By Herrlich H.

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Rooren's equation show that this equation has n solutions (at least when JAI 2). These examples, which are quite explicit for rt --- 3, 5. 7 in Viete's works 164, (Cap. 1 . In later works, such as `{De R ceogn i t i one (On Understanding Equations). published posthumously in 1615, Vlete also stressed the importance of understanding the structure of equations-, meaning by this the relations between roots and eocff icien ts. However, the theoretical tools at his disposal were not sufficiently developed, and he failed to grasp these relations in their full generality.

On the subject of impossible roots, Descartes first seems more cautious than Girard (the emphasis is mine): Every equation can have as many distinct roots as the number of dimensions of the unknown quantity in the equation. [ 16, p. 15). However, Descartes further states: Neither the true nor the false roots are always real; sometimes they are imaginary; that is, while we can always conceive of as many roots for each equation as I have already assigned, yet there is not always a definite quantity corresponding to each root so conceived of.

3) The idea is to determine it in such a way that the right-hand side also becomes a square. 4) V2-ij and, equating the independent terms, we see that this equation holds if'and only if -r + P2 2 +p u+-a 2 _ q2 87L or equivalently, after clearing the denominator and rearranging terms, 8'u'i + 8puL2 + (2p2 - 8r)u - q2 = 0. 4) holds. 2 -+ --P +u2 1 vl-2,u-Y q 2 2u The values of V Care then obtained by solving the two quadratic equations above (one corresponding to the sign + for the right-hand side, the other to the sign ).