history of mathematics - National STEM Centre

More documents

Recommendations

Info

Descartes fiene Descartes (1596- 1650) was a Frenchman. He lived a few kilometres south of Tours, and also in Holland and Sweden. He has been called the father of modern philosophy. Chapter 6 The approach of Descartes Chapter 7 Constructing algebraic solutions Chapter 8 An overview of Lei Geometrie Introduction You can think of the algebra of the 12th to 16th centuries in Europe as being based on Latin translations of Arabic work, and being broadly algorithmic in nature. It was not until the 16th century that the Greek mathematical texts became available in comprehensible Latin. It was then that the transformation of algebra under the influence of Greek geometry begun by the Arabic mathematicians was continued in Europe, first by a French mathematician, Viete. Another Frenchman Descartes, by discarding some of Viete's ideas and building on others, was able to make the major breakthrough you will study in this unit. In Chapter 6 of this unit you will learn how Descartes used the introduction of algebra to make Greek construction problems considerably easier to solve. In Chapter 7 you will meet his methods, which show his full genius. Thanks to his mathematical ideas, an algebraic approach for all geometric problems became available, and the basis for modern algebraic geometry was laid. In Chapter 8 Descartes's work is put into perspective. This unit is designed to take about 15 hours of your learning time. About half of f* this time will be outside the classroom. Work through the chapters in sequence. Many of the activities are in a different style from usual. They ask you to read a translation of Descartes's work, and to answer questions to ensure that you have understood it. There are summaries and further practice exercises in Chapter 12. ||| Mathematical knowledge assumed • the contents of Chapter 3 in The Greeks unit is particularly relevant to this unit • for Activity 6.13, you will need to know some of the angle properties of circles; in particular you should know the 'angle at the centre is twice the angle at the circumference' and 'the angles in the same segments are equal' theorems. 7/
72 6.1 Read about Descartes The approach of Descartes 6.12 VanSchooten's problem 6.2 Two geometrical construction problems 6.3 Reviewing the solution 6.4 An algebraic problem 6.5 Viete's legacy 6.6 Reflecting on Descartes, 1 6.7 Reflecting on Descartes, 2 6.8 Reflecting on Descartes, 3 6.9 Reflecting on Descartes, 4 6.10 Reflecting on Descartes, 5 6.11 Reflecting on Descartes, 6 In this chapter, you will be introduced to the kind of problems in which Descartes was interested, and the difficulties and frustrations he experienced when trying to solve them. In Activity 6.1 you will read about Descartes, and make a brief summary of his life. Activities 6.2 and 6.3 introduce you to two apparently similar geometric problems with very different solutions. By contrast, Activity 6.4 is an algebraic problem, leading to the law of homogeneity, an important influence on the development of mathematics. Activity 6.5 shows you a different approach to algebra. Activities 6.6 to 6.11 are short activities based on the translation of passages from Descartes's book. Finally, Activities 6.12 and 6.13 introduce you to Descartes's method in the context of two geometric problems. Start by reading the introduction. You can work on Activity 6.1 at any time. Activities 6.2 and 6.3 are related and you should work them in sequence, followed by Activities 6.4 and 6.5. If you can, work on them in a group. Activities 6.6 to 6.11 are also related, and should be worked in sequence. You should finish with Activities 6.12 and 6.13, which you can work in either order. 6.13 Trisecting an angle
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 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
Page 67 and 68: 62 The Arabs Activity 5.6 Horner's
Page 69 and 70: 64 The Arabs Indian text. For over
Page 71 and 72: D B P N M K Figure 5,9a The Arabs D
Page 73 and 74: 68 The Arabs Figure 5.10 Activity 5
Page 75: The Arabs Practice exercises for I
Page 79 and 80: Descartes Activity 6.1 Head about D
Page 81 and 82: 76 Descartes Figure 6.5 Figure 6.5
Page 83 and 84: Descartes fViete (Vieta in Latin) w
Page 85 and 86: Descartes The next two sections con
Page 87 and 88: 82 Descartes Activity 6.9 a 3 - b 3
Page 89 and 90: i——— Figure 6.12 Figure 6.13
Page 91 and 92: Descartes Practice exercises for th
Page 93 and 94: Figure 7.2 Descartes Activity 7.1 i
Page 95 and 96: N- i- L Descartes Thus I have In th
Page 97 and 98: A Descartes Activity 7.7 Reflecting
Page 99 and 100: 94 Descartes 23 It is true that the
Page 101 and 102: 96 Descartes Figure 7.11 shows a ge
Page 103 and 104: 98 Descartes It is clear that the p
Page 105 and 106: Descartes ractice exercises lor thi
Page 107 and 108: 102 Descartes Activity 8.1 Activity
Page 109 and 110: 104 Descartes In the next section y
Page 111 and 112: Descartes Today we call these numbe
Page 113 and 114: Descartes Activity 8.9 Normals Figu
Page 115 and 116: Descartes Practice exercises for th
Page 117 and 118: 772 The beginnings of calculus 9.1
Page 119 and 120: Figure 9.4 774 Calculus you can wri
Page 121 and 122: Calculus including Descartes, under
Page 123 and 124: 118 Calculus Activity 9.7 Cavalieri
Page 125 and 126: Figure 9. 7 / 120 Calculus dy 1 Wri
Page 127 and 128:
Calculus JPractiee exercises this c
Page 129 and 130:
10.1 Visualising negative numbers 1
Page 131 and 132:
126 Searching for the abstract For
Page 133 and 134:
Searching for the abstract Activity
Page 135 and 136:
Searching for the abstract John Wal
Page 137 and 138:
Searching for the abstract 31. Hith
Page 139 and 140:
134 Searching for the abstract If y
Page 141 and 142:
Searching for the abstract Practice
Page 143 and 144:
138 Searching for the abstract Extr
Page 145 and 146:
Searching for the abstract /esseTs
Page 147 and 148:
Searching for the abstract d Do you
Page 149 and 150:
144 Searching for the abstract math
Page 151 and 152:
146 Searching for the abstract As a
Page 153 and 154:
Searching for the abstract How does
Page 155 and 156:
Searching for the abstract ; Practi
Page 157 and 158:
752 12 Summaries and exercises Two
Page 159 and 160:
754 12 Summaries and exercises 5 Ar
Page 161 and 162:
156 12 Summaries and exercises usin
Page 163 and 164:
158 Hints Activity 2.3, page 15 2 F
Page 165 and 166:
13 Hints Activity 7.3, page 89 1 Wr
Page 167 and 168:
outcome 1 step to the right Figure
Page 169 and 170:
14 Answers 3 0;6,40 and 0;2,13,20 4
Page 171 and 172:
14 Answers 3 If AD intersects the l
Page 173 and 174:
74 Answers 2 Suppose that you can f
Page 175 and 176:
14 Answers 4 If he had a daughter t
Page 177 and 178:
14 Answers Activity 6.5, page 79 1
Page 179 and 180:
14 Answers 6 The equation z 2 + az
Page 181 and 182:
14 Answers b This meets the ellipse
Page 183 and 184:
14 Answers To get an interpretation
Page 185 and 186:
14 Answers of multiplication by neg
Page 187 and 188:
14 Answers b The required number is
Page 189 and 190:
14 Answers operator has the effect
Page 191 and 192:
14 Answers A i- B 5 E %2 5)-25 = 39
Page 193 and 194:
Index Ptolemy 49, 55, 58 Pythagoras
show all

history of mathematics - National STEM Centre

Create successful ePaper yourself

Delete template?

Save as template?