the vertex angles were acute led to a contradiction. In retrospect, had he
chosen the second option, he could possibly have advanced the discovery
of non-Euclidean geometries by decades—but he went with the first.
Saccheri also was the first to realize an important property of non-Eucli-
dean geometries: the assumption that the vertex angles were acute led to
the conclusion that the sum of the angles in a triangle must be less than
180 degrees. Most investigations of whether the universe is Euclidean
or non-Euclidean involve measuring the angles of a triangle—the larger
the triangle the better—to see if this measurement will reveal the under-
lying geometry of the universe. A triangle the sum of whose angles is less
than 180 degrees, and such that the result is outside the range for experi-
mental error, would unquestionably show that the universe was non-Eu-
clidean. However, a triangle the sum of whose angles is close to 180
degrees would only provide confirming evidence that the universe was
Euclidean, and would not constitute a definitive result.
Another Visit from the Dancing Angels
Saccheri published his results in 1733. Some thirty years later, the German
mathematician Johann Lambert, a colleague of Leonhard Euler and Joseph
Lagrange, took another shot at the problem using a very similar approach.
Instead of using Saccheri quadrilaterals (two right angles with two equal
sides), he looked at a quadrilateral with three right angles and deduced con-
clusions about the fourth angle using postulates one through four. Like
Saccheri, he disposed of the possibility that the fourth angle could be ob-
tuse, but unlike Saccheri, he recognized that no contradiction could be ob-
tained if one assumed that the fourth angle were acute. Under the assumption
that the fourth angle was acute, Lambert managed to prove several impor-
tant propositions about models for non-Euclidean geometry—much as
George Seligman, my undergraduate algebra teacher, had managed to prove
results about algebras of dimension 16. However, Lambert did not construct
models for non-Euclidean geometries, so at the time of his death it wasn’t
clear whether angels could dance on the head of this particular pin. Lam-
bert would be more fortunate than Seligman, as help was on its way—but
the final verdict would not be in for nearly a century.
An Unpublished Symphony from the Mozart of Mathematics
At the turn of the nineteenth century, three mathematicians were to
travel essentially the same path toward the construction of non-Eucli-
dean geometries—and they all did it in essentially the same fashion, byNever the Twain Shall Meet 105