REMs) 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-evident 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, including 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 centuries, and by Western scholars of the Renaissance.
In 1733 Italian teacher and scholar GIROLAMO SACCHERI (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 postulate is also true, then assuming the four postulates
together with the negation of the fifth postulate should
lead to a logical inconsistency. Unfortunately, in following this tact, Saccheri never came across a contradiction.
In 1795 Scottish mathematician and physicist John
Playfair (1748–1819) proposed an alternative formulation of the famous fifth postulate (today known as PLAYFAIR’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 mathematician
NICOLAI
IVANOVICH
LOBACHEVSKY
(1792–1856) and Hungarian mathematician JÁNOS
BOLYAI (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 exploring the geometries that result in assuming that the fifth
postulate does not hold, scholars were led to the discovery of NON-EUCLIDEAN GEOMETRY.
In the late 1800s the German mathematician DAVID
HILBERT (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 EUCLIDEAN GEOMETRY; HYPERBOLIC
GEOMETRY; SPHERICAL GEOMETRY.
Euclid’s proof of the infinitude of primes Around
the third century B.C.E., EUCLID 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 THE ELEMENTS, and he
was the first to recognize and prove this fact about
prime numbers.
Euclid’s proof relies on the observation that any
number N is 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,
…, p contains all the prime numbers. But consider 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 N is 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
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-evident 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, including 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 centuries, and by Western scholars of the Renaissance.
In 1733 Italian teacher and scholar GIROLAMO SACCHERI (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 postulate is also true, then assuming the four postulates
together with the negation of the fifth postulate should
lead to a logical inconsistency. Unfortunately, in following this tact, Saccheri never came across a contradiction.
In 1795 Scottish mathematician and physicist John
Playfair (1748–1819) proposed an alternative formulation of the famous fifth postulate (today known as PLAYFAIR’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 mathematician
NICOLAI
IVANOVICH
LOBACHEVSKY
(1792–1856) and Hungarian mathematician JÁNOS
BOLYAI (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 exploring the geometries that result in assuming that the fifth
postulate does not hold, scholars were led to the discovery of NON-EUCLIDEAN GEOMETRY.
In the late 1800s the German mathematician DAVID
HILBERT (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 EUCLIDEAN GEOMETRY; HYPERBOLIC
GEOMETRY; SPHERICAL GEOMETRY.
Euclid’s proof of the infinitude of primes Around
the third century B.C.E., EUCLID 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 THE ELEMENTS, and he
was the first to recognize and prove this fact about
prime numbers.
Euclid’s proof relies on the observation that any
number N is 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,
…, p contains all the prime numbers. But consider 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 N is 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