What did al-Khwarizmi's "restoring and balancing" actually do?

They did two simple moves that tidy an equation into a standard shape. Al-jabr, "restoring", means moving a deficient quantity to the other side of the equation: in al-Khwarizmi's example, x² = 40x − 4x² becomes 5x² = 40x. Al-muqabala, "balancing", means subtracting the same positive quantity from both sides: x² + 5 = 40x + 4x² becomes 5 = 40x + 3x². Repeat them and negative terms disappear and each kind of quantity (square, root, number) appears at most once. The first move gave algebra its name.

Why does that matter? Because once every equation is tidied, you only need to handle a short list. Al-Khwarizmi's book classifies quadratic equations into six basic types and gives algebraic and geometric methods to solve them. Mathematicians of his world did not deal with negative numbers, so what looks to us like one equation, rearranged, counted as three different types with all coefficients positive.

And there was not a single symbol. In the historian Carl Boyer's words his algebra was thoroughly rhetorical: even the numbers were written out in words. One problem asks you to divide ten into two parts so that one multiplied by itself equals the other taken eighty-one times, and the solution is a paragraph of sentences ending "There remains one, and this is one of the two parts."

The book was no ivory-tower treatise. It was written with the encouragement of Caliph al-Ma'mun as a popular work, with examples from trade, surveying and legal inheritance. Centuries later, in Don Quixote, a Spanish bone-setter was still called an algebrista: a "restorer".