history of mathematics - National STEM Centre

More documents

Recommendations

Info

8 An overview of La Geometrie The importance of Descartes's work Wfitii 8.2 Different curves 8.3 Reflecting on Descartes, 21 8.4 Reflecting on Descartes, 22 8.5 Reflecting on Descartes, 23 8.6 Reflecting on Descartes, 24 8.7 Reflecting on Descartes, 25 { 8.8 The rule of signs 8.9 Normals 8.10 Reflecting on Descartes, 26 8.11 Coordinate geometry This chapter gives an overview of Descartes's work on curves. It then goes on to show how this led Descartes into some work on the solution of equations, the Descartes's rule of signs, which enables you to gain some information about whether the roots are positive or negative. The first of the final two sections deals with finding normals to curves. The second compares the mathematical work of Descartes and his contemporary Frenchman, Fermat. The work of Fermat and Descartes together forms the starting point of analytical geometry. Activity 8.1 is concerned with the general importance of Descartes's work that you have seen so far. Then in Activities 8.2 to 8.4 you will learn about his attempts to classify curves. Activities 8.5 to 8.8 are about the work that Descartes did on finding the number of roots of an algebraic equation from the number of sign changes of its coefficients. Activity 8.9 shows how Descartes found the equations of normals to conies. Activities 8.10 and 8.11 compare the work of Descartes with that of his contemporary, Fermat. Work through the activities in sequence. Descartes's La Geometrie is a revolutionary work in the history of mathematics. Its purpose - finding a generally applicable method for geometrical constructions - was not new. The revolutionary aspects of the work lie primarily in the new methods he pioneered. The work itself, on geometric constructions, really only supported and illustrated Descartes's philosophical thinking, even though it changed the mathematics of the time completely. However, its effect was that geometrical constructions, which had been a subject of study for centuries, received less, rather than more, attention soon after La Geometrie was published. Here are the four major new features of Descartes's method. • Using algebra to analyse geometrical construction problems. Reduce the geometrical problem to a set of given and required line segments. Give letters to these line segments and translate the problem into algebraic relations between these letters. This removes the need to draw every new construction step by step. Instead, you can find the final construction by manipulating algebraic equations. 101
102 Descartes Activity 8.1 Activity 8.2 • Introducing a unit and relaxing the law of homogeneity. By choosing a line segment as the unit, you can consider areas, volumes, and all other quantities with a dimension different from that of a line segment, as line segments. In this way algebra is released from its geometric constraints and becomes a powerful tool. • Translating algebraic equations into corresponding geometrical constructions. If the algebraic equation has been reduced to its simplest form, the final construction of the unknown line segment follows immediately. The line and the circle are no longer the only construction curves. Other curves are now equally acceptable, provided that they can be described by a sequence of continuous movements which are in the end directly and clearly coupled to a straight or to a circular movement. In the final construction, the most suitable curve must be used. • A modern algebraic notation. Apart from a few exceptions, Descartes's notation is the same as today's. The importance of Descartes's work Be prepared to bring your answer to this activity to your next review. 1 Which of the features claimed as revolutionary has impressed you most? Write your answer with reasons in five sentences at most. Include: • breaking with old traditions • the importance for today's mathematics. La Geometric contains even more revolutionary ideas. Some of them will be mentioned in this chapter, if only briefly. It is remarkable that the third point above - translating algebraic equations into geometric constructions - receives the most attention in La Geometric. Descartes did much new work in this field, which was at the frontiers of research. Here are some questions which arise from this work. • Which curves, besides the circle and the line, are acceptable as construction curves? • What is the simplest form of an algebraic equation and how do you find it? • What curves are most suitable for a given geometrical construction? A paragraph of La Geometric, quoted in paragraph 26 of Chapter 7 of this book, shows that Descartes had started looking for a kind of classification of curves. Different curves Read once again paragraph 26 from La Geometric. 1 According to Descartes, what property must be the distinguishing factor between the different classes of curves?
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 and 76: The Arabs Practice exercises for I
Page 77 and 78: 72 6.1 Read about Descartes The app
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: Descartes ractice exercises lor thi
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?