history of mathematics - National STEM Centre

Activity 6.5 Viete's legacy
6 The approach of Descartes
As you saw in The Arabs unit, their work on algebra was carried out using special
cases. For example, ABC could represent any triangle, but there was no equivalent
way of representing all possible quadratic equations.
Viete introduced a notation in which vowels represented quantities assumed
unknown, and consonants represented known quantities.
In modern notation he would have written BA 2 + CA + D = 0 for quadratic
equations. However, he used medieval rather than the current notation for powers,
such as 'A cubus, A quadratus', and 'In' (Latin) instead of 'x', 'aequatur' instead of
'equals'.
1 Translate the following equations from Viete's work into modern notation.
Assume that the word 'piano' means 'plane'.
a A quadratus + B2 in A, aequatur Z piano
b D2 in A - A quadratus, aequatur Z piano
c A quadratus - B in A 2, aequatur Z piano
d Explain how the three equations in parts a, b and c fit Viete's ideas about the
law of homogeneity.
e Why would Viete have needed to solve each of the equations in pans a, b and c
separately?
Descartes's breakthrough
When you cannot use algebra, you cannot use a generally applicable method to
solve geometrical problems. If you can use algebra, a second difficulty emerges:
how do you translate this algebraic solution into a geometric construction?
It was also impossible to solve the harder problems algebraically without violating
the law of homogeneity.
It was his frustration with these difficulties that led Descartes to write La Geometric,
which was his response to a number of questions related to geometrical construction
methods that had so far remained unanswered.
• Is it possible to find a generally applicable solution method for the problems
that, until Descartes's time, had been solved in a rather unstructured manner?
• How can you tell whether or not it is possible to carry out a particular
construction with a straight edge and a pair of compasses?
• What method can you use if it is not possible to carry out a construction with a
straight edge and a pair of compasses?
Descartes worked to find a single strategy to solve these geometrical problems. The
strategy that he gives in La Geometric is an elaboration of his general philosophical
thinking: a method leading to truth and certainty.
La Geometric comprises three parts, which Descartes called 'books'.
• Book I 'Construction problems requiring only lines and circles'
• Book II 'About the nature of curves'
• Book III 'About constructions requiring conic sections and higher-order curves'
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