Saccheri, Girolamo (1667–1733) Italian Geometry
Born on September 5, 1667, in San Remo, Genoa,
Girolamo Saccheri is remembered for his 1733 publica-
tion Euclides ab omni naevo vindicatus (Euclid cleared
from every stain), in which he attempted to prove
Euclid’s
PARALLEL POSTULATE
as a consequence of the
remainder of E
UCLID
’
S POSTULATES
. Although Saccheri
failed in this attempt, he came narrowly close to being
the first to discover a
NON
-E
UCLIDEAN GEOMETRY
.
Saccheri was ordained as a Jesuit priest in 1694
and taught philosophy and theology at various Jesuit
colleges throughout Italy. He pursued mathematics as
an outside interest for a number of years before being
appointed chair of mathematics at Pavia in 1699.
As many scholars throughout the millennia had
also believed, Saccheri was convinced that Euclid’s fifth
postulate, the parallel postulate, could be deduced as a
logical consequence of the remaining four axioms.
Although he was aware that no one had succeeded in
showing that this was indeed the case, Saccheri realized
that another approach, an indirect one, could be
adopted. He reasoned as follows:
Imagine a geometry in which the first four of
Euclid’s axioms are true, but the fifth axiom is
false. If the fifth postulate can indeed be
proved from the first four, then we would have
a geometry with an inherent
CONTRADICTION
:
the fifth axiom is both true and false. Thus
one should study a theory of geometry in
which the parallel postulate is false and look
for a contradiction.
Saccheri followed this plan and wrote about it in his
famous 1733 text, but never found the contradiction he
sought. Just over a century later Hungarian mathemati-
cian J
ÁNOS
B
OLYAI
(1802–60) and Russian scholar
N
IKOLAI
I
VANOVICH
L
OBACHEVSKY
(1792–1856), with
their discoveries of
HYPERBOLIC GEOMETRY
, indepen-
dently proved that no contradiction will ever be discov-
ered. Although Saccheri had developed the mathematical
theory of hyperbolic geometry to a significant extent, he
cannot be credited for its discovery: he never realized
that it is a consistent mathematical theory in its own
right. Nonetheless the work of Saccheri is appropriately
honored in the field, with many results and theorems
properly attributed to him. He died in Milan, Italy, on
October 25, 1733.
sample See
POPULATION AND SAMPLE
.
sample space The set of all possible outcomes from
an experiment or an action is called a sample space.
For example, in casting a die, six scores are possible,
yielding the sample space the set {1, 2, 3, 4, 5, 6}. The
sample space for tossing a coin twice is the set
{head|heads, heads|tails, tails|heads, tails|tails}.
A sample space can be infinitely large. For exam-
ple, the action of recording heights of individuals has a
continuous array of values for its sample space—values
between 12 in. and 100 in., say—with all fractional
values in between possible.
See also
EVENT
;
PROBABILITY
.
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