number theory. Gauss, too, eventually discovered her
true identity, but this did not diminish his respect for
her as a fine scholar. In 1808 the Institut de France set a
competition to find a mathematical theory for the the-
ory of elasticity. Although the necessary mathematics to
solve the challenge was not available to scholars at the
time, Germain devoted a decade of work toward solv-
ing the problem and managed to make significant steps
in developing a beginning theory on the subject. Unfor-
tunately, much of her work was ignored by the Institut.
Germain never married nor obtained a professional
position; she was supported financially by her father
throughout her life. Germain died of breast cancer in
Paris, France, on June 27, 1831. Her death certificate
listed her occupation merely as “property holder.”
glide reflection A reflection in a line followed by a
translation in a direction parallel to that line is called a
glide reflection.
See also
GEOMETRIC TRANSFORMATION
.
gnomon The L-shaped figure that remains when a
PARALLELOGRAM
(usually a square) is removed from
the corner of a larger similar parallelogram is called a
gnomon. For example, in a square 2 ×2 array of
dots, the three dots surrounding any corner dot form
the shape of a gnomon. (Notice, incidentally, that 4 =
1 + 3.) Adding a gnomon of five dots to the 2 ×2
array produces a square 3 ×3 array (and we have that
9 = 1 + 3 + 5), and adding to this a gnomon of seven
dots yields a 4 ×4 array (yielding 16 = 1 + 3 + 5 + 7).
This process shows that, in general, the nth square
number equals the sum of the first nodd numbers.
The pointer on a sundial is also called a gnomon.
It, and the shadow it casts, together form the shape of
an L.
See also
FIGURATE NUMBERS
.
Gödel, Kurt (1906–1978) Austrian-American Logic
Born on April 28, 1906, in Brünn, Austria-Hungary
(now Brno, Czech Republic), Kurt Gödel is today con-
sidered the most important logician of the 20th cen-
tury. His famous pair of incompleteness theorems
stunned the mathematical community, dashing the
hopes of all those who had been fervently searching for
a set of fundamental axioms from which all mathemat-
ics could be logically deduced.
Gödel exhibited a talent for academic work at an
early age and had completed the equivalent of a univer-
sity curriculum before leaving secondary school. In
1923 he entered the University of Vienna to pursue a
degree in physics, but changed to mathematics in 1926
when he was introduced to the field of
FORMAL LOGIC
.
It soon became clear to the faculty of the department
that Gödel would make considerable contributions to
the field. In the summer of 1929 Gödel completed a
doctoral dissertation on the topic and was awarded a
Ph.D. the following year. His thesis outlined the details
of his first famous discovery, but the revolutionary
impact of his work was not fully understood by the
larger mathematical community until he published the
result a year later.
Gödel had proved that any mathematical system
sufficiently sophisticated to incorporate the principles
of arithmetic will always contain statements that can
neither be proved nor disproved. He later also showed
that no mathematical system could be proved to be
consistent, that is, free from
CONTRADICTION
, by mak-
ing use of just the axioms of the system. It had been a
200-year-long dream, since the time of G
OTTFRIED
W
ILHELM
L
EIBNIZ
in fact, to place the whole of mathe-
matics on a firm axiomatic base. Gödel had proved, in
essence, that such a dream can never be realized.
After joining the faculty of the University of Vienna
for three years, Gödel accepted, in 1933, a position at
the Institute for Advanced Study in Princeton, New Jer-
sey. After suffering a nervous breakdown, he returned
to Vienna after only a year, but later returned to the
Institute in 1942 to escape war-torn Europe. Gödel
became an American citizen that same year.
Many awards were bestowed upon him throughout
his life. In 1950 he was one of two recipients of the
first Einstein Award, and he was awarded honorary
doctorates from Yale University in New Haven, Con-
necticut, and Harvard University in Cambridge, Mas-
sachusetts, in the two years that followed. He was
elected to the National Academy of Sciences in 1955.
It is said that Gödel held fixed opinions about many
matters of life, and felt himself to always be right—
especially in the disciplines of mathematics and
medicine. Gödel was of a nervous disposition, and after
enduring severe bleeding from a duodenal ulcer, he
decided to maintain an extremely strict diet of his own
228 glide reflection