the properties of lines and shapes drawn on the surface
of a
SPHERE
. The word geometry comes from the Greek
words ge meaning “earth” and metria meaning “mea-
sure.” As the origin of the word suggests, the study of
geometry evolved from very practical concerns with
regard to the accurate measurement of tracts of land,
navigation, and architecture.
The Greek mathematician E
UCLID
(ca. 300–260
B
.
C
.
E
.) formalized the study of geometry to one of pure
logical reasoning and deduction. In his famous work,
T
HE
E
LEMENTS
, Euclid collated a considerable volume
of Greek knowledge on the subject and showed that all
the results known at the time could be deduced from a
very small collection of self-evident truths or
AXIOM
s,
which he stated explicitly. Any result that can be
deduced from these axioms is today described as
Euclidean. Euclid’s rigorous approach to geometric
investigation remained the standard model of study for
two millennia.
See also E
UCLIDEAN GEOMETRY
; E
UCLIDEAN SPACE
;
E
UCLID
’
S POSTULATES
;
HISTORY OF GEOMETRY
(essay);
NON
-E
UCLIDEAN GEOMETRY
;
PARALLEL POSTULATE
;
PROJECTIVE GEOMETRY
.
Germain, Marie-Sophie (1776–1831) French Num-
ber theory, Mathematical physics Born on April 1,
1776, in Paris, France, scholar Marie-Sophie Germain is
remembered in mathematics for her significant contri-
butions to the topic of
NUMBER THEORY
. Most notably,
226 Germain, Marie-Sophie
History of Geometry
The study of
GEOMETRY
is an ancient one. Records show that
Egyptian and Babylonian scholars of around 1900
B
.
C
.
E
. had
developed sound principles of measurement and spatial rea-
soning in their architecture and in their surveying of land.
Both cultures were aware of P
YTHAGORAS
’s
THEOREM
and had
developed tables of P
YTHAGOREAN TRIPLES
. (The Egyptians
used knotted ropes to construct “3-4-5 triangles” to create
RIGHT ANGLES
.) Ancient Indian texts on altar construction and
temple building demonstrate sophisticated geometry knowl-
edge, and the famous volume J
IUZHANG SUANSHU
(Nine chap-
ters on the mathematical art) from ancient China also
includes work on the Pythagorean theorem.
In ancient Greece, mathematical scholars came to
realize that many properties of shapes and figures could be
deduced logically from other properties. In his epic work
THE ELEMENTS
the Greek geometer E
UCLID
(ca. 300–260
B
.
C
.
E
.)
collated a large volume of knowledge on the subject and
showed that each and every result could be logically
deduced from a very small set of basic assumptions (self-
evident truths) about how geometry should work. Euclid’s
work was rigorous and systematic, and the notion of a logi-
cal
PROOF
was born. E
UCLID
’
S POSTULATES
and the process of
logical reasoning became the model of all further geomet-
ric investigation for the two millennia that followed. His
method of compiling and organizing all mathematical
knowledge known at his time was a significant intellectual
achievement. Euclid’s rigorous approach was, and still is,
modeled in other branches of mathematics. Scholars in
SET
THEORY
, the
FOUNDATIONS OF MATHEMATICS
, and
CALCULUS
, for
instance, all seek to follow the same process of formal rea-
soning as the correct approach to achieve proper under-
standing of these topics.
The next greatest breakthrough in the advancement of
geometry occurred in the 17th century with the discovery
of C
ARTESIAN COORDINATES
as a means to represent points as
pairs of real numbers and lines and curves as algebraic
equations. This approach, described by French mathemati-
cian and philosopher R
ENÉ
D
ESCARTES
(1596–1650) in his
famous 1637 work La géométrie (Geometry), united the
then-disparate fields of algebra and geometry. Unfortu-
nately, Descartes’s interests lay only in advancing methods
of geometric construction, not in developing a full alge-
braic model of geometry. This latter task was pursued by
French mathematician P
IERRE DE
F
ERMAT
(1601–65), who had
also outlined the principles of coordinate geometry in an
unpublished manuscript that he had circulated among
mathematicians before the release of La géométrie. Fermat
later published the work in 1679 under the title Isagoge ad
locus planos et solidos (On the plane and solid locus). The
application of algebra to the discipline provided scholars a
powerful new tool for solving geometric problems, and also
provided them with a large number of different types of
curves for study.
Fermat’s work in geometry inspired work on the the-
ory of
DIFFERENTIAL CALCULUS
and, later, led to the study of
“differential geometry” (the application of calculus to the
study of shapes and surfaces). This was developed by the
German mathematician and physicist C
ARL
F
RIEDRICH
G
AUSS
(1777–1855).
Neither Descartes nor Fermat permitted negative val-
ues for distances. Consequently, neither scholar worked
with a full set of coordinate axes as we use them today. The