The mathematics of unknowns, from Arabic al-jabr, the reunion of broken parts, the bone-setting of numbers, title word of al-Khwarizmi's ninth-century book.
Documented
The book
Documented: Muhammad ibn Musa al-Khwarizmi, working at Baghdad's House of Wisdom, wrote Kitab al-Jabr wa'l-Muqabala around 820 CE, the compendious book on calculation by completion and balancing. Al-jabr, completion, restoring, is the operation of moving a subtracted term across the equals sign; muqabala is balancing. Latin translators made the title word the science's name; the author's name became algorithm.
Assessment: Documented: the treatise, the title's two operations, the double legacy. One book named two pillars of computing.
Stories
The bone-setters
The word's other life: Spanish kept al-jabr's older concrete meaning, the setting of broken bones, and medieval Spanish barber-surgeons advertised as algebristas, algebraists, bone-menders. The sign Algebrista y Sangrador, bone-setter and blood-letter, hung outside shops where equations were never solved.
Assessment: Documented: the Spanish usage is attested in Cervantes' era. For centuries, algebraist meant the man who fixes your arm.
Interesting claims
The balancing theology
The deeper structure: al-jabr wa'l-muqabala, restoration and balancing, is a philosophy of debt. Subtract here, restore there; the universe of the equation is a ledger that must balance. Medieval Arabic mathematics grew beside inheritance law, where estates were divided by exactly such completion and balancing, and the math still carries the jurist's logic.
Assessment: Interesting: the inheritance-law context is documented. Every solved equation is a small estate, settled fairly.
Sources and further reading
Muhammad ibn Musa al-Khwarizmi, Kitab al-Jabr wa'l-Muqabala, c. 820 CE, Baghdad (primary source)
Latin translations of al-Khwarizmi, twelfth century (algebra from the title; algorithm from the author's name)
Spanish algebrista, bone-setter, attested in the early modern period (the word's surgical life)