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Descartes also notes that it is possible to shift the roots of any equation by a certain value, even if those roots are unknown. If you wish to increase the value of each root by 3, simply substitute y − 3 into the given equation wherever x occurs; the new equation in powers of y will, by definition, have roots y = x + 3, shifted as required.

Compare this with its modern form: 3 √ 3 √ x= 108 + 10 − 108 − 10. In contrast, Vi`ete writes “A cubus + B plano 3 in A, aequari Z solido 2” for the modern x3 + 3B 2 x = 2Z3 . He uses vowels for unknowns, consonants for coefficients, “cubus” or “solido” for “cubed,” and “plano” for “squared”; thus, “A cubus” is x3 , “B plano 3” is 3B 2 , and “Z solido 2” is 2Z3 . Note that he uses the modern + and − signs and also the radical √ sign . ” Moreover, the manner of expression of the algebraists left much to be desired.

With his master’s honor at stake, Ferrari challenged Tartaglia to a mathematical duel that ended with Ferrari’s victory, or so it seems from subsequent events, for Ferrari obtained many flattering offers, including a professorship in Bologna, while Tartaglia’s account reveals that he left even before the contest was over. How, then, was the cubic equation solved? The method turns out to be an early instance of a powerful mathematical strategy: to solve a more difficult problem, reduce it to a simpler problem you have already solved.

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A Completeness Theorem in Modal Logic (paper) by Kripke Saul


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