Download Muhammad ibn Musa al-Khwarizmi Abu Kamil ibn Aslam Abu `Ali al

MATH 300 History of Mathematics
Figures in Islamic Mathematics
Seventh Century
The Prophet Muhammad made the Hejira from Mecca to Medina (622)
Within a few decades, Persepolis, Ctesiphon, Damascus, Jerusalem, Alexandria, Tangier, Seville
all fell to Muslim armies and a vast empire was built from India to Spain as Islam spread across
the region
Eighth Century
Caliph al-Mansur established his capital at Baghdad (766)
Ninth Century
Caliph Harun al-Rashid and his son, al-Mamun, set up the Bayt al-Hikma (House of Wisdom)
in Baghdad (800?)
Bayt al-Hikma collected Greek and Hindu texts, prepared editions in Arabic
Muhammad ibn Musa al-Khwarizmi
• Kitab al-jam w’al tafriq bi hisab al-Hind
(Book on Addition and Subtraction after the Method of the Hindus)
Introduces readers to the use of Hindu numerals for calculation
• Al kitab al-muhtasar fi hisab al-jabr w’al muqabala
(Condensed Book on the Method of Restoration and Balance)
Arabic al-jabr = Latin algebra
A manual for solving equations
Provides a classification of quadratic equations into six types and rules for solving
each type
Abu Kamil ibn Aslam
• Kitab fi al-jabr w’al muqabala (Book on algebra and muqabala)
Includes a commentary on Euclid’s “geometric algebra” in Elements, Book II
Displays an independence of number from magnitude characteristic of Greek traditions
Tenth Century
Abu ’Ali al-Hasan ibn al-Hasan ibn al-Haytham (Alhazen)
• Misahat al-mujassam al-mukafi’ (The Measurement of a Paraboloidal Solid)
From a computation of the sum of fourth powers (an early version of a type of integration), he computed the volume of a solid of revolution formed from rotating a
parabola about a line perpendicular to its axis
Mohammad Abu ’l-Wafa al-Buzjani
• Zij al-Majisti (The Great Astronomical Tables)
In a work of spherical trigonometry, proposed the Rule of Four, used to prove the
Spherical Law of Sines
Eleventh Century
Abu al-Rayhan Muhammad ibn Ahmad al-Biruni
• Kitab fi ifrad al-Maqal fi ’amr al-azlal (Exhaustive treatise on shadows)
In a work on mathematical astronomy, all six trignonometric quantities (sine, cosine, tangent, cotangent, secant, cosecant) appear in the geometry of shadow-casting
’Umar ibn Ibrahim al-Khayyami
More well known today as a Persian poet (Rubaiyat) than as a mathematician!
• Maqalah fi al-jabr w’al muqabala (Treatise on Demonstration of Problems of
Algebra)
Paralleling work of al-Khwarizmi on quadratic equations, a study of cubic equations
Cubic equations classified into 14 types
Solution (sometimes more than one) constructed geometrically via the intersection
of conic curves
• Sharh ma’ashkala min musadarat kitab Uqlidis (Explanations of the difficulties
in the postulates in Euclid’s Elements)
Proposed this alternative to Euclid’s Post. V: Any two convergent lines must intersect
Proved Post. V from his axiom
Later scholars in the Islamic tradition extend this study
Twelfth Century
Sharaf al-Din al-Tusi
• al-Mu’adalat (On equations)
Provided conditions on the coefficients of a cubic equation for when 1, 2, or 3 (positive) solutions can be found
Ibn Yahya al-Samaw’al
• al-Bahir fi’l-jabr (The Brilliant in Algebra)
Studies the general arithmetic (algebra) of exponents
Works out an arithmetic of polynomial expressions
Displays twelve rows of the “Pascal Triangle” and states an equivalent of the Binomial Theorem
Pn
Pn
2
Determines values of sums of powers, e.g., i=1 i3 = ( i=1 i) , via a primitive form
of mathematical induction
Thirteenth Century
Ahmad ibn Munim al-Ab’dari
• Fiqh al-hisab (The Laws of Mathematics)
Devotes a chapter to combinatorial problems in a general
work on computation
Works out properties of combinatorial numbers Ckn = nk and Pnn = n!
Abu-l’Abbas Ahmad ibn Al-Banna al-Marrakeshi
• Raf al-Hijab (Lifting the Veil)
Shows how continued fractions can be used to approximate square roots
for combinatorial numbers
Derived the formula Ckn = n(n−1)···(n−k+1)
k(k−1)···1