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In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon … - David Eugene Smith
" "In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation <math>x^2 + ax + b = 0</math> he placed <math>u + z</math> for <math>x</math>. He then had<math>u^2 + (2z + a)u +(z^2 + az + b) = 0.</math>He now let <math>2z + a = 0,</math> whence <math>z = -\frac{1}{2}a,</math>and this gave<math>u^2 - \frac{1}{4}(a^2 - 4b) = 0.</math>
<math>u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.</math>and<math>x = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}.</math>
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About David Eugene Smith
(January 21, 1860 – July 29, 1944) was an American mathematician, educator, and editor.
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Additional quotes by David Eugene Smith
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra.
Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.
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