the entire opus was a monumental achievement.
Another Parallel
It is fascinating how often history repeats itself—even the history of
mathematics. We have seen that the story of the continuum hypothesis is
much like the story of the parallel postulate. An axiomatic system is out-
lined, and the status of an additional axiom is in doubt—is it provable
from the original axioms, or not? In both cases, the additional axiom
turned out to be independent of the original set—the inclusion of either
the additional axiom or its negation resulted in consistent systems of axi-
oms. What is just as fascinating is how the stories parallel each other—a
great mathematician (Kronecker for Cantor, Gauss for Bolyai) either de-
liberately (Kronecker) or inadvertently (Gauss) prevents a lesser mathe-
matician from achieving the recognition he deserves, and it is left to
posterity to bestow the accolades. Meanwhile, the effect is to ruin a life.Never the Twain Shall Meet 111