REM
s) about geometric figures. The systematic approach
he followed and the rigor of reasoning he introduced
was hailed as a great intellectual achievement. His model
of mathematical exploration became the standard for all
mathematical research for the next 2,000 years.
Euclid’s fifth postulate was always regarded with
suspicion. It was never viewed as simple and as self-evi-
dent as his remaining four postulates, and Euclid himself
did his utmost to avoid using it in his work. (Euclid did
not invoke the fifth postulate until his 29th proposition.)
Over the centuries scholars came to believe that the fifth
postulate could be logically deduced from the remaining
four postulates and therefore did not need to be listed as
an axiom. Many people proposed proofs for it, includ-
ing the fifth-century Greek philosopher Proclus, who is
noted for his historical account of Greek geometry.
Unfortunately his proof was flawed, as were the proofs
proposed by Arab scholars of the eighth and ninth cen-
turies, and by Western scholars of the Renaissance.
In 1733 Italian teacher and scholar G
IROLAMO
S
AC
-
CHERI
(1667–1733) believed that because Euclid’s
axioms model the real world, which he thought to be
consistent, they cannot lead to a
CONTRADICTION
. If the
first four postulates do indeed imply that the fifth pos-
tulate is also true, then assuming the four postulates
together with the negation of the fifth postulate should
lead to a logical inconsistency. Unfortunately, in follow-
ing this tact, Saccheri never came across a contradiction.
In 1795 Scottish mathematician and physicist John
Playfair (1748–1819) proposed an alternative formula-
tion of the famous fifth postulate (today known as P
LAY
-
FAIR
’
S AXIOM
). This version of the axiom is considerably
easier to handle, and its negation is easier to envision. In
an attempt to follow Saccheri’s approach, Russian math-
ematician N
ICOLAI
I
VANOVICH
L
OBACHEVSKY
(1792–1856) and Hungarian mathematician J
ÁNOS
B
OLYAI
(1802–1860), independently came to the same
surprising conclusion: the first four of Euclid’s postulates
together with the negation of Playfair’s version of the
fifth postulate will not lead to a contradiction. This
established, once and for all, that the fifth postulate is an
INDEPENDENT AXIOM
and cannot be deduced from the
remaining four postulates. More important, by explor-
ing the geometries that result in assuming that the fifth
postulate does not hold, scholars were led to the discov-
ery of
NON
-E
UCLIDEAN GEOMETRY
.
In the late 1800s the German mathematician D
AVID
H
ILBERT
(1862–1943) noted that, despite its rigor,
Euclid’s work contained many hidden assumptions. He
also realized, despite Euclid’s attempts to describe them,
that the notions of “point,” “line,” and “plane” cannot
be properly defined and must remain as undefined
terms in any theory of geometry. In his 1899 work
Grundlagen der Geometrie (Foundations of geometry)
Hilbert refined and expanded Euclid’s postulates into a
list of 28 basic assumptions that define all that is needed
in a complete account of Euclid’s geometry. His axioms
are today referred to as Hilbert’s axioms.
See also E
UCLIDEAN GEOMETRY
;
HYPERBOLIC
GEOMETRY
;
SPHERICAL GEOMETRY
.
Euclid’s proof of the infinitude of primes Around
the third century
B
.
C
.
E
., E
UCLID
proved that there is no
such thing as a largest
PRIME
number, meaning that the
list of primes goes on forever. He presented his proof as
Proposition IX.20 in his book T
HE
E
LEMENTS
, and he
was the first to recognize and prove this fact about
prime numbers.
Euclid’s proof relies on the observation that any
number Nis either prime, or factors into primes. His
argument proceeds as follows:
Suppose to the contrary that there is a largest
prime number p. Then the finite list 2, 3, 5, 7,
…, pcontains all the prime numbers. But con-
sider the quantity:
N= 2 ×3 ×5 ×7 ×…×p+ 1
It is not divisible by any of prime numbers in
our list (it leaves a remainder of one each
time), and so it has no prime factor. It must be
the case then that Nis prime. Thus we have
created a new prime number larger than the
largest prime p. This absurdity shows that our
assumption that there are only finitely many
primes must be false.
Euclid’s argument is a classic example of a
PROOF BY
CONTRADICTION
. His argument also provides an
ALGORITHM
for generating new primes from any finite
list of primes. For example, from the list of primes 2,
3, 7, Euclid’s argument yields N= 2 · 3 · 7 + 1 = 43 as
a new prime, and from the list 2,3,7,43, we have N=
2 · 3 · 7 · 43 + 1 = 1807 = 13 ×139, yielding 13 as a
new prime.
Euclid’s proof of the infinitude of primes 171