The thousand years between the fall of Rome and the European Renaissance are sometimes dismissed as a mathematical blank spot. They were anything but. The torch passed from Greek to Arabic mathematicians in Baghdad, Cairo, and Córdoba, then back to a slowly reawakening Europe. Along the way, the discipline acquired its modern algebraic vocabulary and most of its numerical notation.
The Islamic Golden Age
Between roughly 800 and 1200 CE, the Islamic world was the center of mathematical research. In Baghdad’s House of Wisdom, scholars translated Greek, Persian, Sanskrit, and Syriac texts and extended them substantially.
Al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi (ca. 780–850) wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala — the book that gave us the word algebra (from al-jabr). He classified quadratic equations into six standard forms and described systematic solution procedures for each. His name also gave us the word algorithm.
Omar Khayyam
The Persian polymath Omar Khayyam (1048–1131) — famous in the West as a poet — was also a first-rate mathematician. He classified cubic equations by the number and nature of their positive roots, and found solutions using intersections of conic sections. His work on the parallel postulate anticipated non-Euclidean geometry by seven centuries.
Thabit ibn Qurra and Al-Kindi
Thabit ibn Qurra (836–901) contributed to number theory (generalized Euclid’s formula for amicable numbers) and to mechanics. Al-Kindi (801–873) introduced Indian numerals to the Islamic world and made major contributions to cryptanalysis — essentially inventing frequency analysis.
Ibn al-Haytham
Ibn al-Haytham (965–1040), known in the West as Alhazen, worked in optics, number theory, and geometry. His seven-volume Book of Optics was one of the most important scientific works of the Middle Ages and influenced European science for centuries.
India and the Hindu-Arabic numeral system
Indian mathematicians made two contributions that would change all of mathematics: place-value notation and zero.
Brahmagupta
Brahmagupta (598–668) wrote the first known systematic treatment of zero as a number — not just a placeholder — and gave rules for arithmetic with negative numbers. He solved general quadratic equations (explicitly using both positive and negative roots) and contributed to cyclic quadrilaterals and Diophantine equations.
Madhava and the Kerala school
The Kerala school in southern India, led by Madhava of Sangamagrama (ca. 1340–1425), developed infinite series for trigonometric functions — including what we now call the Leibniz series for — centuries before Newton or Leibniz. Knowledge of the Kerala school reached Europe only much later; whether it influenced Newton or Leibniz directly remains debated.
The Latin West awakens
Between roughly 1100 and 1300, European scholars began translating Arabic mathematical works into Latin. This “Great Translation Movement” made available to Europe the accumulated mathematics of antiquity and the Islamic world.
Fibonacci
Leonardo of Pisa (ca. 1170–1250), known as Fibonacci, grew up in a merchant family with North African trading connections. His 1202 Liber Abaci (“Book of Calculation”) introduced Hindu-Arabic numerals and al-Khwarizmi’s algebra to European merchants and scholars. The book also contained the Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21… — arising from a problem about rabbit populations.
It took several centuries for Hindu-Arabic numerals to displace Roman numerals in European practice. Italian merchants adopted them quickly; universities held out longer. By 1500 the numerals were standard.
The Renaissance solution of the cubic
In the 16th century, Italian mathematicians made the first genuinely new mathematical advance since antiquity: the general solution of the cubic equation.
The story is a bizarre one. Scipione del Ferro (1465–1526) found a method for solving depressed cubics () but kept it secret. On his deathbed he passed the method to his student Fiore, who challenged Tartaglia to a public contest. Tartaglia, working against the clock, rediscovered the method and won. He then taught it to Cardano under an oath of secrecy — which Cardano broke by publishing it in his 1545 Ars Magna.
The book also contained the quartic solution, found by Cardano’s student Lodovico Ferrari, and the first systematic use of complex numbers, introduced (reluctantly) because they appeared in the cubic formula even when all roots were real.
Legacy
By 1600, mathematicians in Europe had inherited the Greek tradition, absorbed the Islamic and Indian advances, and added their own: the cubic and quartic, the beginnings of algebraic notation, the first uses of complex numbers. The stage was set for the scientific revolution — for Newton, Leibniz, and the birth of calculus.
Frequently asked
Did Europe have any mathematics in the Middle Ages?
Not much that was new until around 1200. What kept mathematics alive was the Islamic Golden Age (roughly 800–1200), which absorbed Greek, Persian, and Indian traditions and made major advances in algebra, trigonometry, and numeracy. Latin Europe received this legacy via translations in the 12th century.
Who brought algebra to Europe?
Leonardo of Pisa (Fibonacci), whose 1202 book Liber Abaci introduced Hindu-Arabic numerals and al-Khwarizmi's algebraic techniques to European merchants and scholars. By the 14th century they were standard in commercial practice, though academic mathematicians held out longer.