If the width of a rectangle is four units and the
length of its diagonal is five units, what is its
breadth?
Four times four is 16, and five times five is 25.
Subtract 16 from 25 and there remains nine.
What times what equals nine? Three times
three is nine. The breadth is three.
The Babylonians used Pythagoras’s theorem to compute
the diagonal length of a square, and they found an
approximation to the square root of two accurate to five
decimal places. (It is believed that they used a method
analogous to H
ERON
’
S METHOD
to do this.) Babylonian
scholars were also interested in approximating the areas
and volumes of various common shapes by using tech-
niques that often invoked Pythagoras’s theorem.
Most remarkable is a tablet that lists 15 large
P
YTHAGOREAN TRIPLES
. As there is no apparent practi-
cal need to list these triples, this strongly suggests that
the Babylonians did indeed enjoy mathematics for its
own sake.
See also
BASE VALUE OF A NUMBER SYSTEM
.
Banach, Stefan (1892–1945) Polish Analysis, Topol-
ogy Born on March 30, 1892, in Kraków, now in
Poland, Stefan Banach is noted for his foundational
work in
ANALYSIS
and for generalizing the notion of
a
VECTOR SPACE
to a general theory of a space of
functions. This fundamental work allows mathemati-
cians today to develop a theory of F
OURIER SERIES
, in
some sense, in very abstract settings. Banach is also
remembered for his work leading to the famous
Banach-Tarski paradox that arises in the study of
AREA
and volume.
Banach began his scholarly career with a university
degree in engineering from Lvov Technical University.
His academic plans were interrupted, however, with the
advent of World War I. During this time Banach was
forced to work building roads, although he did manage
to find time to also teach at local Kraków schools dur-
ing this period. Soon after the war Banach joined a
mathematics discussion group in Lvov and soon
impressed mathematical scholars with his abilities to
solve mathematical problems. Within a week of joining
the group, Banach had drafted a coauthored research
paper on the topic of measure theory, a theory that
generalizes the concept of area. Banach continued to
produce important results in this field at an extremely
rapid rate thereafter.
Banach was offered a lectureship at Lvov Techni-
cal University in 1920 and quickly set to work on a
doctoral thesis. Despite having no previous official
university qualifications in mathematics, Banach was
awarded a doctorate in 1922 by the Jan Kazimierz
University in Lvov.
Banach’s contributions to mathematics were sig-
nificant. His generalized work on Fourier series
founded a branch of mathematics now called func-
tional analysis. It has connections to the fields of
measure theory, integration, and
SET THEORY
. He and
his colleague Alfred Tarski presented their famous
paradoxical result in 1926 in their paper “Sur la
décomposition des ensembles de points en partiens
respectivement congruent” (On the decomposition of
figures into congruent parts). His 1932 paper,
“Théorie des opérations linéaires” (Theory of linear
operators), which develops the notion of a normed
VECTOR SPACE
(that is, a vector space with a notion
of length attached to its vectors), is deemed his most
influential work. As well as conducting research in
mathematics, Banach also wrote arithmetic, algebra,
and geometry texts for high-school students.
In 1939 Banach was elected as president of the Pol-
ish Mathematical Society. Banach was allowed to main-
tain his university position during the Soviet occupation
later that year, but conditions changed with the 1941
Nazi invasion. Many Polish academics were murdered,
but Banach survived, although he was forced to work in
a German infectious diseases laboratory, given the task
to feed and maintain lice colonies. He remained there
until June 1944, but he became seriously ill by the time
Soviet troops reclaimed Lvov. Banach died of lung can-
cer on August 31, 1945.
Banach’s name remains attached to the type of vec-
tor space he invented, and research in this field of func-
tional analysis continues today. The theory has
profound applications to theoretical physics, most
notably to quantum mechanics.
bar chart (bar graph) See
STATISTICS
:
DESCRIPTIVE
.
Barrow, Isaac (1630–1677) British Calculus, Theol-
ogy Born in London, England, (his exact birth date is
Barrow, Isaac 35