The History of Mathematics: A Brief Course

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and how others were changed. A description of such a procedure, based partly on

the work of his predecessors, was given by Leon Battista Alberti (1404-1472) in a

treatise entitled Delia pictura, published posthumously in 1511.

Sixteenth-century Italy produced a group of sometimes quarrelsome but always

brilliant algebraists, who worked to advance their science for the sheer pleasure of

making new mathematical achievements. As happened in Japan a century later,

each new advance brought a challenge for further progress.

Scipione del Ferro. A method of solving a cubic equations was discovered by a

lector (reader, that is, a tutor) at the University of Bologna, Scipione del Ferro

(1465-1525), around the year 1500. He communicated this discovery to another

mathematician, Antonio Maria Fior (dates unknown), who then used the knowledge

to win mathematical contests.

Niccolo Tartaglia. Fior met his match in 1535, when he challenged Niccolo Fontana

(1500-1557) of Brescia, known as Tartaglia (the Stammerer) because a wound he

received as a child when the French overran Brescia in 1512 left him with a speech

impediment. Tartaglia had also discovered how to solve certain cubic equations

and so won the contest.

Girolamo Cardano. A brilliant mathematician and gambler, who became rector

of the University of Padua at the age of 25, Girolamo Cardano (1501-1576) was

writing a book on mathematics in 1535 when he heard of Tartaglia's victory over

Fior. He wrote to Tartaglia asking permission to include this technique in his work.

Tartaglia at first refused, hoping to work out all the details of all cases of the cubic

and write a treatise himself. According to his own account, Tartaglia confided the

secret of one kind of cubic to Cardano in 1539, after Cardano swore a solemn oath

not to publish it without permission and gave Tartaglia a letter of introduction to

the Marchese of Vigevano. Tartaglia revealed a rhyme by which he had memorized

the procedure.

Tartaglia did not claim to have given Cardano any proof that his procedure

works. It was left to Cardano himself to find the demonstration. Cardano kept

his promise not to publish this result until 1545. However, as Tartaglia delayed

his own publication, and in the meantime Cardano had discovered the solution of

other cases of the cubic himself and had also heard that del Ferro had priority

anyway, he published the result in his Ars magna (The Great Art), giving credit

to Tartaglia. Tartaglia was furious and started a bitter controversy over Cardano's

alleged breach of faith.

Ludovico Ferrari. Cardano's student Ludovico Ferrari (1522-1565) worked with

him in the solution of the cubic, and between them they had soon found a way of

solving certain fourth-degree equations.

Rafael Bombelli. In addition to the mathematicians proper, we must also mention

an engineer in the service of an Italian nobleman. Rafael Bombelli (1526-1572) is

the author of a treatise on algebra that appeared in 1572. In the introduction to

this treatise we find the first mention of Diophantus in the modern era. Bombelli

said that, although all authorities are agreed that the Arabs invented algebra, he,

having been shown the work of Diophantus, credits the invention to the latter. In

making sense of what his predecessors did he was one of the first to consider the