publication of his famous 1615 piece Nova stereome-
tria doliorum (New stereometry of wine barrels). This
work was later developed further by B
ONAVENTURA
C
AVALIERI
(1598–1647).
In 1614 J
OHN
N
APIER
published his account of log-
arithms, a mathematical device he invented with the
aim of helping astronomers with their large numerical
calculations. Kepler was delighted to learn of Napier’s
work and helped provide a mathematical explanation
of Napier’s method. Kepler also published tables of log-
arithmic values accurate to eight decimal places.
Kepler died on November 15, 1630, in Regens-
burg, Germany. His second law of planetary motion
played a crucial role in shaping Sir Isaac Newton’s
thinking in the development of his theory of mechanics.
Although Newton made no direct mention of Kepler’s
work in his early writings, he credited Kepler as the
source of inspiration for his ideas in a lecture he gave
to the members of the R
OYAL
S
OCIETY
before the
release of his famous 1687 work Principia.
al-Khw–
arizm–
ı, Muhammad ibn M–
us–
a(ca. 780–850)
Arab Algebra Born in Khw–
arizm, (now Khiva), Uzbe-
kistan, Arab scholar Muhammad ibn M–
us–
a al-Khw–
arizm
–
ı
is remembered for writing two extraordinarily influential
texts. The first, Al-jam’ w’al-tafriq ib hisab al-hind (Addi-
tion and subtraction in Indian arithmetic), introduced the
Indian system of numerals to the West, and a second
work that sparked the development of algebra. The title
of his second work, Hisab al-jabr w’al-muq–
abala (Calcu-
lation by restoration and compensation), when translated
into Latin, produced the very word algebra we use today.
Al-Khw–
arizm
–
ı was a scholar at the House of
Wisdom in Baghdad, a learning academy and library
founded by the Caliph al-Mamun in 813. (This institu-
tion was the first library constructed anywhere in the
world since the destruction of the famous Library of
Alexandria.) Housing a large collection of Greek philo-
sophical and scientific texts, the library hired Arab
scholars to translate manuscripts into Arabic and to
conduct further research in the topics covered.
Al-Khw–
arizm
–
ı was one such scholar who took particular
interest in the works on mathematics and astronomy.
In his first famous pieces, Al-jam’ w’al-tafriq ib
hisab al-hind, al-Khw–
arizm
–
ı described and explained
the advantages of the decimal-place system as used in
India for writing numbers and doing arithmetic. This
work was translated into Latin some 300 years later
and became the primary source for Europeans who
wanted to learn the new numeration system. It is inter-
esting to note that although this work uses a symbol
for zero (in Arabic called sifr) al-Khw–
arizm
–
ı did not
regard it as a number and used it only as a placeholder
(to distinguish 203 from 23, for instance).
Al-Khw–
arizm
–
ı intended that his second famous
piece, Hisab al-jabr w’al-muq–
abala, also be used as a
teaching guide. It aimed to offer an array of techniques
and methods for solving very practical problems in
matters of trade, inheritance, law, surveying, and archi-
tecture. Beginning at an extraordinarily elementary
level, al-Khw–
arizm
–
ı first defined the natural numbers
and the act of counting to then move on to the main
topic of the first section of the piece, namely, solving
elementary linear and quadratic equations. It is impor-
tant to note that all of al-Khw–
arizm
–
ı’s mathematics
was done entirely in words, and no symbols were ever
used. He called a number a “unit,” an unknown a
“root,” and a quantity squared a “square.”
To solve equations al-Khw–
arizm
–
ı used two opera-
tions which he called al-jabr and al-muq–
abala. The
first, “completion,” is the process of removing any neg-
ative terms from an equation. In modern notation, this
converts x2– 5x= 3 to x2= 3 + 5x, for instance. The
second, “balancing,” is the process of subtracting posi-
tive terms of the same power when they occur on each
side of an equation. In modern notation, this means
rewriting 5x2+ 7 = 3x2+ x, for instance, as 2x2+ 7 =
x. With these two maneuvers, al-Khw–
arizm
–
ı reduces all
linear and quadratic equations into six basic types,
which he lists. The remainder of this part of the text is
devoted to solving these equations. Today’s standard
practice of
COMPLETING THE SQUARE
is described in
this section.
The next part of al-Khw–
arizm
–
ı ’s text consists of
applications and worked examples. Because many of
the problems were geometric in nature, al-Khw–
arizm
–
ı
would dismiss negative solutions to problems and
allow only positive answers. He recognized that
quadratic equations may yield two solutions to a prob-
lem. Al-Khw–
arizm
–
ı also computed areas of simple geo-
metric figures, as well as the volume of a sphere, the
cone, and the pyramid, in this section. The final section
of the text focuses on the complicated Islamic rules for
inheritance and does not develop any new mathemati-
cal content.
al-Khw–
arizm–
ı, Muhammad ibn M–
us–
a 293