Book Review: An Imaginary Tale: The Story of $\sqrt{-1}$, Volume 46, Number 10

was considered insoluble. Such an interpretation was quite sensible in the Greek and Arab schools of algebra, in which algebraic equations were generally expressions of geometric relationships. The need for imaginary numbers did not manifest itself until the sixteenth century discovery of Cardan’s Formula for the roots of the cubic equation. The seeds for imaginary numbers were planted in the twelfth century when Arab algebra was introduced into Italy through the Latin translation of al-Khwārizmı̄’s great treatise, Al-jabr wā’lmuqābala. In 1225 Leonardo of Pisa (Fibonacci) published an approximate solution of a specific cubic and showed that the exact solution could not have a certain form. Incorrect solutions of the cubic were published in 1328 and 1344 by Paolo Gerardi and by Maestro Dardi of Pisa. By the end of the fourteenth century, however, a crucial step was taken. In two anonymous Florentine manuscripts there appears the linear change of variable that transforms a general cubic equation into the so-called depressed cubic