Algebra

Born, Not Borrowed

Problems that we would express with equations are older than writing; Babylonian scribes solved quadratics, and Diophantus manipulated equations with virtuosity. What al-Khwarizmi created around 820 was algebra as a discipline: a named subject (al-jabr wa-l-muqabala) with its own primitive objects (the number, the root/thing shay, the square mal), a complete classification of its basic problems (the six equation types of degree ≤ 2), general solution procedures stated for all cases, proofs that the procedures work, and an applications program (inheritance, commerce, surveying). The difference between a bag of solved problems and a discipline is exactly this architecture, and it is why the subject carries his book's word, algebra, and not a Greek or Sanskrit one.

The Development, Stage by Stage

Stage 1 — Foundation (al-Khwarizmi, c. 820). Six types, rhetorical style (everything in words), geometric proofs by completing literal squares. Positive coefficients and positive roots only; a quadratic like \(x^2 + 21 = 10x\) is recognized as having two solutions (3 and 7).

Stage 2 — Power (Abu Kamil, c. 900). Abu Kamil works freely with irrational coefficients and solutions (\(x = \sqrt{5} - 1\) style answers), higher powers, and intricate systems. Algebra can now digest any number the geometry of the day can produce.

Stage 3 — Arithmetization (al-Karaji, c. 1000; al-Samawal, c. 1150). al-Karaji detaches algebra from geometric proof and treats expressions in \(x\) by the rules of arithmetic; al-Samawal completes the program: polynomial long division, negative exponents with the law \(x^m x^n = x^{m+n}\), coefficient tables that are one notational step from writing polynomials as coefficient vectors, and induction-style proofs. This is the algebra of "operating on the unknown as if it were known", the subject's defining mental move, stated as such.

Stage 4 — The cubic frontier (al-Mahani to Khayyam to Sharaf al-Din, 860–1200). The sphere-division problem yields a cubic (al-Mahani); al-Khazin solves it with conics; Omar Khayyam systematizes: all fourteen irreducible cubic types solved by intersecting conic sections, with an explicit statement that arithmetic solutions are lacking and desired. Sharaf al-Din al-Tusi then analyzes existence and number of roots via the maximum of the cubic expression and approximates roots numerically by a Horner-type digit scheme. Between them, the two Persians hold the world's most advanced theory of equations until sixteenth-century Italy.

Stage 5 — Symbolism (Maghreb, 12th–15th c.). The Maghrebi school develops written signs for the unknown, its powers, roots, and equality, and computes with them, a genuine symbolic algebra, codified by al-Qalasadi (d. 1486), independent of and earlier than Viète.

What Europe Received, and What It Re-Grew

Europe received Stage 1 directly: Robert of Chester's and Gerard of Cremona's Latin al-Khwarizmi (12th c.), plus Abu Kamil in Latin and Hebrew, plus Fibonacci's massive synthesis (1202), which carries al-Khwarizmi's classification and Abu Kamil's problems into Italian mathematical culture. The Italian maestri d'abaco chewed on cubics for three centuries inside exactly this framework until del Ferro (c. 1515) and Tartaglia cracked the arithmetic solution Khayyam had wished for, published with due machinery in Cardano's Ars Magna (1545), whose opening page credits the subject's origin to "Mahomet the son of Moses the Arab" (al-Khwarizmi). Stages 3–5 were largely re-grown on European soil (Stifel, Cardano, Bombelli, Viète, Descartes) with limited direct textual transmission, a repeated pattern: the Islamic tradition demonstrates which developments were mathematically natural, even where influence cannot be documented.

Connection to Modern Mathematics

Modern algebra's core reflexes trace here: treating equations as objects to classify and transform (al-jabr and al-muqabala are the first named equation-transformations, the ancestors of "add to both sides"); the polynomial as a formal object with coefficient arithmetic (al-Karaji–al-Samawal); the interplay of algebra and geometry in solving equations by curve intersection (Khayyam → Descartes' Géométrie, the founding text of analytic geometry, which explicitly solves equations by intersecting conics); and root-counting by extremum analysis (Sharaf al-Din), the germ of the discriminant-and-derivative viewpoint. Even the \(x\) on every whiteboard has a candidate ancestry in shay ("thing"), via Old Spanish transliteration xay, though historians treat that etymology as plausible-but-unproven.

Sources: Rashed, The Development of Arabic Mathematics; Berggren, Episodes, ch. 4; Katz; Cardano, Ars Magna (Witmer translation); Oaks & Alkhateeb's studies of rhetorical algebra.