By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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**Extra resources for A selection of problems in the theory of numbers **

**Example text**

Viète treated all his examples of avulsed three-term equations in similar fashion. In each case he gave instructions for calculating bounds for the roots. He also gave rules for calculating the second root from the first, and confirmed the correctness of the rules by extracting the second root directly. Nowhere, however, did he give any derivations or explanations. With De resolutione, even more than with De aeqationum, one is left feeling that Viète had done far more work behind the scenes than he was prepared to explain.

Bombelli’s treatment of quadratic, cubic, and quartic equations is in Book II. Like Cardano and other contemporary writers he wrote equations as relationships between positive terms, and dealt with each possible case separately: 3 cases for quadratics, 13 cases for cubic, and 43 cases for quartics. His treatment of quadratics was standard. For cubics, right from the beginning, he taught the transformations by which equations of one type could be changed to equations of another. These were exactly those given by Cardano: replace x by k=x or by x ˙ k for some suitable value of k.

Harriot, however, was one such reader, and by chance was also fortunate enough to acquire most of Viète’s published work. 10 His notes on Viète’s ‘positively affected’ powers fill twelve manuscript pages. Those on Viète’s ‘negatively affected’ powers fill a further twelve pages, in which the letter ‘b’ has been added to the pagination. The ‘avulsed’ powers are on eighteen pages marked with the letter ‘c’. The most obvious differences between Harriot’s re-writing and Viète’s original are changes in notation.