The six types of equation (al-Khwarizmi)

The six types are the classes into which al-Khwarizmi divided everything needed in reckoning by al-jabr and al-muqabala, at the opening of his Book of al-Jabr and al-Muqabala. ¹ They rest on three kinds of number: the root, anything multiplied by itself, from one upward or among the fractions below it; the square (mal), what results from a root multiplied by itself; and the simple number, any number stated without relation to a root or a square. ²

• Squares equal roots: "a square equals five of its roots", so the root is five and the square twenty-five. ²
• Squares equal number: "five squares equal eighty", so one square is a fifth of eighty, sixteen. ³
• Roots equal number: "four roots equal twenty", so the root is five and the square twenty-five. ³

In each type, more or less than one square is reduced to a single square, and whatever equals it is treated likewise. ²

• Squares and roots equal number: "a square and ten of its roots equal thirty-nine dirhams"; the procedure is to halve the roots, square the half, add it to the number, take the root of the sum and subtract half the roots, giving the root three and the square nine. ^{3 4}
• Squares and number equal roots: "a square and twenty-one equal ten of its roots", which has two answers: root three and square nine, or root seven and square forty-nine; he says this type alone "works by both addition and subtraction". ⁵ If the square of half the roots is less than the number, the problem is "impossible"; if equal, the root equals half the roots. ⁶
• Roots and number equal squares: "three roots and four equal a square", so the root is four and the square sixteen. ⁶

Al-Khwarizmi says that roots are not halved in the first three types, and that for each of the remaining three he made "a figure from which the cause is shown". ⁶ For the square and ten roots he draws the square as a figure of unknown side, divides the ten roots into four rectangles placed on its sides, then completes the four corners with small squares totalling twenty-five, so that the large square is sixty-four with side eight. ⁷ He insists that everything worked by al-jabr and al-muqabala "must bring you to one of the six chapters". ⁸ He then sets six model problems, one for each type, in the "chapter of the six problems". ⁹

The editors of the 1939 Cairo edition write these types in modern symbols and explain that al-Khwarizmi split quadratic equations into three compound kinds because he worked only with positive numbers. ^{1 3} In their account the name "impossible" he gave to the rootless case remained in use among mathematicians until the late eighteenth century, when the study of imaginary quantities began. ⁶