history of mathematics - National STEM Centre

7 Constructing algebraic solutions
6 In the previous passages Descartes has considered the following equations.
z 2 +az-b 2 = 0
z 2 -az-b 2 =0
z 2 -az + b 2 =0
Why does Descartes not consider the equation z 2 + az + b 2 = 0 ?
Subsequently, Descartes compares his work to that of the mathematicians of
antiquity.
Activity 7.6 Reflecting on Descartes, 11
18 These same roots can be found by many other methods. I have given
these very simple ones to show that it is possible to construct all the
problems of ordinary geometry by doing no more than the little covered in
the four figures that I have explained. This is one thing which I believe the
ancient mathematicians did not observe, for otherwise they would not
have put so much labour into writing so many books in which the very
sequence of the propositions shows that they did not have a sure method
of finding all, but rather gathered together those propositions on which
they had happened by accident.
1 According to Descartes, what is missing from the work of previous
mathematicians when solving geometrical construction problems?
In the remaining part of Book I, Descartes addresses a complex problem from
antiquity, known as Pappus's problem. In treating Pappus's problem, Descartes
drops a remark which is particularly apt as a conclusion for this section.
19 Here I beg you to observe in passing thatthe considerations that
forced ancient writers to use arithmetical terms in geometry, thus making
it impossible for them to proceed beyond a point where they could see
clearly the relation between the two subjects, caused much obscurity and
embarrassment in their attempts at explanation.
Solutions which are curves
Descartes had previously said that a construction problem has a finite number of
solutions if the algebraic formulation of the problem has as many equations as it has
unknown quantities. In one of the previous quotations, paragraph 8 in Chapter 6, he
touched briefly upon the problem that arises when there are more unknown
quantities than there are equations.
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