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4 27 Thus, x= 3 b + 2 b2 a3 + − 4 27 3 b − + 2 b2 a3 + . 4 27 When a and b are positive, the root x is thus also positive. The calculation presented here follows only the idea of Cardano’s argument. He himself argued geometrically: If we divide a cube of side β = α + x by planes, parallel to its faces, into one cube of side α and one of side x, then in addition to those two cubes we obtain three rectangular parallelepipeds with sides α, α, x and three with sides α, x, x. , pp. 8–9. —Transl. 6 François Viète, or Vieta (1540–1604), lived after Cardano but Cardano essentially knew this result, now known as Vieta’s theorem.

He spent a great deal of effort on carefully verifying and substantiating the rule. From our standpoint it is not easy to understand the difficulty: Just substitute into the equation and verify it! But the absence of a well-developed algebraic notation made what any schoolchild today can do automatically, accessible to only a select few. Without knowing the original texts from that time we cannot appreciate how much the algebraic apparatus “economizes” thought. The reader must always keep this in mind, so as not to be deluded by the “triviality” of the problems over which passions seethed in the 16th century.

In addition he had two similar pendula, but of rather different lengths. He observed that while the short one made a certain number of oscillations, for example, 300 along its longest arcs, in the same time that the long one always made the same number, say 40, both along its longest arcs and its shortest ones; repeating this several times. . , he concluded from this that the time to go back and forth is the same for the same pendulum, the longest or the shortest, and that there are almost no notable differences in this which must be attributed to interference by the air, which resists a faster moving heavy object more than a slowly moving one.

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A brief history of mathematics by Fink K.

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