About twelve hundred years ago a scholar was born in Khwarazm, in what is now Uzbekistan. We know very little about his life. We know his work better than almost any other book of its age.
He worked in Baghdad, at the library the caliph had built, and some time around the year eight hundred and twenty he finished a book about solving equations. Its title contained the word al-jabr, which meant moving a term from one side of an equation to the other. Europe borrowed that word and never gave it back. We call the whole subject algebra.
His name gave us a second word. He wrote a book of step-by-step procedures for calculating, and Latin writers called such a procedure by his name. Every time a programmer says algorithm, they are saying "al-Khwarizmi" with the corners worn off.
Here is the part that surprises students. He had no symbols. There was no letter standing for the unknown, no plus sign, no equals sign — none of these had been invented yet. He wrote every equation as a sentence, in words, exactly the way this reading is written.
And he had no negative numbers, so he could not write one general recipe. He had to treat several cases separately, and he proved each one with a diagram — a real square, drawn on the page, with smaller rectangles added to its sides until it was complete.
That drawing is the reason the method is still called completing the square. He was completing an actual square.