history of mathematics - National STEM Centre

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Abu Kamil 65-7 al-Hayatham 40,67 al-Kashi 59 al-Khwarizmi 59,64-5,127,129 al-Tusi, Sharafal-Din 69 algebra 74,77,78,101-2 Arab 64-7,79 Babylonian 22-4 Boolean 143 Clifford 144-8 amicable numbers 61-2 analogy, arguing by 138,139,141 angle bisecting 31 -2 trisecting 34, 85, 99, 106 Antiphon 34,35 Apollonius of Perga 49-51, 94 Arab mathematics 22, 53-69 Archimedes 46-8, 60, 67, 113-14 Aristotle 35, 39, 40 Arnauld, Antoine 130,131 astronomy 13,49,55,60 axioms 39, 141-2, 143, 149 see also under Euclid B Babylonian mathematics 3-27, 55, 63, 125-6 Barrow, Isaac 120 Bel), AH 60 binomial series 121 Boole, George 143 Brahmagupta 60-61, 127 Bryson 34,35-6 bullae 9-10 calculus 109-10,111-22 Cardano, Geronimo 77,125 'cattle' problem 60 Cavalieri, Bonaventura 117, 118 Cayley, Arthur 142 'chess-board' problem 59, 60 Chin Chiu Shao 62 Chinese mathematics 20,56,64, 125-6, 127 see also Chiu-chang suan-shu Chiu-chang suan-shu 62,63,126-7 circle 98 squaring 34-6 clay tablets 8, 10-20, 22, 24-7 Clifford, William Kingdon 143-4 commutative operations 147 complex numbers 105, 139-40, 142, 143, 146 conic sections 49-51, 67-8, 94 constructions 30-32, 74-7, 79, 88-91 Descartes 's solution strategy 79-84,87-8, 101-2 Greek classification 5 1 , 94 instruments for 95, 96 solution curves 91,92 coordinate systems 51,92,108 counting, evolution 8-9 cube, doubling 34, 89 curves 49-51,93-5,98-9,102 Descartes' s classification 102, 103 equations of 99,102,103,109 instruments for drawing 95-8 normals to 103-4,107-8 see also under constructions D De Morgan, Augustus 142-3 Descartes, Rene 73-110,125 diagonal of square 1 9-2 1 , 45 Diophantus 52, 125 E ellipse 50,51,94,95,98 normal to 107,108 equations 62-4, 84-5, 104-5 Arab algebra 64-8 cubic 67-9, 89 of curves 99, 102, 103, 109 imaginary solutions 105-6 indeterminate 60-61 negative solutions 105,129 quadratic 22-4, 65-7 simultaneous 17-18, 126 Euclid 33, 36-8, 42-4, 55, 66, 128 axioms 37, 38, 39-40 definitions 38-9,113 'Euclid's rules' 30,31-2,36 finding mean proportional 80-81 Euclidean algorithm 43 Euler, Leonard 60, 62, 1 3 1 -3, 1 38-9 extrapolation 129, 138, 140, 141 F Fermat, Pierre 49,61, 108, 109-10, 114-16 fractions 17, 56 Frend, William 140 G geometry 74, 78, 80 analytical 98,106,108,109-10 Babylonian 24-7 coordinate 109 Greek30-32 Girard, Albert 129 Greek mathematics 29-52, 55, 63, 77, 1 12-13, 127-8 H Hamilton, WR 142,143,148 Hankel, Hermann 141 Heron of Alexandria 45-6, 63, 66-7 Index Hilbert, David 141 Hipparchus 57-8 homogeneous quantities 78 see also law of homogeneity Homer's method 62-3,69 'hundred fowls' problem 59, 60 Huygens, Christiaan 96 Hypatia of Alexandria 51-2 hyperbola 50,51,94,98,109 / Ibn al-Banna 61 incommensurable quantities 44-5 Indian mathematics 55-6,58,125-6, 127 indivisibles 117,118 infinity 17 inheritance problems 56-7 intuition 138-9 irrational numbers 45,105,141 L law of homogeneity 78 relaxation 81-2,90,102 Leibniz, Gottfried von 118-21 line segment, bisecting 3 1 -2, 36-7 livestock problem 126 Lobachevsky, Nicholas 40 M maxima and minima 114-15 mean proportional 33,80-81 Menaechmus 67 multiplication tables 14-15,19 N negative numbers 124-6,129-31 combining 126-9,131-5,141 Newton, Isaac 118,120-22 number systems Clifford model 144-6 decimal 55-6 Pythagorean 44 sexagesimal 13, 15-18 0 Omar Khayyam 40, 67-8 P n (pi) 46-8, 97 Pappus 49, 94 Pappus's problem 91,92 parabola 50,51,94,96,98 area under 48-9, 1 17 parallel postulate 40 Peacock, George 140 Pell equations 60-61 perpendicular, drawing 3 1 prime numbers 43-4 proof by contradiction 1 1 3 187
Index Ptolemy 49, 55, 58 Pythagoras's theorem 20-21, 25-7, 44 Q quadrature 32-4, 117-19 quaternions 143, 148 fl rebus principle (early writing) 11 reciprocals 19 right-angled triangles 37-8, 80-81 see also Pythagoras's theorem Robinson, Abraham 120 rule of double false 63-4 Saunderson, Nicholas 131-3 sign rule sine (term) spirals square roots Stifel, Michael subtracted numbers tangents Tartaglia Thabit ibn Qurra token system trigonometry turning operators u units Text and illustration credits We are grateful to the following for permission to reproduce copyright material: Chelsea Publishing Company, Inc. for an extract from The Mathematical Work of John Wallis by J F Scott; Dover Publications, for extracts from Elements, from Euclid, the Thirteen Books of the Elements, 2nd edn. unabridged, by Sir Thomas Heath, La Geometrie by Descartes, 1954 and The Geometry of Rene Descartes, 1954; McGraw Hill Publishers, for an extract from A Source Book in Mathematics by D E Smith; Merlin Press Ltd for extracts from The Life of Henry Brulard, by Stendahl, translated by Jean Stewart and BCJG Knight; Open University Press, for extracts from The History of Mathematics: A Reader by J Fauvel and J Gray; Oxford University Press, for an extract from Mathematical Thought from Ancient to Modern by Kline; Routledge, for extracts from Early Mesopotamia by J N Postgate; J Wiley & Sons Inc. for an extract from A History of Mathematics, 2nd edn.; University of Wisconsin Press, Madison, Wisconsin for an extract from Kushyar Ibn Labban, Principles of Hindu Reckoning, edited and translated by Martin Levey and Marvin Petruck. We are grateful to the following for permission to reproduce photographs and other copyright material: Figure 1.2, Musee du Louvre; Figures 1.3a and 2.1, Karl Menninger, Number Words and Number Symbols, Massachu setts Institute of Technology 1969, p. 163; Figures 1.3b and 1.8, O Neugebauer, Vorleusugen Uber Geschichte del Anbiken Mathematischen Wissenschaften (Erster Band) 1969, p.51, Springer-Verlag GmbH & Co. KG; Figure 1.4, Georges Ifrah, De Wereld van Het Getal, 1988, p. 132, Servire. © 1985 Editions Robert Laffont S.A., Paris; Figure 1.5, Denise Schamandt-Besserat, Archaeological Magazine, Archaeological Institute of America; Figure 1.6, Lauros-Giraudon; Figure 1.7, The Trustees of the British Museum; Figure 2.3, Prof. A H Aaboe, Episodes from the Early History of Mathematics, 188 105-6 58 96-7, 113-14 20-22,45-6,81 125 125-8, 140 112-17 77 61 8-10 57-9 147-8 80, 102 V van Schooten, Frans vectors Viete, w 49, 74, 84-5, 88, 96 114, 146, 147, 148 49, 78-9, 128-9 Wallis, John 130 Wantzel, Pierre-Laurent 85 Weierstrass 120 Wessel, Caspar 139-40 writing, evolution 8-11 z zero 16 Random House, 1964; Figure 2.4a, E M Bruins & M Rutten, Memoires de la Mission Archeologique en Iran, Tome XXXIV, Textes Mathematiques de Suse, 1961, Plate 6 Tablet K, Paris Librairie Orientaliste Paul Geuthner; Figures 2.4b and, 2.8b O Neugbauer, The Exact Sciences in Antiquity, 1969, Dover Publications/Constable & Co; Figure 2.6, we are unable to trace the copyright holder of this figure and would appreciate receiving any information that would enable us to do so; Figure 2.7, Lucas N H Bunt, Phillip S Jones and Jack D Bedient, The Historical Roots of Elementary Mathematics, Dover Publica tions/Constable & Co; Figure 2.8a, Yale Babylonian Collection; Figure 2.10, John Fauvel, Mathematics through History, QED Books, York; Figure 2.12, O Neugebauer and A Sachs, Mathematical Cuneiform Texts, 1945, (American Oriental Series vol.29). American Oriental Society, American School of Oriental Research, New Haven, Connecticut (we are unable to trace the copyright holder of this figure and would appreciate receiving any information that would enable us to do so); Figure 2.13, E M Bruins & M Rutten, Memoires de la Mission Archeologique en Iran, Tome XXXIV, Textes Mathematiques de Suse, 1961, Plate 1 p. 23, Paris Librairie Orientaliste Paul Geuthner; Figure 2.14, O Neugebauer, Mathematische Keilscriftexttexte 3. band, dritter teil, Verlag van Julius Springer, 1935/1937. (Springer-Verlag GmbH & Co. KG); Figure 5.1, George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, 1992, Penguin Books; Page 73, The Geometry of Rene Descartes, title page, Dover Publications, inc; Figure 12.1, J Fauvel and J Gray, History of Mathematics: A Reader, The Macmillan Press Ltd. Adapted from O Neugebauer and A Sachs, Mathematical Cuneiform Texts, \ 945, American Oriental Society, American School of Oriental Research, New Haven, Connecticut (we are unable to trace the original copyright holder of this figure and would appreciate receiving any information that would unable us to do so).
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Nuffield Advanced Mathematics Histo
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Contents Introduction 7 Unit 1 The
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Introduction This book, about the h
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The Babylonians Chapter 1 Introduct
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Introduction to the Babylonians On
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Figure 1.2 Stele inscribed with Ham
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Token V Figure 1.5 7 Introduction t
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There are no practice exercises for
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A sexagesimal number system is a sy
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ff m yfr < n Figure 2.5 «« Activi
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Activity 2.5 Babylonian fractions 2
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Activity 2.6 Babylonian arithmetic
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Figure 2.10 Activity 2.9 2 Babyloni
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Type Quadratic Table 2.1 x +ax = b
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Activity 2.12 Geometry Figure 2.13a
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Practice exercises for this chapter
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3 An introduction to Euclid fthese
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The Greeks Activity 3.3 B Figure 3.
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34 The Greeks Activity 3.4 g Now us
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36 The Greeks inscribes another wit
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38 The Greeks Activity 3.7 therefor
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The Greeks Interpret postulate 3 in
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42 More Greek mathematics 4.1 Some
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The Greeks Q.E.D. stands for 'quod
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Figure 4.3 46 The Greeks • treat
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Figure 4.6 48 The Greeks together w
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The Greeks Activity 4.8 The cone Th
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The Greeks Practice exercises for t
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5 Arab mathematics You should think
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The Arabs Western Arabic, or Gobar,
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The Arabs Activity 5.2 Figure 5.3a
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7776 Arabs Activity 5.4 This proble
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62 The Arabs Activity 5.6 Horner's
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64 The Arabs Indian text. For over
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D B P N M K Figure 5,9a The Arabs D
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68 The Arabs Figure 5.10 Activity 5
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The Arabs Practice exercises for I
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72 6.1 Read about Descartes The app
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Descartes Activity 6.1 Head about D
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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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Page 157 and 158: 752 12 Summaries and exercises Two
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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
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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
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Page 187 and 188: 14 Answers b The required number is
Page 189 and 190: 14 Answers operator has the effect
Page 191: 14 Answers A i- B 5 E %2 5)-25 = 39
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