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When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to sol… - David Eugene Smith
" "When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation <math>ax^2 + bx + c = 0</math>, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the science was known about 1800 B.C., and probably still earlier.<
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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
Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants.
Although Cardan reduced his particular equations to forms lacking a term in <math>x^2</math>, it was Vieta who began with the general form<math>x^3 + px^2 + qx + r = 0</math>and made the substitution <math>x = y -\frac{1}{3}p,</math> thus reducing the equation to the form<math>y^3 + 3by = 2c.</math>He then made the substitution<math>z^3 + yz = b,</math> or <math>y = \frac{b - z^2}{z},</math>which led to the form<math>z^6 + 2cz^2 = b^2,</math>a sextic which he solved as a quadratic.
The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.
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