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Activity 7.14 Reflecting on Descartes, 19 Figure 7.14 Figure 7.15 7 Constructing algebraic solutions 1 Descartes mentions the circle, the parabola, the hyperbola and the ellipse as the representatives of the simplest class of curves (those with no terms higher than the second degree). Did he forget one? If so, which one? To conclude this chapter, here is an example of a geometrical construction corresponding to an equation of the third degree: the trisection of an angle, which you saw at the end of Chapter 6. Descartes shows this construction in Book III. First his analysis: 27 Again, let it be required to divide the angle NOP, or rather, the circular arc NQTP, into three equal parts. Let NO = 1 be the radius of the circle, NP = q be the chord subtending the given arc, and NQ = z be the chord subtending one-third of that arc; then the equation is z 3 =3z-q. For, drawing NQ, OQ and OT, and drawing QS parallel to TO, it is obvious that NO is to NQ as NQ is to QR as QR is to RS. Since NO = 1 and NQ = z,then QR = z 2 and RS = z 3 ; and since NP or q lacks only RS or z 3 of being three times NQ or z, we have q - 3z - z 3 or z 3 = 3z - q. In a way similar to that in Activity 6.13, Descartes arrives at the equation z 3 =3z-g. Descartes uses a parabola and a circle in his construction of the length z. The method is similar to that of Omar Khayyam, as described in Chapter 5. It is shown in modern notation in Figure 7.15. The figure shows the graph of the parabola y = —x and the circle centre E (-•j
Descartes ractice exercises lor this chapter are on WO how to construct solutions to the following: y 2 = -ay + b 2 Z 2 =az-b 2 Preparing for your next review • Reflect on the 'What you should know list' for this chapter. Be ready for a discussion on any of the points. • Answer the following check questions. 1 Make sure that you can explain the construction for solving the equation Z 2 =az + b 2 . 2 Give an example of a problem which cannot be solved by using a straight edge and compasses only, but can be solved by using other curves. 3 What other curves are allowed by Descartes?
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Nuffield Advanced Mathematics Histo
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Contents Introduction 7 Unit 1 The
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Introduction This book, about the h
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The Babylonians Chapter 1 Introduct
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Introduction to the Babylonians On
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Figure 1.2 Stele inscribed with Ham
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Token V Figure 1.5 7 Introduction t
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There are no practice exercises for
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A sexagesimal number system is a sy
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ff m yfr < n Figure 2.5 «« Activi
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Activity 2.5 Babylonian fractions 2
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Activity 2.6 Babylonian arithmetic
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Figure 2.10 Activity 2.9 2 Babyloni
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Type Quadratic Table 2.1 x +ax = b
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Activity 2.12 Geometry Figure 2.13a
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Practice exercises for this chapter
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3 An introduction to Euclid fthese
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The Greeks Activity 3.3 B Figure 3.
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34 The Greeks Activity 3.4 g Now us
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36 The Greeks inscribes another wit
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38 The Greeks Activity 3.7 therefor
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The Greeks Interpret postulate 3 in
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42 More Greek mathematics 4.1 Some
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The Greeks Q.E.D. stands for 'quod
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Figure 4.3 46 The Greeks • treat
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Page 59 and 60: 5 Arab mathematics You should think
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Page 65 and 66: 7776 Arabs Activity 5.4 This proble
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Page 129 and 130: 10.1 Visualising negative numbers 1
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Searching for the abstract ; Practi
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752 12 Summaries and exercises Two
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754 12 Summaries and exercises 5 Ar
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156 12 Summaries and exercises usin
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158 Hints Activity 2.3, page 15 2 F
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13 Hints Activity 7.3, page 89 1 Wr
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outcome 1 step to the right Figure
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14 Answers 3 0;6,40 and 0;2,13,20 4
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14 Answers 3 If AD intersects the l
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74 Answers 2 Suppose that you can f
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14 Answers 4 If he had a daughter t
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14 Answers Activity 6.5, page 79 1
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14 Answers 6 The equation z 2 + az
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14 Answers b This meets the ellipse
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14 Answers To get an interpretation
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14 Answers of multiplication by neg
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14 Answers b The required number is
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14 Answers operator has the effect
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14 Answers A i- B 5 E %2 5)-25 = 39
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Index Ptolemy 49, 55, 58 Pythagoras
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