arl Friedrich Gauss (1777–1855) lived in Go
and worked in many areas, ranging from complex
numbers to astronomy and electricity. He gave the
ﬁrst satisfactory proof that every polynomial equation has a
complex root, and in number theory he initiated the study
of congruences and proved the law of quadratic reciprocity.
He also determined which regular polygons could be con-
structed with ruler and compasses: his regular 17-gon is
shown on the East German stamp.
Gauss was also interested in the foundations of geome-
try. Euclid’s Elements commences with ﬁve ‘‘postulates.’’
Four are straightforward, but the ﬁfth is different in style. It
is equivalent to the parallel postulate: given any line l and
any point p not lying on l, there is a unique line through p
that is parallel to l. For more than 2000 years, mathemati-
cians tried to deduce this result from the ﬁrst four postulates,
but they were unable to do so.
This is because there are ‘‘non-Euclidean geometries’’
satisfying the ﬁrst four postulates but not the ﬁfth. Such
geometries have inﬁnitely many lines through p that are
parallel to l, and they were discovered independently