al-Kashi, Jamshid Mas’ud (ca. 1380–1429) Iranian
Arithmetic, Astronomy Born in Kashan, Iran, alKashi is best remembered in mathematics for his early
use of decimals to represent fractions and to approximate real numbers, two centuries before European
scholars developed the same technique. His famous
text, The Key to Arithmetic, completed March 2, 1427,
describes all the mathematics necessary for those studying astronomy, surveying, architecture, and accounting.
Al-Kashi developed iterative techniques for extracting
nth roots and for solving CUBIC and QUARTIC EQUATIONs, and gave an estimate for the value of π correct
to 16 decimal places. (This far surpasses the achievement of the ancient Greek and Chinese scholars who
approximated π as a fraction correct, in decimal notation, to six decimal places.)
That al-Kashi and his contemporaries chose to
represent quantities as decimals indicates an important
and sophisticated shift in the historical understanding
of “number.” For instance, writing π explicitly as a
decimal, just as any number can be so written,
changed the status of π from a physical geometric construct (the ratio of the circumference of a circle to its
diameter) to an ordinary member of the class of REAL
NUMBERS. Al-Kashi adopted a perspective where all of
mathematics was thus unified by the study of the realnumber system.
Al-Kashi was very interested in astronomy and
viewed much of his mathematical work as a tool for
assisting astronomical calculations. Techniques of
TRIGONOMETRY played a prominent role. In his text
The Treatise on the Chord and Sine, al-Kashi presented
tables of sine values and listed a value of sin(1°) accurate to 16 decimal places. This allowed for extremely
accurate astronomical calculations. On a purely mathematical note, al-Kashi also worked on the TRISECTING
AN ANGLE problem, noting that the obstacle behind it
reduces to one of solving a cubic equation. Applying
iterative techniques, al-Kashi was able to provide an
innovative approach to the problem, one that would
find approximate solutions to any prescribed degree of
accuracy. European scholars working on the same
problem throughout the Middle Ages and the Renaissance were unaware of al-Kashi’s work.
Kendall’s method See RANK CORRELATION.
Kepler, Johannes (1571–1630) German Astronomy,
Calculus Born on December 27, 1571, in Württemberg, Germany, Johannes Kepler is best remembered
for his three laws of planetary motion: that planets
travel in elliptical orbits with the Sun at one focus;
that they do so at a rate so that the line joining the
planet to the Sun sweeps out equal areas in equal units
of time; and that the square of the period of a planetary orbit is proportional to the cube of the width of
the ELLIPSE it traverses. These laws were all based on
detailed observation of the orbit of Mars. As a mathematician, Kepler is noted for giving the first clear
proof of how LOGARITHMs work, for his study of the
291
K
Arithmetic, Astronomy Born in Kashan, Iran, alKashi is best remembered in mathematics for his early
use of decimals to represent fractions and to approximate real numbers, two centuries before European
scholars developed the same technique. His famous
text, The Key to Arithmetic, completed March 2, 1427,
describes all the mathematics necessary for those studying astronomy, surveying, architecture, and accounting.
Al-Kashi developed iterative techniques for extracting
nth roots and for solving CUBIC and QUARTIC EQUATIONs, and gave an estimate for the value of π correct
to 16 decimal places. (This far surpasses the achievement of the ancient Greek and Chinese scholars who
approximated π as a fraction correct, in decimal notation, to six decimal places.)
That al-Kashi and his contemporaries chose to
represent quantities as decimals indicates an important
and sophisticated shift in the historical understanding
of “number.” For instance, writing π explicitly as a
decimal, just as any number can be so written,
changed the status of π from a physical geometric construct (the ratio of the circumference of a circle to its
diameter) to an ordinary member of the class of REAL
NUMBERS. Al-Kashi adopted a perspective where all of
mathematics was thus unified by the study of the realnumber system.
Al-Kashi was very interested in astronomy and
viewed much of his mathematical work as a tool for
assisting astronomical calculations. Techniques of
TRIGONOMETRY played a prominent role. In his text
The Treatise on the Chord and Sine, al-Kashi presented
tables of sine values and listed a value of sin(1°) accurate to 16 decimal places. This allowed for extremely
accurate astronomical calculations. On a purely mathematical note, al-Kashi also worked on the TRISECTING
AN ANGLE problem, noting that the obstacle behind it
reduces to one of solving a cubic equation. Applying
iterative techniques, al-Kashi was able to provide an
innovative approach to the problem, one that would
find approximate solutions to any prescribed degree of
accuracy. European scholars working on the same
problem throughout the Middle Ages and the Renaissance were unaware of al-Kashi’s work.
Kendall’s method See RANK CORRELATION.
Kepler, Johannes (1571–1630) German Astronomy,
Calculus Born on December 27, 1571, in Württemberg, Germany, Johannes Kepler is best remembered
for his three laws of planetary motion: that planets
travel in elliptical orbits with the Sun at one focus;
that they do so at a rate so that the line joining the
planet to the Sun sweeps out equal areas in equal units
of time; and that the square of the period of a planetary orbit is proportional to the cube of the width of
the ELLIPSE it traverses. These laws were all based on
detailed observation of the orbit of Mars. As a mathematician, Kepler is noted for giving the first clear
proof of how LOGARITHMs work, for his study of the
291
K