Mathematics is no different from most other human endeavors, in that
there are individuals of estimable achievement but substantially less than
estimable character.
Eugenio Beltrami and the Last Piece of the Puzzle
There was one final obstacle that had not yet been surmounted: the devel-
opment of a model that would exhibit the wondrous geometric properties
that Gauss, Bolyai, and Lobachevsky had formulated. This was accom-
plished by Eugenio Beltrami, an Italian geometer, who in 1868 wrote a
paper in which he actually constructed such a model. Beltrami was defi-
nitely trying to find a concrete realization for the theory that the three
early non-Euclidean geometers had developed, for he wrote in this pa-
per that “We have tried to find a real foundation to this doctrine, instead
of having to admit for it the necessity of a new order of entities and
concepts.”^13 Beltrami also played an important role in the history of non-
Euclidean geometry, as it was he who first focused attention on the work
Saccheri had done.
Many interesting curves in mathematics have resulted from the analy-
sis of a physical problem. One such curve is the tractrix, which is the
curve generated by the following situation. Imagine that the string of a
yo-yo is completely extended and the free end fastened to a model train
traveling on a straight track. The train moves at a constant velocity, keep-
ing the string taut. The tractrix represents the curve traced out by the
center of the yo-yo; it gets closer and closer to the track, but never quite
reaches it.
If the tractrix is rotated round the railroad track, the track represents a
central axis of symmetry for the resulting surface, which is known as
a pseudosphere. The pseudosphere is the long-sought model for a non-
Euclidean geometry, and every triangle drawn on its surface has the sum
of the angles less than 180 degrees.
Is the Universe Euclidean or Non-Euclidean?
Gauss’s experiment in measuring the sum of the angles in a triangle
whose sides were approximately 40 miles long was the first to try to deter-
mine whether the geometry of the universe could be non-Euclidean. Re-
call that Gauss found, to within experimental error, that his measurement
was consistent with a Euclidean universe. This is still a question that fas-
cinates astronomers, and so experiments have continued up through the
present day, with the lengths being employed now on the order of billions
112 How Math Explains the World