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Activity 2.12 Geometry Figure 2.13a Figure 2.13b 2 Babylonian mathematics The Old Babylonian tablet shown in Figure 2.13 was found in Susa, and appears to show an isosceles triangle and its circumscribed circle. The number on the base of the triangle is 60, and the numbers on the equal sides are 50. 1 Explain why it is necessary to calculate the radius of the circle before you can construct the diagram. 2 Calculate the radius of the circle, and check that your answer agrees with that on the tablet. Babylonian students also used Pythagoras's result to solve problems about ladders leaning against walls. Here is an example from an Old Babylonian text. I A ladder of length 0; 30 stands against a wall. How far will the lower end move out from the wall if the upper end slips down a distance of 0; 6 units? A later, modified, version of this problem can be found on the right-hand side of the New Babylonian tablet whose transcription is shown in Figure 2.14. 25
26 777e Babylonians Figure 2.14 Activity 2.13 Applying Pythagoras's theorem ^mm^w ^^jSriti&^.O'Z&r4:: Here is a translation of Figure 2.14. The lines refer to lines of the right-hand column of the tablet. Line 17 A bar set up against a wall. If the top slips down 3 yards, Line 18 the lower end slides away 9 yards. What is the bar, what the wall? The units are not known. For example, 3 times 3 is 9. 9 times 9 is 1,21. To 9 you will add 1,21. Line 19 1,30 times 0; 30. 45 is it. The reciprocal of 3 is 0; 20. 0; 20 times 45, Line 20 15 is the bar. What is the wall? 15 times 15 is 3,45. 9 times 9 is
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 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 and 46: The Greeks Interpret postulate 3 in
Page 47 and 48: 42 More Greek mathematics 4.1 Some
Page 49 and 50: The Greeks Q.E.D. stands for 'quod
Page 51 and 52: Figure 4.3 46 The Greeks • treat
Page 53 and 54: Figure 4.6 48 The Greeks together w
Page 55 and 56: The Greeks Activity 4.8 The cone Th
Page 57 and 58: The Greeks Practice exercises for t
Page 59 and 60: 5 Arab mathematics You should think
Page 61 and 62: The Arabs Western Arabic, or Gobar,
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 73 and 74: 68 The Arabs Figure 5.10 Activity 5
Page 75 and 76: The Arabs Practice exercises for I
Page 77 and 78: 72 6.1 Read about Descartes The app
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76 Descartes Figure 6.5 Figure 6.5
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Descartes fViete (Vieta in Latin) w
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Descartes The next two sections con
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82 Descartes Activity 6.9 a 3 - b 3
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i——— Figure 6.12 Figure 6.13
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Descartes Practice exercises for th
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Figure 7.2 Descartes Activity 7.1 i
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N- i- L Descartes Thus I have In th
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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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Searching for the abstract Activity
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Searching for the abstract John Wal
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Searching for the abstract 31. Hith
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134 Searching for the abstract If y
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Searching for the abstract Practice
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138 Searching for the abstract Extr
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Searching for the abstract /esseTs
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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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