Era IV — The Twelfth Century (1100–1200)¶
The century in which Europe began translating this tradition into Latin (see The Translation Movement) while the tradition itself pressed on: symbolic-style polynomial arithmetic, an astonishing proto-analytic treatment of cubics, and the flowering of Maghrebi calculation.
al-Khazini¶
Abd al-Rahman al-Khazini (fl. c. 1115–1130) · Merv · mechanics, astronomy
A former slave of Byzantine origin who became a leading scientist at the Seljuk court in Merv. His Book of the Balance of Wisdom (Kitab Mizan al-Hikma, 1121/22) is the most comprehensive medieval treatise on mathematical statics and hydrostatics: theory of centers of gravity (building on al-Quhi and Ibn al-Haytham), the hydrostatic balance, and tables of specific gravities of metals, gems, and liquids measured to impressive precision. It treats gravity as directed toward the center of the Earth and varying with distance in a qualitative sense, mechanics as a mathematical science, four centuries before Stevin and Galilei retraced parts of this ground. His Sanjari Zij served practical astronomy.
Sources: DSB (R. E. Hall, "al-Khazini"); partial English translation by N. Khanikoff (1859).
Jabir ibn Aflah¶
Abu Muhammad Jabir ibn Aflah (fl. c. 1120–1150) · Seville · astronomy, trigonometry
Latinized as Geber (a persistent source of confusion with the alchemist Jabir ibn Hayyan). His Islah al-Majisti ("Correction of the Almagest") criticized Ptolemy, notably re-ordering Venus and Mercury above the Sun, and, importantly for mathematics, opened with a self-contained treatise on spherical trigonometry organized around the sine rule and the rule of four quantities, discarding the Menelaus theorem entirely. Gerard of Cremona translated it into Latin; through this channel Jabir's trigonometry reached Regiomontanus, whose De triangulis omnibus borrows results from Geber (Cardano later pointedly accused Regiomontanus of insufficient acknowledgment). "Geber's theorem" in spherical trigonometry (\(\cos B = \cos b \sin A\) for a right spherical triangle, in one common statement) carries his Latin name to this day.
Sources: DSB (R. Lorch, "Jabir ibn Aflah"); Lorch's studies of the Islah; Van Brummelen.
al-Samawal¶
al-Samawal ibn Yahya al-Maghribi (c. 1130 – c. 1180) · Baghdad, Persia · algebra, arithmetic
Born to a Jewish family in Baghdad (he converted to Islam in adulthood and wrote about it), a prodigy who composed his masterpiece al-Bahir fi al-Jabr ("The Dazzling in Algebra") around age nineteen, consolidating and extending the al-Karaji school.
Polynomial arithmetic. Al-Samawal manipulates polynomials via tables of coefficients indexed by power, including negative powers, performing addition, multiplication, and crucially long division of polynomials exactly as taught today, and carrying quotients into infinitely continuing expansions (e.g., dividing to obtain what is in effect a power series in \(1/x\)), with an explicit statement of the rule \(x^m \cdot x^n = x^{m+n}\) spanning negative exponents: "operating on unknowns with all the arithmetical tools the arithmetician applies to the known." Historians (Rashed) treat this as the clearest medieval statement of the program of arithmetizing algebra.
Induction-style proof. His proofs of the binomial coefficient recurrence and of identities like the sum of squares proceed by regression from \(n\) to \(n-1\) down to a base case, the recognizable skeleton of mathematical induction before its formalization (Pascal, 1654).
Approximation. In al-Qiwami he computes with decimal fractions and approximates roots by methods equivalent to keeping arbitrarily many decimal places, aware of controlled precision.
Sources: Rashed & Ahmad's edition of al-Bahir (1972); Rashed, Development of Arabic Mathematics; Katz; DSB.
Sharaf al-Din al-Tusi¶
Sharaf al-Din al-Muzaffar ibn Muhammad al-Tusi (c. 1135 – c. 1213) · Tus, Damascus, Aleppo, Mosul · algebra, geometry
(Not to be confused with Nasir al-Din al-Tusi of the next century.) His Treatise on Equations takes Khayyam's classification of cubics and asks the next question: when do solutions exist, and how many? For cubic types that can lack positive solutions, he determines the condition by locating the maximum of the cubic expression: for \(x^3 + c = bx\) (his style, rendered in modern notation), he identifies the critical value \(x = \sqrt{b/3}\), where the expression \(bx - x^3\) is greatest, and compares \(c\) with that maximum to count roots. The critical value is precisely the root of what we would call the derivative, and the historian Roshdi Rashed has argued this amounts to a local-maximum analysis anticipating differential ideas; other historians (Hogendijk, Berggren) offer more conservative readings via Euclidean tools. Either way, the treatise also contains a remarkable numerical method for approximating cubic roots digit by digit (related to what is now called the Ruffini–Horner scheme, independently central in Chinese mathematics and formalized in Europe in 1804–1819). He also invented the linear astrolabe ("al-Tusi's staff"), a rod-form astrolabe.
Sources: Rashed's edition, Sharaf al-Din al-Tusi: Oeuvres mathématiques (1986); J. P. Hogendijk, "Sharaf al-Din al-Tusi on the number of positive roots of cubic equations," Historia Mathematica 16 (1989); Berggren's review discussions; DSB.
al-Hassar¶
Abu Bakr Muhammad al-Hassar (fl. 12th century) · Maghreb (Ceuta or Fez region) · arithmetic
Author of Kitab al-Bayan wa-l-Tadhkar, a practical arithmetic of the Maghrebi school whose lasting monument is notational: al-Hassar's text contains the earliest known systematic use of the horizontal fraction bar, writing a fraction as numerator over denominator separated by a bar, the notation the entire world now uses. Fibonacci, educated partly in Bugia (in the Maghreb), uses precisely this notation in the Liber Abaci, the plausible route into Europe. The treatise was translated into Hebrew (by Moses ibn Tibbon, 1271), evidence of its circulation.
Sources: Lamrabet, Introduction à l'histoire des mathématiques maghrébines; Djebbar's studies of Maghrebi mathematics; Katz.
Ibn al-Yasamin¶
Abu Muhammad Abdallah ibn al-Yasamin (d. c. 1204) · Maghreb, Marrakesh, Seville · arithmetic, algebra, pedagogy
A Berber scholar famous for mathematical didactic poems, above all the Urjuza fi al-jabr ("Poem on Algebra"), which versified the rules of algebra so students could memorize them; it was recited, memorized, and commented on across North Africa for centuries, a reminder that this tradition invested seriously in pedagogy. His school's manuscripts also display the developing Maghrebi algebraic symbolism (letters/marks for the unknown, its powers, roots, and equality) that al-Qalasadi would later systematize.
Sources: Djebbar; Abdeljaouad's studies of Maghrebi symbolism; Lamrabet.