history of mathematics - National STEM Centre

Activity 6.3 Reviewing the solution
Recall from Chapter 3,
that a quadrature
problem involved finding
a square which was
equal in area to another
figure.
Cardano and Tartaglia
were both involved in the
solution of cubic
equations. If you would
like to read about their
rivalry, see 'Tartaglia, the
stammerer' in the
Mathematics reader.
6 The approach of Descartes
1 Compare your own attempt at the solution of the AX = 2BX problem, and the
solution of some fellow students, with that presented in the text above. Which of the
solutions do you judge to be the most apt and practical?
2 Why do you think it the most apt? In what respect is it more practical than the
other solutions?
3 Did any of you use algebra in analysing the problem? Is algebra used in the
solution presented in the text above?
4 In the light of the remarks made just after Activity 6.2 and at the beginning of
this section, reflect briefly on the solution in the text to question 3 in Activity 6.2,
comparing it with your solution to question 1 in Activity 6.2.
Interest in the culture of the ancient Greeks was one of the leading themes in these
times. Mathematicians had a keen interest in Greek geometry, and a Latin
translation of Euclid's Elements was published in 1482. Geometrical construction
problems, not least quadrature problems, were studied on a large scale.
You may have noticed that one of the important subjects in present day
mathematics, algebra, has hardly been mentioned up to now in this unit. People used
algebra in very early times, for example, in Mesopotamia in about 1800 BC, as you
have seen in The Babylonians unit, in the later Greek period and in Arab
mathematics. In 16th century Italy, through the work of Cardano, Tartaglia and
others, algebra became valued and popular. Cardano, who wrote a famous algebra
book, Ars Magna, in 1545, considered it to be a 'great art'. But, before Descartes,
geometry and algebra were separate disciplines with almost nothing in common.
The 17th century was to change that significantly.
The law of homogeneity
Activity 6.4 An algebraic problem
Figure 6.8
Consider a cube of side a, shown in Figure 6.8. Three numbers associated with this
cube are the total length, L, of the edges of the cube, the total surface area, A, of the
cube, the volume, V, of the cube.
1 For which value of a is A = L7
2 For which value of a is V = L1
3 For which value of a is V + A + L = 90 ?
Give your answers correct to two decimal places. Use a graphics calculator if
necessary.
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