By Paul Drijvers, Aad Goddijn, Martin Kindt (auth.), Paul Drijvers (eds.)
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Extra resources for Secondary Algebra Education: Revisiting Topics and Themes and Exploring the Unknown
This gives the following three numbers as a After finding the value of x = 1365 762 solution: 58 --------484 1878 -----------484 7338 -----------484 As shown by the final steps of his treatment, Diophantus no longer wants to use fractions in the numerator or denominator, but in this case he does use the common quadratic numerator, so that the fractions are not given in their simplest form. ” It is certainly evident, and calculation shows that the solution is correct. Nevertheless, it is striking that Diophantus does not use the word ‘proof’ until this point in the text.
Viète: Si A cubus – B – D – G in A quad. + B in D + B in G + D in G in A, aequatur B in D in G: A explicabilis est de quadlibet illarum trium, B, D, vel G. In today’s language: 3 2 If A + – B – D – G A + BD + BG + DG A = BDG then A is equal to one of these three: B, D or G. However, here we are given the strong impression that A, B, D, and G have equivalent roles in the equation. After all, A is equal to one of the three. Thomas Harriot (1560 – 1621) made this much more explicit by ascertaining that if a, b and c are the solutions of a third-degree equation, then the equation must be: x – a x – b x – c = 0 39 AAD GODDIJN Later on (in 1673), in a letter to Huygens, Leibniz used pure algebra to shown that if a, b and c are given by the solution formulas of Cardano/Bombelli, then the thirddegree equation can indeed be found.
The rule only indicates how to deal with ‘difference times difference’. The avoidance of negative numbers is apparent in the solution to the above problem 39. iv Cubic and quartic equations In the above examples, we have seen that before the year 1000, people were working on second-degree equations in various ways. We saw special examples with the Babylonians, and with Diophantus we found indeterminate equations, where whole or rational solutions were sought. Later on, we will address the geometric methods, which Euclid and the Arabic mathematicians used for quadratic equations in one unknown.