In the history of mathematics, algebra existed for centuries without a single letter. In ninth-century Baghdad, Muḥammad ibn Mūsā al-Khwārizmī wrote a manual for merchants and heirs to estates that gave the field its name, from al-jabr, “restoration,” one of two operations for balancing an equation. His methods were argued entirely in prose and proved with hand-drawn geometric diagrams. Eventually, mathematicians in the same tradition pushed that symbol-free algebra further than most people realize was possible, one compressing the whole subject into a fifty-four-line poem, another solving equations no one before him could by intersecting curves cut from cones. Then, in 1591, a French lawyer named François Viète decided that letters could stand for known and unknown quantities alike.
Alex Garnick, a PhD candidate in Harvard’s Department of the History of Science, studies how Arabic mathematical texts and techniques circulated into seventeenth-century Europe. We trace this history from al-Khwārizmī’s geometric proofs to Descartes’s reworking of the relationship between algebra and geometry, and he explains why calling any single moment “the birth of modern algebra” misses what actually changed.
