Mathematical Heritage

From al-Khwārizmī
to the classroom.

Twelve scholars of the medieval Arabic mathematical tradition, in chronological order, whose work seeded the techniques our students will meet in olympiad halls today, algebra, trigonometry, conic sections, the iterative root, the decimal fraction.

The Academy continues a mathematical tradition carried in Arabic across several centuries. Its teaching draws on the same habits of generalisation, geometric reasoning, recursion, and iterative refinement.

From Baghdad to Marāgha, from Cairo to Samarkand, scholars writing in Arabic gave algebra its name, formalised plane and spherical trigonometry, classified the cubic equations by conic intersection, refined the iterative root, and pushed the decimal fraction into sixteen place computation.

These twelve scholars make that intellectual line visible. Their work helps teachers and students place each problem within a thousand year arc.

This page provides the narrative arc. Historical Profiles provides the evidence layer: one scholar per page, with dates, works, claims, and source links presented for checking.

Twelve scholars

A short genealogy.

Arranged roughly in chronological order. Each entry names one signature contribution that still appears in olympiad and classroom mathematics today.

i

الخوارزمي

c. 780 to c. 850

Muḥammad ibn Mūsā al-Khwārizmī

Baghdad · court of the Abbasid caliph al-Maʾmūn

Al-Khwārizmī's Kitāb al-jabr wa-l-muqābala gave algebra both its name and its first systematic treatment as a discipline apart from arithmetic. He set out the six canonical forms of the quadratic and showed how completing the square, reducing, and balancing recover the unknown. From his Latinised name we still draw the word algorithm.

Signature contribution The quadratic, solved by completing the square, the procedure every olympiad student inherits before they know its history.

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ii

ثابت بن قُرّة

836 to 901

Thābit ibn Qurra

Ḥarrān · Baghdad

A translator, astronomer and geometer of the Banū Mūsā circle, Thābit produced one of the earliest general proofs of the Pythagorean theorem and gave a rule for generating amicable numbers, pairs (m, n) in which the proper divisors of each sum to the other. His work on incommensurables anticipated later developments in number theory by several centuries.

Signature contribution Thābit's rule for amicable numbers, the first systematic generator of a nontrivial number theoretic family.

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iii

البَتّاني

c. 858 to 929

Abū ʿAbdullāh al-Battānī

Ḥarrān · Raqqa

Al-Battānī replaced the Greek chord with the sine and tabulated it as a ratio. This shifted trigonometric calculation away from length. In right angled trigonometry he gave the relation MacTutor records as b sin A = a sin(90° − A), the shape in which the sine and cosine relationship first enters the textbook line. He refined the length of the solar year to within minutes of the modern value. His astronomical tables, the Zīj al-Ṣābiʾ, were read in Latin Europe for half a millennium.

Signature contribution Trigonometry as a calculus of ratios, sine and cosine, calibrated against Ptolemy's chord.

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iv

أبو الوفاء البوزجاني

940 to 998

Abū al-Wafāʾ al-Būzjānī

Būzjān · Baghdad

Al-Būzjānī introduced the tangent, cotangent, secant and cosecant as named functions, and he gave a rigorous proof of the spherical law of sines, whether it is originally his or independently Abū Naṣr Manṣūr's remains a live question in the historiography. He also worked on geometric constructions with a single rule and a compass with a fixed radius, a constraint that still surfaces in olympiad problems on constructibility.

Signature contribution Tangent, cotangent, secant, cosecant, the complete trigonometric family, named and tabulated.

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v

الكرجي

13 April 953 to c. 1029

Abū Bakr al-Karajī

Baghdad · the “mountain countries”

Al-Karajī pushed algebra away from geometric figures and toward the manipulation of powers of an unknown as symbols in their own right. His al-Fakhrī gave the sum of cubes identity 1³ + 2³ + … + n³ = (1 + 2 + … + n)² and one of the earliest inductive arguments in the surviving record, preserved in detail by al-Samaw'al a century later, who credits the technique back to him.

Signature contribution An arithmetised algebra of powers, and the inductive method whose surviving record runs through al-Samaw'al a century later.

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vi

ابن الهيثم

c. 965 to c. 1040

Ibn al-Haytham (Alhazen)

Basra · Cairo

Ibn al-Haytham worked in optics, geometry and number theory, but he is most relevant to us as an early articulator of what is now called Wilson's theorem, that (p − 1)! ≡ −1 (mod p) for every prime p. His geometric studies of conic reflection and the problem of finding the point on a spherical mirror that reflects light between two fixed points (now called Alhazen's problem) remain canonical.

Signature contribution An early statement of Wilson's theorem, and a sustained study of conic reflection.

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vii

البيروني

15 September 973 to 13 December 1048

Abū al-Rayḥān al-Bīrūnī

Khwārazm · Ghazna · fieldwork in India

Al-Bīrūnī brought a working scientist's discipline to the measurement of the world. From a single elevated observation at Nandana Fort in the Punjab he estimated the Earth's radius using nothing more than a quadrant, a horizon angle, and spherical trigonometry, a method still worth teaching for its economy. Across his works, an insistence on reporting the error alongside the measurement.

Signature contribution A trigonometric measurement of the Earth's radius from one observation, radius, and a career long standard of observational precision.

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viii

عمر الخيّام

1048 to 1131

Omar Khayyām

Nīshāpūr · Isfahan

In his Treatise on the Proofs of Problems of Algebra, Khayyām classified the cubic equations into fourteen canonical forms and gave a geometric solution to each through the intersection of conic sections. He worked toward a proof of Euclid's parallel postulate; the constructions he used in that attempt would later inform non-Euclidean geometry.

Signature contribution Geometric solution of the cubic, by intersecting conics, nearly five centuries before Cardano's algebraic treatment.

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ix

السموأل

c. 1130 to c. 1180

Al-Samaw'al al-Maghribī

Baghdad · Marāgha

Al-Samaw'al is the reason we know what al-Karajī did. His Al-Bāhir fi'l-jabr preserves the detailed inductive argument, credits it back to al-Karajī explicitly, and adds the sum of squares identity in the form olympiad students still meet. His combinatorics problem of ten unknowns and 210 equations formed from sums of six quantities anticipates a modern family of problems about symmetric data.

Signature contribution The sum of the first n squares, in closed form, and the surviving record of al-Karajī's inductive method.

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x

شرف الدين الطوسي

c. 1135 to 1213

Sharaf al-Dīn al-Ṭūsī

Ṭūs · Damascus · Aleppo · Mosul · Baghdad

Sharaf al-Dīn returned to the cubic and studied the conditions for a positive solution. For the five contested cubic cases he locates the point where a cubic polynomial takes its maximum and reads off the number of positive roots from a comparison with the constant. Rashed reads the maximum computation as a derivative argument; Hogendijk reads it as Euclid Book II. The archive entry keeps both readings in view. He also gave a Ruffini Horner style iteration for cubic roots, six centuries early.

Signature contribution Existence conditions for the cubic derived from its maximum, and a numerical scheme that anticipates Horner.

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xi

نصير الدين الطوسي

1201 to 1274

Naṣīr al-Dīn al-Ṭūsī

Ṭūs · Marāgha Observatory

Naṣīr al-Dīn detached trigonometry from astronomy and treated it as an independent mathematical discipline. He gave a clean statement of the law of sines for plane triangles and developed the geometric device now called the Ṭūsī couple, a pair of circles whose combined motion produces linear oscillation, a construction that would reappear, unattributed, in Copernicus.

Signature contribution Trigonometry as an autonomous mathematical subject, and the Ṭūsī couple.

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xii

جمشيد الكاشي

c. 1380 to 1429

Jamshīd al-Kāshī

Kāshān · Samarkand Observatory

At Ulugh Beg's observatory in Samarkand, al-Kāshī computed sin 1° to ten correct sexagesimal places by an iterative method that is, in substance, Newton Raphson applied to a cubic. In his Risāla al-muḥīṭīya (Treatise on the Circumference, July 1424) he gave 2π to nine sexagesimal, equivalent to sixteen decimal, places, a record that stood until van Ceulen computed twenty decimal places at the turn of the seventeenth century. His Miftāḥ al-Ḥisāb (Key to Arithmetic, 1427) is the encyclopedic teaching companion to that work.

Signature contribution An iterative root method now associated with Newton and Raphson, and 2π to sixteen decimal places.

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The same problems, passed forward.

Every cubic our students solve, every triangle they inscribe, every iterative root they refine, they are doing, again, what was done in Baghdad and Marāgha and Samarkand. The Academy's task is to keep that line visible, and to put it back into the classroom.

See the curriculum Our mission