history of mathematics - National STEM Centre

82
Descartes
Activity 6.9
a 3 - b 3 + ab 2 , 1 write Va 3 -b 3 + ab 2 , and similarly for other roots. Here it
must be observed that by a 2 , b 3 , and similar expressions, I ordinarily
mean only simple lines, which, however, I name squares, cubes, etc., so
that I may make use of the terms employed in algebra.
Reflecting on Descartes, 4
1 Descartes introduces an efficient way of working so that it is no longer necessary
to draw all line segments. What is this way of working?
2 From the above extract it is clear that Descartes introduces a kind of revolution.
He does not obey the law of homogeneity. This is evident in one of the sentences in
the text. Which sentence?
3 In the texts before this one he has, in some places, also pushed aside the law of
homogeneity. Give two examples.
To continue:
Activity 6.10 Reflecting on Descartes, 5
5 It should also be noted that all parts of a single line should always be
expressed by the same number of dimensions, provided unity is not
determined by the conditions of the problem. Thus, a 3 contains as many
dimensions as ab 2 or b 3 , these being the component parts of the line
which I have called ^Ja 3 - b 3 + ab 2 It is not, however, the same thing
when unity is determined, because unity can always be understood, even
where there are too many or too few dimensions; thus, if it be required to
extract the cube root of a 2b 2 -b, we must consider the quantity a 2b2
divided once by unity, and the quantity b multiplied twice by unity.
1 How does Descartes remedy the problem of having terms of different
dimensions in one equation?
The solution strategy
At this point Descartes describes the strategy needed to solve a geometrical
construction problem.
6 Finally, so that we may be sure to rememberthe names of these lines, a
separate list should always be made as often as names are assigned or
changed. For example, we may write, AB = 1, that is AB is equal to 1;
GH = a, BD = b, and so on.
7 If, then, we wish to solve any problem, we first suppose the solution
already effected, and give names to all the lines that seem needful for its