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There were (I think) no number theoretic or relation to science that time, and the only impression I get from reading books is that they did it too one-up other persons on competitions.

That's not a mathematical Motivation. What was the mathematical Motivation ?

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  • $\begingroup$ Why must there be a mathematical motivation? At the time the cubic formula was discovered mathematical challenges to one-up someone were an activity going on among some people in Europe. Also, it seems to me that there is a pretty obvious motivation within math anyway: find an extension of the quadratic formula to polynomials of higher degree (once the concept of a polynomial is formulated). $\endgroup$ – KCd Jul 24 '17 at 10:20
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    $\begingroup$ Could you explain what you mean by "mathematical motivation"? What was the "mathematical motivation" of Diophantes in solving Diophantine equations? $\endgroup$ – Alexandre Eremenko Jul 24 '17 at 11:27
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    $\begingroup$ There was a number of geometric problems inherited from ancients that were known by that time (largely due to Arabs) to reduce to cubics. Omar Khayyám systematically solved all of them by geometric means (intersecting conic sections), but it was generally felt that algebraic solution would be preferable. On that Pacioli opined that "solving the cubic is as impossible as the quadrature". By the time of del Ferro and Cardano the symbolism was developed enough to make it possible, see How was geometry historically used to solve polynomial equations? $\endgroup$ – Conifold Jul 25 '17 at 2:31
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This appears to have arisen out of the interaction of two men, Tartaglia and Cardano. Of the two, Tartaglia was the first to address the problem, but Cardano's solutions probed the matter deeper, covered more cases, and was basically the more "general" method.

It is noteworthy that Cardano's lasting legacy is that one of his students, Ferrari solved the quartic shortly afterward.

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