The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book *Ars Magna*. The method, often called Cardano’s method, gives the solution of a depressed cubic equation *t*^{3}* + p t + q = *0. The general cubic equation can be reduced to this form by a simple linear transformation of the dependent variable. The solution is given by

Cardano assumed that the discriminant Δ = ( *q */ 2 )^{2} + ( *p */ 3 )^{3}, the quantity appearing under the square-root sign, was positive.

Raphael Bombelli made the psychedelic leap that Cardano could not make. He realised that Cardano’s formula would still give a solution when the discriminant was negative, provided that the square roots of negative quantities were manipulated in the correct manner. He was thus the first to properly handle complex numbers and apply them with effect.

**Solving the Cubic**

Gerolamo Cardano, with reluctant help from Niccolo Tartaglia, found the solutions of a general cubic equation *x*^{3} + *a x*^{2} + *b x + c* = 0. Cardano noted that, in some cases, his method produced square roots of negative quantities. He could formally proceed and obtain a correct answer, but he regarded the intermediate quantities as meaningless. He was the first person to present a calculation with complex numbers. Solving a cubic with negative discriminant, he wrote: *“Dismissing mental tortures and multiplying 5 + **√** *( *-15*)* by 5 – **√** *(*-15*) *the product is 40″*. However, he dismissed this process as meaningless.

It was to take a huge leap of the imagination to realize that complex numbers are meaningful. It could be described as psychedelic, or mind-opening. Complex numbers allowed us to break free of the real number line into the complex plane, and led to enormously fruitful developments in mathematics. It was Raphael Bombelli who made this great leap.

**Raphael Bombelli**

Raphael Bombelli (1526–1572) was the first mathematician to consider roots of negative quantities as having logical significance and mathematical utility was. Bombelli received no formal university education but his home-town was Bologna, and he would have followed the public contests and arguments between mathematicians there. He was nine years old at the time of the competition between Tartaglia and Fior, and nineteen when Cardano’s *Ars Magna* was published. He must have studied this work closely.

Bombelli worked as an engineer on several projects, including an unsuccessful attempt to drain the Pontine Marshes. During this time he began writing a textbook on algebra. Cardano’s masterpiece was understandable only to those with an extensive knowledge of mathematics. Bombelli aimed to produce a more accessible textbook. He also worked on translating Diophantus’ *Arithmetica*.

His own book, *Algebra*, was heavily influenced by Diophantus. The first three volumes appeared in 1572, the year of his death. The remaining two volumes remained in manuscript form until they were published in early twentieth century. Bombelli’s book was very influential and was praised by no less a personage that Leibniz, who called Bombelli “an outstanding master of the analytical art”.

**Bombelli’s Algebra**

Bombelli’s book was a comprehensive account of algebra as known at the time of its publication. It also contained Bombelli’s innovative work on complex numbers. He was amongst the first mathematicians to work explicitly with negative numbers. He included a geometric demonstration of how minus times minus gives plus, something that continues to perplex students to this day. Bombelli was the first person to state the rules for manipulation of complex numbers. He gave explicit rules for addition, subtraction and multiplication of these quantities. Bombelli showed how the Cardano-Tartaglia formula for the roots of cubic equations gave correct real roots even when square roots of negative numbers were involved.

Bombelli’s book on algebra was very influential and, given his account of imaginary quantities, it is reasonable to regard him as the inventor of complex numbers. In MacTutor, Edmund Robertson writes that the book is “one of the most remarkable achievements of 16^{th} century mathematics”, and credits Bombelli for understanding the importance of complex numbers at a time when nobody else did.

The extract from Bombelli’s *Algebra* shown above suggests that he was a man of diplomacy, making no undue claims regarding the value of his own work and anxious to avoid the conflicts that characterised many of the mathematical developments of the time.

**Sources**

MacTutor History of Mathematics archive. O’Connor, John J and Edmund F Robertson: Biography of Raphael Bombelli. __http://www-history.mcs.st-and.ac.uk/Biographies/Bombelli.html__

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