Era II — The Tenth Century (900–1000)¶
The century of consolidation and specialization. Decimal fractions appear, the six trigonometric functions are completed, geometric construction becomes a refined art, and the tradition spreads west to Córdoba.
Abu Kamil¶
Abu Kamil Shuja ibn Aslam (c. 850 – c. 930) · Egypt · algebra
"The Calculator from Egypt," the essential bridge between al-Khwarizmi and everything after. His Book on Algebra (Kitab fi al-jabr wa-l-muqabala) extends al-Khwarizmi decisively: he works comfortably with irrational coefficients and irrational solutions, manipulates surds with rules equivalent to \(\sqrt{a}\pm\sqrt{b} = \sqrt{a + b \pm 2\sqrt{ab}}\), handles powers of the unknown up to \(x^8\), and solves systems and indeterminate problems with real technical power. His treatise on the pentagon and decagon applies algebra to geometric construction problems, and his book on indeterminate problems overlaps the Diophantine tradition.
Transmission. Abu Kamil is a principal, documented source of Fibonacci's Liber Abaci (1202); many of Fibonacci's problems are Abu Kamil's, sometimes with the same numbers. A Latin translation and a Hebrew translation (by Mordecai Finzi, 15th c.) carried him into Europe independently.
Sources: Rashed's edition of Abu Kamil (2012); Berggren, Episodes; Katz; DSB.
al-Nayrizi¶
Abu al-Abbas al-Fadl ibn Hatim al-Nayrizi (c. 865 – c. 922) · Baghdad · geometry, astronomy
Latinized as Anaritius. Author of the most important early Arabic commentary on Euclid's Elements, which preserves lost Greek material (including parts of Heron's and Simplicius's commentaries) and contains his own attempted proof of the parallel postulate. Gerard of Cremona translated the commentary into Latin, through which it influenced medieval European geometry teaching, including Albertus Magnus. He also wrote on the mathematics of the spherical astrolabe and on qibla determination.
Sources: DSB; Lo Bello's English editions of Anaritius; Rosenfeld.
Sinan ibn Thabit and Ibrahim ibn Sinan¶
Sinan ibn Thabit (c. 880–943) and his son Ibrahim ibn Sinan (908–946) · Baghdad · geometry, astronomy
Thabit ibn Qurra's son Sinan was chiefly a physician and administrator of Baghdad's hospitals, but his son Ibrahim, dead at 38, was a geometer of the first rank. Ibrahim's quadrature of the parabola is simpler than Archimedes' own (a fact he advertised, family honor being at stake, since critics said his grandfather's proof was too long) and is arguably the most elegant pre-calculus parabola quadrature ever given, using an affine-invariance argument of striking modernity. His treatise On Analysis and Synthesis is one of the most sophisticated methodological works of the tradition, a study of mathematical discovery itself, and his work on drawing the conic sections gives practical pointwise constructions of ellipse, parabola, and hyperbola.
Sources: Berggren, Episodes; Rashed & Bellosta, Ibrahim ibn Sinan: Logique et géométrie au Xe siècle (2000); DSB.
al-Uqlidisi¶
Abu al-Hasan Ahmad ibn Ibrahim al-Uqlidisi (fl. c. 952) · Damascus · arithmetic
His name means "the Euclidean," probably because he earned a living copying Euclid manuscripts. His Kitab al-Fusul fi al-Hisab al-Hindi ("Chapters on Hindu Reckoning," written 952/953 in Damascus) is the earliest surviving Arabic arithmetic on Indian numerals, and it contains the earliest known appearance of decimal fractions, complete with a notational mark (a stroke over the units digit) to mark the decimal place: he halves 19 five times to get 0.59375, written in pure decimal form. He also deliberately adapted the algorithms from the Indian dust board to pen and paper, an underappreciated step toward mathematics as a written, checkable discipline.
Historians long credited decimal fractions to al-Kashi (15th c.); A. S. Saidan's 1966 study and 1978 translation of al-Uqlidisi moved the date back nearly five centuries. Al-Kashi remains the one who made decimal fractions a systematic computational instrument.
Sources: A. S. Saidan, The Arithmetic of al-Uqlidisi (1978); Berggren, Episodes, ch. 2; Katz.
al-Khazin¶
Abu Ja'far al-Khazin (d. c. 971) · Khurasan · algebra, geometry, astronomy
Solved "al-Mahani's equation," the cubic arising from Archimedes' sphere-division problem, by means of intersecting conic sections, the first known success of the method that Omar Khayyam would later systematize for all cubics. Also worked on number theory, including a treatment of the impossibility side of what we would call Fermat-type problems for certain cases (his proofs were incomplete), and on the isoperimetric problem.
Sources: Rashed; DSB; Berggren.
Abu al-Wafa al-Buzjani¶
Abu al-Wafa Muhammad ibn Muhammad al-Buzjani (940–998) · Baghdad · trigonometry, geometry, arithmetic
One of the greatest of all the tradition's mathematicians, and the central figure in the completion of trigonometry.
Trigonometry. Abu al-Wafa worked with all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant), established the identities relating them, proved addition and double-angle theorems in a clean modern-style form (equivalent to \(\sin(a \pm b) = \sin a\cos b \pm \cos a\sin b\)), computed sine tables of unprecedented accuracy using a refined interpolation, and stated the spherical law of sines (priority contested with his contemporaries Abu Nasr Mansur and al-Khujandi; the three worked in overlapping circles and the sources do not settle it).
Practical geometry. His Book on What Is Necessary from Geometric Construction for the Artisan treats constructions with straightedge and a rusty compass (fixed opening), exact and approximate constructions of regular polygons, and the dissection problems behind the girih tile patterns of Islamic ornament. This work links high geometry to craft practice.
Arithmetic. His practical arithmetic for scribes and merchants includes the first recorded systematic treatment in Arabic of negative quantities as debts in a computational rule.
Transmission. His trigonometric consolidation flowed into every later zij and, through the Latin and Byzantine transmission of Islamic astronomy, into European trigonometry, where Regiomontanus's De triangulis (1464, printed 1533) organizes essentially the same material.
Sources: Berggren, Episodes, ch. 5; Katz; DSB (A. P. Youschkevitch, "Abu'l-Wafa"); Kennedy, Survey.
al-Quhi¶
Abu Sahl Wayjan ibn Rustam al-Quhi (fl. c. 970–1000) · Tabaristan, Baghdad · geometry, astronomy
Regarded by contemporaries as the leading geometer of his day and director of the Buyid observatory in Baghdad (988). A virtuoso of Archimedean and Apollonian methods: he solved sphere-division and related problems via conics, studied the perfect compass (an instrument drawing any conic in one sweep), investigated centers of gravity, and pushed beyond Archimedes on problems of inscribing solids. His correspondence with Abu Ishaq al-Sabi is a rare surviving record of working mathematical discussion in the period.
Sources: Berggren's studies and editions of al-Quhi; Rashed, Geometry and Dioptrics; DSB.
al-Sijzi¶
Abu Said Ahmad ibn Muhammad al-Sijzi (c. 945 – c. 1020) · Sijistan, Shiraz · geometry
A specialist in the geometry of conic sections and geometric problem-solving; his treatise on the construction of the regular heptagon by conics is among the classics of the genre (the heptagon, impossible by straightedge and compass, was a proving ground for the tradition's conic-section methods). His methodological essay on making geometric problems tractable is an early "how to think about hard problems" text. He is also reported to have constructed an astrolabe based on the assumption of a rotating Earth, evidence such hypotheses were discussable.
Sources: Hogendijk's studies of al-Sijzi; Berggren; DSB.
al-Khujandi¶
Abu Mahmud Hamid ibn al-Khidr al-Khujandi (c. 940–1000) · Rayy · astronomy, trigonometry
Builder of a giant sextant at Rayy (the largest observing instrument of its time) for measuring the obliquity of the ecliptic, an instance of the tradition's insight that instrument scale buys precision, later pushed further at Maragha and Samarkand. He is one of the claimants (with Abu al-Wafa and Abu Nasr Mansur) to the spherical law of sines, and he asserted (without a complete proof) the impossibility of integer solutions to \(x^3 + y^3 = z^3\), an early appearance of a Fermat's Last Theorem case in the literature.
Sources: DSB; Kennedy; Rashed.
Ibn Sahl¶
Abu Sa'd al-Ala' ibn Sahl (fl. c. 984) · Baghdad · optics, geometry
Rescued from obscurity by Roshdi Rashed's 1990 reconstruction of his treatise On Burning Instruments: Ibn Sahl there analyzes refraction in lenses and states, geometrically, the constant-ratio relationship now called Snell's law, some six centuries before Snellius and Descartes. He used it to design plano-convex and biconvex burning lenses via the anaclastic problem. His work fed directly into Ibn al-Haytham's Optics. The finding rewrote the history of geometrical optics.
Sources: R. Rashed, "A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses," Isis 81 (1990); Rashed, Geometry and Dioptrics in Classical Islam.
al-Saghani¶
Abu Hamid al-Saghani (d. 990) · Baghdad · astronomy, geometry
Astronomer at the Buyid observatory and instrument maker (his name is associated with fine astrolabes). Mathematically he is remembered for work on the trisection of the angle, one of the three classical Greek problems, by neusis-type and conic methods, part of the tradition's systematic treatment of problems beyond straightedge and compass.
Sources: DSB; Berggren.
Maslama al-Majriti¶
Maslama ibn Ahmad al-Majriti (c. 950–1007) · Córdoba · astronomy, mathematics
The scholar who naturalized eastern mathematical astronomy in al-Andalus. He adapted al-Khwarizmi's zij to the meridian of Córdoba and the Hijra calendar (this adaptation, in Adelard of Bath's Latin translation, is the version that survives, the original being lost), edited Ptolemy's Planisphaerium (again surviving through the Arabic-Latin line), and founded the school from which Andalusi mathematics and astronomy descend: his students' students include the makers of the Toledan Tables. Through Córdoba, the mathematics of Baghdad began its westward journey toward Latin Europe.
Sources: DSB; Samsó, Las ciencias de los antiguos en al-Andalus; Kennedy, Survey.
Ibn Yunus¶
Abu al-Hasan Ali ibn Abd al-Rahman ibn Yunus (c. 950–1009) · Cairo · astronomy, trigonometry
Court astronomer to the Fatimid caliphs; his enormous Hakimi Zij rests on decades of observations from Cairo, many still usable by modern astronomers studying the secular acceleration of the Moon (Simon Newcomb used his eclipse records in the 19th century). Mathematically, his tables display extraordinary computational technique, and he is associated with a prosthaphaeresis-type identity, equivalent to \(\cos a\cos b = \frac{1}{2}[\cos(a-b)+\cos(a+b)]\), of the kind that sixteenth-century European astronomers later used to convert multiplication into addition, the immediate precursor idea to logarithms. (Direct influence of Ibn Yunus on the European prosthaphaeresis school is not documented; the parallel is conceptual.)
Sources: DSB (D. King, "Ibn Yunus"); King, "Islamic Astronomy," in Walker (ed.), Astronomy before the Telescope; Kennedy, Survey.
al-Karaji¶
Abu Bakr Muhammad ibn al-Hasan al-Karaji (c. 953 – c. 1029) · Baghdad · algebra, arithmetic, engineering
(Sometimes written al-Karkhi in older literature.) The pivotal figure in the arithmetization of algebra: where al-Khwarizmi justified algebra geometrically, al-Karaji cut it loose. In al-Fakhri, al-Badi, and al-Kafi fi al-hisab, he treats the arithmetic of polynomial expressions systematically, powers \(x, x^2, x^3, \dots\) and their reciprocals as an indexed sequence, with rules for operating on polynomials as such, effectively founding polynomial algebra. He gave the table of binomial coefficients with the additive generation rule (the "Pascal's triangle" recurrence \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)), known from al-Samawal's report of his lost work, and used an early form of argument by recursion/induction in proving results like the sum-of-cubes formula \(\left(\sum k\right)^2 = \sum k^3\).
His practical side: The Extraction of Hidden Waters, a serious engineering treatise on groundwater and qanat construction, containing applied geometry and surveying.
Transmission. The al-Karaji school (through al-Samawal) parallels and anticipates the algebraic techniques of the European Renaissance; direct textual transmission to Europe is limited, making this a case study in independent re-development along a path the Islamic tradition had already walked.
Sources: Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra; Katz; DSB; Woepcke's 1853 study of al-Fakhri (historic first Western notice).
Kushyar ibn Labban¶
Kushyar ibn Labban al-Jili (c. 971–1029) · Gilan, Baghdad region · arithmetic, astronomy
Author of Principles of Hindu Reckoning (Usul Hisab al-Hind), one of the clearest surviving textbooks of decimal arithmetic, translated into English by Levey and Petruck (1965). It teaches the full algorithm suite including square and cube root extraction, and shows sexagesimal (base-60) computation as used by astronomers alongside decimal, exactly the dual-base fluency the profession required. His zijes were widely used, and his student al-Nasawi continued the arithmetic textbook tradition.
Sources: Levey & Petruck, Kushyar ibn Labban: Principles of Hindu Reckoning (1965); DSB; Kennedy.