history of mathematics - National STEM Centre

More documents

Recommendations

Info

7.1 Practice with constructions 7.2 Reflecting on Descartes, 7 7.3 Reflecting on K Descartes, 8 !•' 7.4 I'" Reflecting on i Descartes, 9 7.5 Reflecting on Descartes. 10 i .._.___. 7.6 | Reflecting on Descartes, 11 I' 7 -7 ?•' Reflecting on Descartes. 12 Constructing algebraic solutions 7.13 Reflecting on Descartes. 18 7.14 Reflecting on Descartes, 19 7.15 Reflecting on Descartes. 20 This chapter delves further into Descartes's work. After reminding you about some simple constructions, it shows how Descartes constructed the geometric lengths corresponding to algebraic expressions, using straight lines and circles. Descartes goes on to argue that other curves should also be allowed. Activity 7.1 gives you practice at constructing geometric lengths corresponding to algebraic expressions. Activities 7.2 to 7.6 give a more systematic method for these constructions. In Activities 7.7 to 7.9 you will see how Descartes considered other curves. In Activities 7.10 to 7.14, you will learn how he argued against the restriction of the curves used for the solution of construction problems to the straight line and circle. Finally, in Activity 7.15, you will return to the problem of trisecting an angle, which you first considered in Chapter 6. "Work on the activities in sequence. agKp^ .^ All the activities are fairly short, and you should try to work several of them in one sitting. • ili|llii|||l/ All the activities are suitable for small group working. Introduction Having introduced algebraic methods for solving geometric problems and also relaxed the law of homogeneity, Descartes had at his disposal a generally applicable method for analysing geometrical construction problems. After an algebraic analysis, the next step in solving geometric problems is the translation of this analysis into a geometrical construction of the solution. You have already seen some simple examples of these 'translations'. Here are some reminders. Example 1 Let a, b, and c be line segments. Construct the line segment z such that ab 87
Figure 7.2 Descartes Activity 7.1 i——— Figure 7.3 Figure 7.4 Solution • Draw an arbitrary acute angle O, as shown in Figure 7.1. • Mark off the distances b and c along one side of the angle such that OB = b and OC = c, and mark off a along the other side so that OA = a. • Draw BZ parallel to AC. • Then OZ is the required line segment with length z. Example 2 Let a be a line segment. Construct the line segment z such that z = Va • Solution • Take the length a and the length of the unit and draw them in line, as in Figure 7.2, so that OA = a and OE = 1. • Draw a circle with diameter AE. • At O, draw a line perpendicular to AE. Call the point where this line intersects the circle, Z. • Then OZ is the required line segment with length z. In the next activity there are two constructions for practice. If you want to use a unit, take 1 cm as the unit. Practice with constructions 1 Let a and b be the two line segments shown in Figure 7.3. Construct the line segment z for z = ^^ab 2 Look again at Van Schooten's problem in Activity 6.12. Figure 7.4 shows a straight line AB, with a point C on it. The problem was to find the point D on AB produced, such that the rectangle with sides AD and DB is equal (in area) to the square with side CD. In Activity 6.12, you analysed this problem algebraically, and should have found i 2 i that x = — — , or — = ——— . Construct x using one of these expressions as a a — b b a — b starting point. Constructions requiring a circle One of the 'rules of the game' applied to geometrical constructions from antiquity was that only a pair of compasses and a straight edge were used to execute the constructions. Descartes discusses this in Book I: 12 I shall therefore content myself with the statement that if the student, in solving these equations, does not fail to make use of division wherever possible, he will surely reach the simplest terms to which the problem can be reduced.
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: Descartes Practice exercises for th
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?