her work paved the way for other scholars in the field
to establish the validity of F
ERMAT
’
S LAST THEOREM
for
all values n< 100, the only substantial step made
toward solving the problem during the 19th century.
Germain also worked to develop a mathematical theory
of elasticity. She is noted as one of the very few women
scholars in mathematics before the 20th century.
At the age of 13, after reading an account of the
life and death of A
RCHIMEDES OF
S
YRACUSE
, Germain
said she felt impelled to become a mathematician. She
pursued her studies by first teaching herself Latin and
Greek and then reading the works of S
IR
I
SAAC
N
EW
-
TON
, L
EONHARD
E
ULER
, and E
UCLID
. As a woman,
Germain was not permitted to enter a university, but
she did manage to continue her study of mathematics
by reading course notes borrowed from students who
attended the École Polytechnique in Paris. Mathemati-
cian J
OSEPH
-L
OUIS
L
AGRANGE
(1736–1813) presented a
course on
ANALYSIS
at the university, and Germain sub-
mitted to him a thesis on the topic under the
pseudonym Louis LeBlanc, a male name. Lagrange was
greatly impressed by the contents of the paper and
sought out its author. He was not at all perturbed to
discover that “M. LeBlanc” was a woman. In fact, he
was so impressed with her work that he offered her
personal counsel in the study of advanced mathematics.
Again under her male pseudonym, Germain later
established a correspondence with the German mathe-
matician C
ARL
F
RIEDRICH
G
AUSS
(1777–1855), with
whom she developed and shared much of her work in
Germain, Marie-Sophie 227
notions of negative distance and negative area were first
put forward by S
IR
I
SAAC
N
EWTON
(1642–1727) and G
OTTFRIED
W
ILHELM
L
EIBNIZ
(1646–1716), the coinventors of
CALCULUS
.
The 19th century saw other major advances in geome-
try. It had long been noted that Euclid’s fifth postulate, the
so-called
PARALLEL POSTULATE
, is not necessary for a great
deal of geometry. Many Arab scholars of the first millen-
nium attempted, without success, to show that the fifth pos-
tulate could be logically deduced from the remaining four
(thereby rendering it unnecessary), as did European schol-
ars of the Renaissance. In 1795 Scottish mathematician
John Playfair showed that the fifth postulate is equivalent to
the statement that, through any point, one can draw one,
and only one, line through that point parallel to a given line.
(This is today called P
LAYFAIR
’
S AXIOM
.) Although not elimi-
nating the need for the fifth postulate, Playfair showed that
it could be understood in a more tractable form.
In 1829 Russian mathematician N
IKOLAI
I
VANOVICH
L
OBACHEVSKY
(1792–1856) took a bold step and considered a
geometric world in which the fifth postulate is false. He
assumed that through a given point more than one line
could be drawn parallel to a given line. In doing this,
Lobachevsky discovered a new, consistent mathematical
system free from contradiction, one as logically valid as the
geometry of Euclid. (This geometry is today called
HYPER
-
BOLIC GEOMETRY
.) The philosophical impact of Lobachevsky’s
work was enormous: he had shown that mathematics need
not be based on a single set of physical truths, and that
other equally valid mathematical systems do exist based on
alternative, carefully chosen axioms. Lobachevsky had also
shown that Euclid’s fifth postulate cannot be established as
a consequence of the remaining four axioms: he had pre-
sented a valid example of a system in which the first four of
Euclid’s postulates hold, but the fifth does not.
Surprisingly some of Lobachevsky’s ideas were antici-
pated well before the 19th century. The great Persian math-
ematician and poet O
MAR
K
HAYYÁM
(ca. 1048–1122)
established a number of results that we recognize today as
non-Euclidean. These results were later translated into
Latin, and extended upon, by Italian priest G
IROLAMO
S
AC
-
CHERI
(1667–1733). Unfortunately, neither scholar discovered
the validity of
NON
-E
UCLIDEAN GEOMETRY
, as each was
focused instead on trying to establish Euclid’s fifth postulate
as a consequence of the remaining four.
The German mathematician B
ERNHARD
R
IEMANN
(1826–66) discovered an alternative form of non-Euclidean
geometry in which Euclid’s fifth postulate fails in a different
way. In a system of
SPHERICAL GEOMETRY
it is never possible
to draw a line through a given point parallel to a given line.
Riemann’s contributions to the advancement of geom-
etry were significant. In his famous 1854 lecture “Über die
Hypothesen welche der Geometrie zu Grunde liegen” (On
the hypotheses that lie at the foundation of geometry), Rie-
mann put forward the view that geometry can be the study
of any kind of space of any number of dimensions, and later
developed the mathematics needed to properly describe
the shape of space. A
LBERT
E
INSTEIN
(1879–1955) later used
this work to develop his theory of relativity.
See also
AFFINE GEOMETRY
; A
RABIC MATHEMATICS
; B
ABYLO
-
NIAN MATHEMATICS
; J
ÁNOS
B
OLYAI
;
DIMENSION
; E
GYPTIAN MATHE
-
MATICS
; T
HE
E
LEMENTS
;
GEOMETRIC
;
TRANSFORMATION
; G
REEK
MATHEMATICS
;
HISTORY OF EQUATIONS AND ALGEBRA
(essay);
I
NDIAN MATHEMATICS
;
PERSPECTIVE
;
POSTULATE
;
PROJECTIVE
GEOMETRY
;
THEOREM
;
TOPOLOGY
.