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. _ Practice exercises tor this chapter are on page 152. 3 An introduction to Euclid • the meaning of 'quadrature' • how to draw a square equal in area to a given polygon • the three classical geometric problems. 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 Use the methods of this chapter to construct a square equal in area to an equilateral triangle. 2 Use Euclid's rules, and Euclid's proposition 1.1, to construct a regular hexagon. 41
42 More Greek mathematics 4.1 Some geometric algebra 4.2 The Euclidean algorithm 4.3 A proposition about prime numbers 4.4 Incommensurables 4.5 Heron's algorithm to find square roots 4.6 Archimedes and it 4.7 The area under a parabolic arc 4.8 The cone 4.9 Naming the conies In this chapter, you will investigate some of the Greeks' achievements in areas other than geometric constructions. In Activities 4.1 to 4.4, you will see more of Euclid's work, including some geometric algebra and some work on numbers. Then, in Activity 4.5, you will meet a square root algorithm. Activities 4.6 and 4.7 look briefly at some of the work of Archimedes; then Activities 4.8 and 4.9 introduce you to the work of Apollonius. All the activities are suitable for small group working. You should work Activities 4.1 to 4.4 in sequence. Book II of Euclid's Elements is short, containing only 14 propositions, each of which is concerned with geometric algebra. Modern mathematics at secondary school level makes extensive use of algebraic notation; magnitudes are represented by letters that are understood to be numbers, and arithmetic operations can be applied to these under the usual rules of algebra. The Greek mathematicians of Euclid's time used line segments to represent magnitudes, and they operated on these using the theorems and axioms of geometry. Activity 4.1 Some geometric algebra 1 The following is proposition 1 from Book II of the Elements. If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.
Page 2 and 3: Nuffield Advanced Mathematics Histo
Page 4 and 5: Contents Introduction 7 Unit 1 The
Page 6 and 7: Introduction This book, about the h
Page 8 and 9: The Babylonians Chapter 1 Introduct
Page 10 and 11: Introduction to the Babylonians On
Page 12 and 13: Figure 1.2 Stele inscribed with Ham
Page 14 and 15: Token V Figure 1.5 7 Introduction t
Page 16 and 17: There are no practice exercises for
Page 18 and 19: A sexagesimal number system is a sy
Page 20 and 21: ff m yfr < n Figure 2.5 «« Activi
Page 22 and 23: Activity 2.5 Babylonian fractions 2
Page 24 and 25: Activity 2.6 Babylonian arithmetic
Page 26 and 27: Figure 2.10 Activity 2.9 2 Babyloni
Page 28 and 29: Type Quadratic Table 2.1 x +ax = b
Page 30 and 31: Activity 2.12 Geometry Figure 2.13a
Page 32: Practice exercises for this chapter
Page 35 and 36: 3 An introduction to Euclid fthese
Page 37 and 38: The Greeks Activity 3.3 B Figure 3.
Page 39 and 40: 34 The Greeks Activity 3.4 g Now us
Page 41 and 42: 36 The Greeks inscribes another wit
Page 43 and 44: 38 The Greeks Activity 3.7 therefor
Page 45: The Greeks Interpret postulate 3 in
Page 49 and 50: The Greeks Q.E.D. stands for 'quod
Page 51 and 52: Figure 4.3 46 The Greeks • treat
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Page 55 and 56: The Greeks Activity 4.8 The cone Th
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Page 59 and 60: 5 Arab mathematics You should think
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Page 63 and 64: The Arabs Activity 5.2 Figure 5.3a
Page 65 and 66: 7776 Arabs Activity 5.4 This proble
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Page 89 and 90: i——— Figure 6.12 Figure 6.13
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A Descartes Activity 7.7 Reflecting
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94 Descartes 23 It is true that the
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96 Descartes Figure 7.11 shows a ge
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98 Descartes It is clear that the p
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Descartes ractice exercises lor thi
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102 Descartes Activity 8.1 Activity
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104 Descartes In the next section y
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Descartes Today we call these numbe
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Descartes Activity 8.9 Normals Figu
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Descartes Practice exercises for th
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772 The beginnings of calculus 9.1
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Figure 9.4 774 Calculus you can wri
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Calculus including Descartes, under
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118 Calculus Activity 9.7 Cavalieri
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Figure 9. 7 / 120 Calculus dy 1 Wri
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Calculus JPractiee exercises this c
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10.1 Visualising negative numbers 1
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126 Searching for the abstract For
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138 Searching for the abstract Extr
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Searching for the abstract d Do you
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144 Searching for the abstract math
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146 Searching for the abstract As a
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Searching for the abstract How does
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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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