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The Essence of Mathematics Through Elementary Problems
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Index

  • 10-gon

    • regular, 210

  • n-gon, 39, 195, 196, 200, 205, 292

    • regular, 15, 28, 47, 131, 196, 199, 201, 205, 211–214, 216, 250, 251, 265, 269

    • regular circumscribed, 213–217, 270

    • regular inscribed, 211, 213–217, 254, 270

  • x-axis, 15

    • positive, 15

  • Ars Magna, 129, 132, 156

  • Astronomia Nova, 199

  • De Divina Proportione, 199

  • De revolutionibus, 132

  • Disquisitiones arithmeticae, 20, 201

  • Elements, 13, 14, 181, 199, 207, 210, 230

  • Liber Abaci, 56

  • The Sand Reckoner, 283

  • Trattato d’aritmetica, 109

  • 10-gon

    • regular, 264

  • abacists, 111

  • acceleration, 96, 106

    • constant, 96, 106, 107

    • uniform, 105, 106

  • addition, 8, 10, 11, 35

  • Al-Khwarizmi, Muhammad ibn Musa (c.780–c.850), 111

  • algebra, x, 2, 10, 51, 111, 112, 145, 156, 171, 172, 197, 198

    • abstract, 173

    • Lie, ix, 109

    • linear, 172, 173

    • mental, 2

    • symbolical, 131

  • algorithm

    • division, 134, 166

    • Euclidean, 6, 9, 133, 134, 166

    • standard written, 172

  • analysis, 173

    • dimensional, viii, 109

    • method of, 197, 205

  • Anaximenes (c. 586–c. 526 BC), 194

  • angle, 4, 14–16, 23–25, 48, 130, 171, 174, 219

    • acute, 189, 262

    • angle sum, 249, 256, 258–260, 319, 320

    • apex, 279

    • dihedral, 209, 263, 265

    • exterior, 180, 184, 185, 234–236

    • obtuse, 185, 262

    • reflex, 235

    • right, 14, 17, 23, 24, 40, 44, 176, 185, 218, 220, 231, 234, 237, 239, 241, 254, 257, 272–275

    • sector, 269

    • solid, 201

    • straight, 176, 179–185, 189, 201, 231, 232, 234–236, 238, 241, 242, 244

    • subtended at the centre of the circle, 267

    • subtended by a chord, 44, 184, 235, 243

    • subtended by a diameter, 44, 194, 254

    • subtended by an arc, 269, 273

  • angles

    • alternate, 183, 237–239, 319

    • base, 47, 178, 250, 260, 263

    • congruent, 28

    • corresponding, 183

    • internal, 182

    • measured in degrees, 23

    • opposite, 17

    • special, 16

    • supplementary, 184, 185, 262

    • vertically opposite, 238, 241–243, 248, 250

  • anthyphairesis, 9

  • antiprism, 205

    • n-gonal, 258

  • apex, 224

    • of an isosceles triangle, 177

    • of a pyramid, 193

  • arc, 24, 25, 184, 211, 214, 218–220, 235, 236, 243, 264, 268, 269, 272

    • circular, 24

    • length of, 274

    • of a great circle, 272

  • arc length, 24

  • Archimedes (c. 287–212 BC), 198, 199, 283

  • area, 211

    • of a circle, 214, 215, 217

    • of a circumscribed regular n-gon, 217

    • of a parallelogram, 97, 186, 188, 237, 243, 244, 272

    • of a quarter circle, 216

    • of a rectangle, 188, 237

    • of a regular n-gon, 214, 215

    • of a regular polygon, 211

    • of a sector, 24

    • of a sector of a circle, 216, 267, 268

    • of a semicircle, 216

    • of a spherical triangle, 27, 45

    • of a trapezium, 193

    • of a triangle, 15, 97, 106, 191, 206, 237, 244

    • of an ellipse, 223

    • of an inscribed regular n-gon, 216

    • surface area of a pyramid, 211

    • surface area of a right circular cone, 216, 269

    • surface area of a right cylinder, 215

    • surface area of a right prism, 211, 216

    • surface area of a right pyramid, 216

    • surface area of a sphere, 26

    • under the graph, 104, 106, 107

  • argument, 130

  • arithmetic, 11, 111, 171

    • inverse, 156

    • mental, 2, 3, 5, 8, 20, 172

    • modular, 101

    • of complex numbers, 42, 129

    • of norms, 20

    • of surds, 9

    • structural, 12, 34

    • written, 2

  • Arnold, Vladimir (1937–2010), viii, 109

  • average, 93, 98

  • averaging, 104

  • Babylonians, 14, 23

  • base, 10, 237

    • of an isosceles triangle, 177

    • of a cone, 244

    • of a pyramid, 193, 218, 244, 245

    • of a rectangle, 106, 188

    • of a right cylinder, 215

    • of a right prism, 216

  • Bernoulli, Jacob (1654–1705), 302

  • bi-pyramid, 353

  • bisector, 171, 186

    • of angle, 186, 237

    • perpendicular, 22, 44, 47, 179, 221, 223, 224, 232, 240, 254–256, 260, 275–277, 281

  • Bolyai, János (1802–1860), 182

  • bracket, 10, 11

  • Bragg, Sir William Lawrence (1890–1971), xiii

  • Bronowski, Jacob (1908–1974), 170

  • Bruno, Giordano (1548–1600), 284

  • calculation, 1

    • direct, 11

    • exact mental, 14

    • mental, xv

  • calculus, 11, 173

  • cancellation, 34

  • Cantor, Georg (1845–1918), 142, 284

  • Cardano, Girolamo (1501–1576), 129, 132, 156

  • Catalan, Eugène Charles (1814–1894), 77

  • caustic, ix, 109

  • centroid, 189

  • chord, 44, 184, 235, 242, 243, 320

  • Christ, Jesus, 14

  • circle, 9, 15, 24, 44, 174

    • area of, 214–217

    • great, 25, 26, 218

    • longitude, 275

    • of Apollonius, 223

    • of longitude, 273

    • perimeter of, 213, 216, 268

    • unit, 15, 16

  • circumcentre, 179, 180, 184, 196, 199, 200, 204, 205, 210, 236, 240, 254–256, 263, 270

  • circumcircle, 22, 184, 185, 210, 236, 241, 254–256, 264, 265

  • circumference, 25, 44, 179, 195, 269

  • circumradius, 180, 199, 210, 213, 264, 265

  • code

    • Gray, 282, 312

  • coefficient, 124

  • concave, 107

  • concavity, 107

  • concept

    • abstract, xv

  • cone, 244

    • apex of, 224

    • generator of, 224

    • right, 216

  • congruence, 171, 174

    • ASA-congruence, 177, 178, 237–239, 250

    • RHS-congruence, 136, 185, 186, 236, 237, 239, 257

    • SAS-congruence, 14, 15, 48, 177, 178, 185, 190, 232, 233, 238, 240, 242, 249

    • SSS-congruence, 47, 177, 178, 186, 231, 232, 238, 257

  • conic sections, 225, 227

  • constant

    • of proportionality, 101

  • construction

    • geometrical, 199, 206

    • of a cube, 227

    • of a parabola, 221

    • of a regular pentagon, 255

    • of an equilateral triangle, 175

    • of regular polygons, 175, 199–201

    • of regular polyhedra, 199

    • ruler and compasses, 9, 174, 175, 179, 181, 183, 193, 199–201, 230

  • converse, 14, 16, 17

    • statement, 16

  • convexity, 107

  • coordinate, 15

  • coordinates

    • Cartesian, 129

  • Copernicus, Nicolaus (1473–1543), 132

  • counterexample, 314

  • Courant, Richard (1888–1972), vii

  • cow

    • spherical, 91

  • Coxeter, Harold Scott MacDonald (1907–2003), ix, 109

  • Cramer, Gabriel (1704–1752), 114

  • criteria

    • congruence, 174, 178, 181, 185, 191, 206

    • parallel, 174, 179, 181, 182, 206

    • similarity, 174, 191–193, 206

  • cube, 28, 29, 48, 49, 201, 203–205, 209, 210, 227, 244, 260

    • cross-section of, 28

    • truncated, 259

    • unit, 209

  • cubes, 2, 3, 56, 61, 119

  • cuboctahedron, 258

  • curriculum, xv

  • curve, ix, 109

    • quadratic, 227

  • cylinder

    • right, 215

  • d’Alembert, Jean le Rond (1717–1783), 283, 286

  • Dürer, Albrecht (1471–1528), 280

  • da Vinci, Leonardo (1452–1519), 199

  • Dandelin, Germinal Pierre (1794–1847), 224

  • de la Vallée Poussin, Charles-Jean (1866–1962), 68

  • de Moivre, Abraham (1667–1754), 130

  • decimal, 172

  • degree, 23

  • del Ferro, Scipione (1465–1526), 132

  • del Fiore, Antonio (1506–??), 132

  • dell’Abbaco, Paolo (1282–1374), xiii, 97, 109, 111

  • denominator, 5, 34, 65, 84, 105

  • derivative, 24

  • Descartes, René (1596–1650), 3, 112, 156, 193, 197, 198

  • diagonal, 4, 15, 16, 48, 49, 187, 188, 196, 206, 238, 239, 249, 250, 255, 262, 263, 316, 363

  • difference, 6

  • digit, 2

    • leading, 56, 288

    • units, 56, 288

  • digit-sum, 55

  • dimension, 2, 35

  • directrix, 221, 224

    • of ellipse, 223

    • of hyperbola, 224

    • of parabola, 221, 280

  • disc

    • circular, 269

  • discount, 8, 35

  • discriminant, 127

  • division, 10, 11, 40

    • with reminder, 134

  • divisor, 6, 134

  • dodecagon

    • regular, 16, 195

  • dodecahedron

    • regular, 205, 260

    • snub, 258

    • truncated, 259

  • Dogan, Serkan, 348

  • Earth, ix

    • circumnavigation of, ix

  • eccentricity, 221

    • of a hyperbola, 226

    • of an ellipse, 226

    • of parabola, 221

  • edge, 312

  • Edwards, Harold M. (1936–), 1

  • Ehrlich, Paul R. (1932–), 111

  • ellipse, 223, 224, 226, 280

    • area of, 223

  • encryption, 6

  • equation, 4

    • cubic, 127–129, 131–133, 156, 165, 166

    • differential, 11

    • linear, 201

    • Pell, 76, 290

    • quadratic, 4, 9, 123, 125, 129, 133, 156, 201, 226, 227

    • quartic, 127, 129, 132, 156, 158, 159

  • equator, 25, 273, 274

  • equidistant, 28, 44, 179, 180, 186, 221, 225, 232, 237, 254

  • Eratosthenes, (c. 276 – 194 BC), 31

  • Euclid, (flourished c. 300 BC), 13, 14, 175, 181, 199, 207, 210, 230, 231

  • Euler, Leonhard (1707–1783), 19, 20, 42, 112, 122, 123, 153, 302, 310, 312

  • expansion

    • base 2, 115

    • base 3, 115

    • decimal, 115

  • exponent, 30

  • face, 311

  • factor, 6, 29, 67, 74, 76, 77, 84, 87, 90, 119–121, 153, 154

    • common, 5, 6, 18, 76, 148, 167

    • enlargement, 8

    • highest common, 33

    • linear, 119, 120, 131, 165

    • linear conjugate, 165

    • prime, 31, 67, 71, 90, 311, 317, 348

    • quadratic, 165

    • scale, 18, 191

  • factorial, 7, 326

  • factorisation, 6, 11, 69, 71, 76, 78, 120, 121, 152, 164

    • algebraic, 72, 73

    • of integers, 6, 32

    • of polynomials, 12

    • prime, 5, 31, 72

    • prime power, 6

  • Fermat, Pierre de (1601–1665), 112, 123, 154, 197, 201, 289

  • Ferrari, Lodovico (1522–1565), 132

  • Fibonacci, Leonardo Pisano (c. 1170 – c. 1250), 56

  • focus, 224

    • of ellipse, 223, 224

    • of hyperbola, 224

    • of parabola, 221

  • formula

    • Euler’s, 312

  • fraction, x, 5, 7

    • continued, 9

    • decimal, 85

    • unit, 65, 172

  • fractions

    • decimal, 172

  • frustum, 193, 245

  • function, 3, 4, 9, 11, 129

    • constant, 104

    • trigonometric, 16, 24, 273

  • Galen (c. 130 – c. 210), ix

  • Galileo, Galilei (1564–1642), 174

  • Gauss, Carl Friedrich (1777–1855), 20, 68, 69, 173, 182, 201

  • geometry, 9, 173

    • algebraic, ix, 109

    • analytic, 173

    • Cartesian, 173

    • coordinate, 172, 173

    • dynamic, 172, 173

    • elementary, 172

    • Euclidean, 170, 172, 173

    • of surfaces, 173

  • gnomon, 293

  • Gog, Julia, 48

  • Goldbach, Christian (1690–1764), 122

  • Golden Ratio, 30, 31, 37, 199

  • Golding, William (1911–1993), xv

  • Grassmann, Hermann Günther (1809–1877), 173

  • Gray, Frank (1887–1969), 356

  • group, ix, 109, 173

    • Coxeter, ix, 109

    • of transformations, 173

  • Hadamard, Jacques (1865–1963), 68

  • Harte, John, 91

  • HCF, 5–7, 10

  • height, 14, 237

    • of a rectangle, 188

    • of a parallelogram, 188

    • of a pyramid, 244, 245

    • of a right cylinder, 215

    • of a right prism, 216

    • of a triangle, 97, 106

  • hemisphere, 274

  • hexagon, 49, 252

    • regular, 16, 28, 29, 49, 196, 210, 231, 247, 251, 253, 264, 265, 281

  • Hilbert, David (1862–1943), 174

  • Hildebrand, Joel Henry (1881–1983), 51

  • Hipparchus, (died c. 125 BC), 208

  • hyperbola, 224, 226, 281

  • hypotenuse, 14, 15, 39, 43

  • icosahedron

    • regular, 204, 205, 209, 210, 257

    • truncated, 260

  • icosidodecahedron, 258

  • identity

    • trigonometric, 5, 27

  • incentre, 186, 187

  • incircle, 9, 113, 181, 187

  • induction

    • mathematical, 57, 78, 88, 118, 145, 147, 162, 270, 271, 282, 285, 288, 290, 292–294, 300, 314, 315, 324

  • inequality

    • triangle, 180, 270, 271, 277

  • infinite descent, 314–316

  • inradius, 187, 213, 216

  • integer part, 54

  • integers, 6, 10

    • consecutive, 6

    • coprime, 5

    • cubes, 3

    • Gaussian, 134, 166–168

    • positive, 3, 5

    • powers of 2, 3

    • relatively prime, 5, 6, 32, 33, 41

    • squares, 3

  • integral, 104, 310

  • integration, 11

  • interval, 115

  • Kapitsa, Peter (1894–1984), xi

  • Kepler, Johannes (1571–1630), 199, 224, 263

  • kinematics, 105

  • Klein, Felix (1849–1925), 173

  • Lagrange, Joseph-Louis (1736–1813), 20

  • law

    • associative, 60, 82

    • commutative, 60, 82

    • distributive, 60, 82

    • index laws for powers, 2

  • LCM, 5

  • Legendre, Adrien-Marie (1752–1833), 20, 68, 248

  • Leibniz, Gottfried Wilhelm (1646–1716), 112, 304, 306, 310

  • length, 9, 211

  • line, 14, 25, 171, 173, 174

    • parallel, 16, 30, 44

    • segment, 25, 49, 174, 175

  • Livio, Mario, 31

  • Lobachevski, Nikolai (1792–1856), 182

  • locus, 179, 180, 186, 198, 220–223, 254, 275–278

  • logarithm

    • natural, 310

  • longitude, 25

  • lune, 26, 27

  • map, 312

  • mapping

    • one-to-one, 10

    • two-to-one, 10

  • mathematics

    • elementary, xi, xiii, 4, 28

  • mean

    • arithmetic, 93, 96, 105, 118, 149, 305

    • geometric, 97, 118, 149, 305

    • harmonic, 96, 105, 108, 118, 149, 305

    • quadratic, 118, 149

  • mechanics, 107

  • median, 189, 241, 261, 262

  • Menelaus, (c. 70–130 AD), 208

  • Mersenne, Marin (1588–1648), 123, 289

  • midpoint, 15, 21, 99, 113, 114, 136, 137, 178–180, 187–189, 193, 203, 231–234, 239, 240, 242, 243, 245, 246, 255–257, 261, 263–265, 267, 268, 270, 308

  • modulus, 42, 115, 130

  • Moise, Edwin (1918–1998), 174

  • Morotzkin, I.V., viii, 109

  • multiple, 5, 10, 30

    • common, 5

  • multiplication, 10, 11, 35

    • long, 64, 72

    • of complex numbers, 129

  • multiplier, 10

  • net, 48

  • Newton, Issac (1642–1727), 112, 125, 224

  • norm, 42, 167

  • notation

    • algebraic, 2, 156

  • number, 2, 9, 171

    • complex, 1, 20, 27, 42, 76, 77, 126–131, 134, 164

    • complex conjugate, 128, 129, 164

    • complex in modulus form, 27

    • complex in polar form, 129, 130

    • Fermat, 123, 289

    • Fibonacci, 30, 56, 58, 285, 294

    • irrational, 63, 126

    • positive, 3, 4

    • prime, 5, 76, 90, 119

    • rational, 63

    • real, 3

    • real positive, 8, 28

    • square triangular, 76

    • triangular, 56, 57, 61, 75–77, 284, 293

  • numeral system

    • binary, 54

    • Hindu-Arabic, 56

  • numerator, 34, 105

  • numerosity, 171

  • octagon, 39

    • regular, 9, 16, 39

  • octahedron, 203, 257

    • regular, 203, 205, 208–210, 257, 258, 262, 263

    • stellated, 263

    • truncated, 260

  • octant, 209

  • operation, 2, 3

    • arithmetic, 11

    • direct, 3, 10

    • inverse, 3

  • origin, 15

  • orthocentre, 188

  • Pólya, George (1887–1985), xiii, 29

  • Pacioli, Luca (c. 1445–1509), 199

  • Pappus, c. 290–c. 350 AD), 198

  • parabola, 221, 224–226, 280

  • parallelogram, 17, 30, 149, 187, 188, 237, 239, 240, 242–244, 246, 254, 272

    • Fibonacci, 59

  • parallels, 171, 174

  • parameter, 108, 112

    • hidden, 108

    • hidden intermediate, 108

  • parity, 18, 41

  • Pascal triangle, 351

  • pattern, 2

  • Pell, John (1611–1685), 290

  • Penrose, Roger (1930 –), 169

  • pentagon, 29, 113, 114, 264

    • convex, 195

    • regular, 4, 9, 28, 30, 49, 124, 195, 196, 205, 209, 210, 249, 250, 252, 255, 263

  • pentagram, 195

  • percentage, 7, 35, 60

  • perimeter, 21, 213, 270–272

    • of a circle, 213, 216, 217, 268

    • of a convex polygon, 217, 270, 271

    • of a regular polygon, 211–213, 216, 217

    • of a sector, 216

    • of a triangle, 43, 189, 240

    • of an ellipse, 223

  • perpendicular, 14, 40, 45, 176, 178–181, 183, 186, 188, 206, 210, 221, 224, 232, 234, 237, 239, 240, 244–246, 254–256, 261, 262

  • Petrie, John Flinders (1907–1972), 49

  • Petrie, Sir William Matthew Flinders (1853–1942), 49

  • Pigeon Hole Principle, 33

  • place value, 2, 51, 171, 172

  • plane, ix, 109, 174

    • complex, 129

    • real, ix, 109

  • Plato, (c. 428–347 BC), 205

  • Poincaré, Henri (1854–1912), 283

  • point, 171, 173, 174

    • decimal, 115, 172

  • polygon, 24, 28, 49, 292

    • angles of, 24

    • convex, 217

    • cyclic, 28

    • Petrie, 49

    • regular, 28, 199, 201, 205, 211, 256

    • regular inscribed, 202

  • polyhedron, 49, 198

    • convex, 201

    • dual, 205

    • regular, 28, 199, 202, 203, 205, 210, 265

    • semi-regular, 205, 206, 259

    • spherical, 311, 312, 352–354

  • polynomial, 126

    • anti-symmetric, 125

    • cubic, 127

    • long division of, 126

    • quadratic, 112, 119, 122, 124–126, 131, 158

    • quartic, 158, 159

    • symmetric, 124

  • postulate

    • parallel, 181

  • powers, 2, 10

    • algebraic, 3

    • numerical, 3

    • of 10, 2, 62, 71, 84, 85

    • of 2, 2, 3, 56–58, 64, 65, 76, 88

    • of 3, 88

    • of 4, 12, 38, 56

    • proper, 77

    • sums of, 20

  • prime numbers, 5, 6, 12, 18–20, 31, 32, 42, 43, 53, 119, 120, 122, 151, 153, 154, 288, 289, 297, 335, 348

    • distribution of, 68

    • Fermat, 123, 154, 201, 290

    • Mersenne, 123

  • prism, 205, 211

    • 3-gonal, 259

    • n-gonal, 260

    • right, 216

  • problems, xiii

    • inverse, 10, 11

    • multi-step, x

    • one-step, x

    • word, 91, 92, 98, 172

  • procedure

    • deterministic, ix

    • direct, 10, 11

    • inverse, 10, 11

  • progression

    • arithmetic, 19, 41

  • projectile, 221

  • projection, 49

  • proof, 1

  • proportion, x, 100, 101, 109

  • Ptolemy, (died 168 AD), 208

  • pyramid, 193, 211, 218, 244, 245

    • right, 216

  • Pythagoras, 13

  • Pythagorean triple, 18

  • Pythagorean triples, 14, 17

    • primitive, 18, 19, 197

  • Pythagoreans, 13

  • quadrant, 15, 16

  • quadrilateral, 17, 113, 114, 137, 174, 184, 238, 242

    • cyclic, 27, 184, 185, 236, 241, 257

    • planar, 257

  • quantifier, 317

  • quantity, 91, 171

    • averaged, 104

    • continuous, 97, 98, 171

    • discrete, 104

    • unknown, 98, 108

  • quotient, 126, 160, 166

  • radians, 24, 25, 196, 205, 214, 216, 248, 249, 262, 273, 292

  • radius, 15, 24, 174, 211

  • ratio, 5, 7, 30, 100

  • recognising structure, 91, 92, 99

  • rectangle, 14, 21, 28, 35, 44, 48, 106, 187, 188, 211, 217, 223, 237, 239, 246, 254, 257, 269

    • Golden, 257

    • similar rectangles, 7

  • rectangular, 28

  • recursion, 54, 284

  • relation

    • Fibonacci recurrence, 56

    • recurrence, 58

  • remainder, 40, 54, 55, 90, 126, 134, 160, 161, 166–168

  • rhombicosidodecahedron

    • great, 260

    • small, 259

  • rhombicuboctahedron, 259

    • great, 260

  • rhombus, 4, 124, 187, 229, 250, 257

  • Riemann, Bernhard (1826–1866), 173

  • Robbins, Herbert (1915–2001), vii

  • root, 3, 4, 10, 124–129

    • complex, 131

    • conjugate, 158, 161

    • cube, 133, 166

    • cube complex, 251

    • of a polynomial, 125

    • of a quadratic equation, 37, 131, 149

    • of a quadric, 124, 125

    • of unity, 130, 131

    • real, 127, 128

    • repeated, 132, 154

    • square, 4, 11, 66, 124, 128, 129, 151, 155, 201

  • Rule

    • Cosine, 5, 17, 170, 206–208, 219, 261, 262, 264, 273, 274

    • Cramer’s, 139

    • Sine, 5, 22, 23, 45, 47, 206–208, 219, 220, 261

  • ruler and compasses, 9, 174, 175, 179, 181, 183, 193, 199–201, 230

  • Sarton, George (1884–1956), 169

  • scales, 112

  • scaling, viii, 23, 109

  • scaling argument, 109

  • secant, 242

  • sector, 267, 269

  • semicircle, 216

  • sequence, 12, 13, 52, 53, 56, 59

    • constant, 57

    • Farey, 117

    • Fibonacci, 56–58, 80, 294

    • of cubes, 56

    • of differences, 57

    • of natural numbers, 56, 57

    • of partial sums, 58

    • of powers, 58

    • of prime numbers, 56

    • of second differences, 57

    • of squares, 56

    • of triangular numbers, 56, 57

    • recurrent, 13

  • series

    • alternating harmonic, 310

    • alternating, 310

    • divergent, 306

    • Farey, 117, 118, 145–149

    • geometric, 121, 122, 141, 142, 297–300, 302, 303, 305, 336

    • harmonic, 305, 308, 310

    • power, 310

  • Shakespeare, William (1564–1616), xi

  • sharing, 92, 172

  • Sieve of Eratosthenes, 31

  • similarity, 171, 174

    • AAA-similarity, 191, 192, 242–245

    • of parabolas, 221

    • SAS-similarity, 190, 191, 240

    • SSS-similarity, 191

  • simplifying, 91, 92

  • singularity, ix, 109

  • speed, 94, 96

    • as a function of time, 104, 107

    • average, 95, 96, 103–107

    • constant, 101, 110

    • negative, 101

    • relative, 108

    • varying, 104

  • sphere, 25

    • surface area of, 26

    • unit, 26, 218, 220

  • square, 3–5, 9, 14–16, 21, 28, 29, 44, 187

    • unit, 227

  • square lattice, 20

  • Square Root Test, 31, 71

  • squares, 2, 3, 56, 119

  • squaring, 3, 11

  • star

    • pentagonal, 195, 248

  • statement

    • converse, 16, 17

  • Steiner, George (1929–), xi, 91, 363

  • stella octangula, 263

  • Stevin, Simon (1548–1620), 112

  • structure, 2, 31, 35, 51, 57, 60, 73, 92, 93, 98

    • recognising, 91, 92, 99

  • subtraction, 10, 11, 35

  • sum, 6

    • alternating, 87, 309, 310

    • of four squares, 20

    • of two squares, 19

    • partial, 58

  • surds, 9, 14, 37, 125, 131, 208, 212, 214

  • Swetz, Frank, 14

  • symbol, 5, 9

  • system

    • numeral, 10

  • SzegÖ, Gábor (1895–1985), 29

  • tables

    • multiplication, 2, 13

    • times, 2

  • tablet

    • clay, 14

  • Tartaglia, NiccolÒ (1499–1557), 132

  • Test

    • Square Root, 31, 71

  • test

    • divisibility, 54

  • tetrahedron, 48, 210, 220, 263

    • regular, 28, 48, 203, 208–210, 258, 261–263

    • truncated, 259

  • Thales (c. 620–c. 546 BC), 193, 194

  • Theorem

    • de Moivre’s, 130

    • Binomial, 336

    • Fermat’s Last, 1, 145

    • Gougu, 14

    • Midpoint, 113, 189, 190, 192

    • Prime Number, 68

    • Ptolemy’s, 27

    • Pythagoras’, 9, 13–17, 39, 40, 170, 186, 188, 195, 206, 207, 217–219, 248, 264, 272, 273

    • Pythagoras’ converse, 17, 40, 264

    • Pythagoras’, spherical version of, 273

    • Remainder, 332

    • Thales’, 193

  • theory

    • elementary number, 17

  • tiling, 199, 253

    • Archimedean, 198

    • regular, 196, 197, 210

    • semi-regular, 198, 205, 252, 253, 258

    • semi-regular tilings, 199

  • time scales, 171

  • times tables, 2

  • TIMSS, 99

  • topology, ix, 109, 172, 173

    • four-dimensional, ix, 109

  • transversal, 182, 183

  • trapezium, 44, 239

    • area of, 193

    • isosceles, 21, 263

  • tree

    • binary, 313

    • Calkin-Wilf, 313

  • triangles, 17, 22, 171, 173, 174, 176, 177

    • acute angled, 189, 207

    • congruent, 45, 136, 176, 177, 181, 185, 191

    • equiangular, 253

    • equilateral, 15, 21, 24, 28, 29, 43, 48, 175, 196, 199, 204, 205, 208, 230, 231, 247, 251, 253, 254, 257, 260, 262, 281

    • isosceles, 30, 39, 47, 177, 178, 185, 231, 233–236, 240, 247, 249, 250, 253–256, 260–263, 267, 268

    • isosceles right angled, 48

    • obtuse angled, 207

    • on a sphere, 26

    • orthic, 188, 189, 240, 241

    • regular, 16

    • right angled, 14, 17, 23, 39, 40, 43, 48, 135, 136, 148, 186, 192, 210, 217–220, 239, 247, 254, 257, 261, 264, 265, 272, 274

    • similar, 4, 190–193, 240

    • spherical, 23, 25, 26, 45, 218–220

    • spherical, area of, 25

  • trigonometry, 14, 192, 195, 206–208

  • triple

    • Pythagorean, 14, 19, 197

  • Ulam’s spiral, 153

  • Ulam, Stanisław (1909–1984), 153

  • unit, 100, 106, 108, 172

    • abstract, 108

    • dimensionless, 109

    • of distance, 106, 107

    • of measure, 23

    • of measurement, 109

    • of speed, 106

    • repeated, 171

  • unitary method, 102

  • valency, 352

  • value

    • approximate, 30

  • variable, 112

  • variety

    • toric, viii, 109

  • VAT, 8

  • vector, 172

    • position, 114

    • product, 272

  • vertex, 312

  • vertex figure, 196–198, 201–206, 259

  • Vesalius, Andreas (1514–1564), ix

  • Viète, François (1540–1603), 112

  • visualisation, xv

  • volume

    • of a cone, 244, 245

    • of a pyramid, 244

  • von Neumann, John (1903–1957), xiii, 299

  • Wagner, Roy, 109

  • Waring, Edward (1736–1798), 20

  • wave, ix, 109

  • Whitehead, Alfred North (1861–1947), 134

  • word, 5

  • Yevdokimov, Oleksiy, 150

  • Zarathustra, 363

  • Zeno of Elea (c. 495 BC – c. 430 BC), 284