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M Figure 4.10 F L Figure 4.11 This result, in the words of Apollonius is: I 'if 4 More Greek mathematics we apply the square LPto a straight line length k, we obtain the abscissa FL', or'the exact parabola of the square of the ordinate on the line segment k gives the abscissa'. This is based on a much earlier use of the word 'parabola', possibly introduced by the Pythagoreans. Hence Apollonius named this curve the parabola. 2 In the second case, FL cuts AB and AC at the points F and J (Figure 4.10). a Use an argument like the one in question 1 to show that LP 2 = LM x LN is proportional to LF x LJ. b Suppose that the constant of proportionality is the ratio of some length FK to the length FJ (Figure 4.11). Show that there is a point Q on KJ such that LP 2 is equal to the rectangle FLQR. In Apollonius's words, this is to say that 'the square of LP is applied to the line FJ in ellipsis by a rectangle LJSQ, similar to the rectangle FJTK'. Here, the term 'ellipsis' means 'deficiency'. Hence this conic section was named the ellipse. 3 In the third case, FL cuts AB at F, and cuts CA produced at J. The Greek word 'hyperbola' means 'in excess' or 'a throwing beyond'. Use an argument similar to that in question 2 to explain why Apollonius named the curve in this third case the hyperbola. The way in which Apollonius worked with the conies, using abscissa and ordinates, suggests that he was very close to developing a system of coordinates. Many of his results can be re-written almost immediately into the language of coordinates, even though no numbers were attached to them. As a result, it has sometimes been claimed that Apollonius had discovered the analytic geometry which was developed some 1800 years later. However, the fact that the Greeks had no algebraic notation for negative numbers was a great obstacle to developing a coordinate system. Greek mathematicians classified construction problems into three categories. The first class was known as 'plane loci', and consisted of all constructions with straight lines and circles. The second class was called 'solid loci' and contained all conic sections. The name appears to be suggested by the fact that the conic sections were defined initially as loci in a plane satisfying a particular constraint, as they are today, but were described as sections of a three-dimensional solid figure. Even Apollonius derived his sections initially from a cone in three-dimensional space, but he went on to derive from this a fundamental property of the section in a plane, which enabled him to dispense with the cone. The third category of problems was 'linear loci', which included all curves not contained in the first two categories. Hypatia Hypatia of Alexandria was born in about AD 370 towards the end of the Greek era. She was the daughter of Theon of Alexandria, known for his commentary on Ptolemy's Almagest and his edition of Euclid's Elements. 51
The Greeks Practice exercises for this chapter are on page 153. 52 Hypatia' s main contribution appears to have been one of continuing the work of her father, writing commentaries on the works of the great Alexandrian mathematicians, Apollonius and Diophantus. She is thought to have written a commentary on an astronomical table of Diophantus, on Diophantus's Arithmetic, on work related to Pappus, on Apollonius's work on conies, and to have written her own work about areas and volumes. She became a respected teacher of the Greek classics. She was invited to be president of the Neoplatonic School at Alexandria, which was at the time one of the last institutions to resist Christianity, and has been described as one of the last pagan, that is, non-Christian, philosophers. She was martyred in AD 415. Possibly because of her martyrdom she occupies an exalted place in the history books. Her writings have been lost, so it is hard to evaluate her work objectively. The commentaries of others of her time have provided the world with one of the best sources of information about the development of Greek mathematics. Since it was hard to understand Euclid's Elements or Apollonius's Conies without help from teachers and commentators, Hypatia's death can be thought of as ending the Greek era. Reflecting on Chapter 4 What you should know • the contributions of Euclid, Archimedes, Apollonius and Hypatia • how to interpret Euclidean theorems of geometric algebra in terms of modern algebra • what is meant by incommensurability • how the Greeks proved that some square roots are incommensurable • why the conic sections are named 'parabola', 'ellipse' and 'hyperbola'. 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 Make brief notes on the lives and contributions of Euclid, Archimedes, Apollonius, and Hypatia. 2 Write in your own words a paragraph about the significance to the Greeks of the discovery of incommensurable quantities. 3 Prepare some examples of each of the three categories of construction problems.
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Nuffield Advanced Mathematics Histo
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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
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Page 55: The Greeks Activity 4.8 The cone Th
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
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Page 95 and 96: N- i- L Descartes Thus I have In th
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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 31. Hith
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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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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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