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