Euclidean geometry
A Greek mathematician performing a geo-
metric construction with a compass, from
The School of Athens by Raphael. (The figure
depicted may be either Archimedes or Euclid,
and despite the painting’s popular name,
neither was Athenian.)
Euclidean geometry is a mathematical sys-
tem attributed to the Greek mathematician
Euclid of Alexandria. Euclid’s Elements is the
earliest known systematic discussion of geo-
metry. It has been one of the most influential
books in history, as much for its method as
for its mathematical content. The method
consists of assuming a small set of intuitively
appealing axioms, and then proving many
other propositions (theorems) from those ax-
ioms. Although many of Euclid’s results had
been stated by earlier mathematicians,[1]
Euclid was the first to show how these pro-
positions could be fit together into a compre-
hensive deductive and logical system.[2] The
Elements begin with plane geometry, still
taught in secondary school as the first axio-
matic system and the first examples of formal
proof. It goes on to the solid geometry of
three dimensions. Much of the Elements
states results of what are now called algebra
and number theory, couched in geometrical
language.[3]
For over two thousand years, the adjective
"Euclidean" was unnecessary because no oth-
er sort of geometry had been conceived. Euc-
lid’s axioms seemed so intuitively obvious
that any theorem proved from them was
deemed true in an absolute sense. Today,
however, many other self-consistent non-Euc-
lidean geometries are known, the first ones
having been discovered in the early 19th cen-
tury. It also is no longer taken for granted
that Euclidean geometry describes physical
space. An implication of Einstein’s theory of
general relativity is that Euclidean geometry
is a good approximation to the properties of
physical space only if the gravitational field is
not too strong.[4]
The Elements
The Elements are mainly a systematization of
earlier knowledge of geometry. Its superior-
ity over earlier treatments