nth-order symmetric group
490
nth root 112
nth root of unity 358
nth-term test 103
n-tuple 358
null angle 14
null set. See empty set
number 358–359
number-base machine 30
number line 359, 441, 442
number-naming puzzle 54–55
number systems 358–359
number theory 82, 139–140,
159, 167, 191, 337–338,
359–360, 429–430
numerable. See denumerable
numerator 204
numerical differentation 360
numerical integration 360–361
O
obelus (%) 146
oblate spheroid 154, 362
oblique 362
oblique angle 362
oblique cone 362
oblique cylinder 115
oblique pyramid 423
obtuse angle 14, 362
obtuse triangle 362, 506
obverse 362
octahedron 396, 397
octant 363
octonions 359, 432
odd functions 177, 490
odd numbers 177–178
odds (probability) 363, 439
Oeuvres completes d’Augustin
Cauchy 67
Oeuvres de Camille Jordan
288
officer problem 302–303
Omar Khayyám (Umar al-
Kha¯yyamı¯) 18, 227,
363–364
“On a Curious Property of
Vulgar Fractions” (Farey)
188
“On a New Method in
Elementary Number Theory”
(Erdös) 167
On Burning Mirrors (Diocles)
136
On Conoids and Spheroids
(Archimedes) 18
On Divisions (Euclid) 169
“one-to-one” function 209
On Floating Bodies
(Archimedes) 18–19
On Reasoning in a Dice Game
(Huygens) 180
On Spirals (Archimedes) 474
On the Economy of
Machinery and Manufactures
(Babbage) 33
“On the Problem of the Most
Efficient Tests of Statistical
Hypotheses” (Neyman) 354
“On the Propagation of Heat
in Solid Bodies” (Fourier)
201
On the Sphere and Cylinder
(Archimedes) 18, 173
“On the Theory of Groups
Depending on the Symbolic
Equation θ
n
= 1” (Cayley)
68
“onto” function 208
open half-plane 243–244
open half-space 243–244
open interval 278
operation 364
operational precedence. See
order of operation
operations research (OR) 33,
364
opposite 364
opposite angles 364
oppositely congruent solids 92
Optica promota (J. Gregory)
240
Optics (Euclid) 169
Optics (Ptolemy) 422
optics, Newton and 353
Optiks (Newton) 353
optimization 133, 161, 285,
316, 364, 364–365
and history of calculus 57
and inscribed triangles 387
and Snell’s law of
refraction 468
Steiner and 484–485
von Neumann and 350
OR. See operations research
orbit (sequence of iterates)
150
order. See permutation
order, of polyomino 406
ordered partition 380–381
ordered set 365–366, 367,
530
order of a matrix (dimension
of a matrix) 258, 366
order of group 366
order of magnitude 329
order of operation (opera-
tional precedence) 366–367
order properties 267, 367
ordinal numbers 367–368
ordinate 62
Oresme, Nicole 63, 208, 234,
368
orientation 369
origami 513
origin 105, 359
orthant 363
orthic triangle 387
orthocenter 12, 354, 387, 507
orthogonal 369, 391
orthogonal. See Cartesian
coordinates
Osborne’s rule 369
osculation (tacnode) 148
osculinflection 492, 493
Oughtred, William 344,
369–370, 370, 466
outlier 370
oval 370
ovals of Cassini 370
Ozanam, Jacques 449
P
packing spheres 472
paddle wheel 174
Paganini, Nicolò 13
pairwise disjoint (independent,
mutually exclusive) 371
Pappus 13, 57, 239, 253,
371–372, 470
Pappus’s problem 372
Pappus’s theorems 372–373,
470, 471
and torus 500
parabola 92–93, 93, 137,
155, 189, 239, 373, 373,
429
as trajectory 213, 503
parabolic cylinder 411
paraboloid 101, 235,
373–374
paradox 374–375, 412, 532
and pure mathematics
422–423
“Paradoxien des Unendlichen”
(Bolzano) 47
parallel 375–376, 376, 377
parallelepiped (parallelopiped)
376
and scalar triple product of
three vectors 512
parallelogram 376, 430, 474
area of 23, 23
cross (vector) product as
110
parallelogram law
(parallelogram rule) 376,
376–377
parallelotope 376
parallel postulate (Euclid’s fifth
postulate) 159, 227, 239,
255, 375, 377, 377–378,
378, 473
and altitudes of triangle
12–13
and angles of triangle 505
converse of exterior-angle
theorem 183
and Euclidean vs. non-
Euclidean geometry
170–171, 318, 355–356
Omar Kayyám and 18,
364
parallel projection 418
parametric equations (freedom
equations) 164, 378, 386
and arc length of curve 22
of cardioid 62
of circle 75, 95
of helixes 95
of Möbius band 340
of torus 500
of unit circle 517
parentheses 50–51, 519
parity 378–379
Parmenides 532
Parmenides (Plato) 532
partial derivative 138, 233,
239, 259, 379
partial differential equations,
d’Alembert and 8–9
partial fractions 162,
379–380
partially ordered set 365–366
partial sum 102, 380, 459
and limit 312
partition 358, 380–381, 436
partition function 381
Pascal, Blaise 74, 189,
381–382, 382, 383
on cycloid 310
and history of calculus 57
and Pappus’s theorem
372–373
and probability theory
413, 414, 417
Pascal’s distribution 44
Pascal’s triangle 45, 65, 195,
196, 382–385, 384, 497
and combination 80–81
pathological functions 529
Pathwaie to Knowledge
(Recorde) 443
payoff matrix 217
Peano, Giuseppe 265, 385
Peano’s curve 385, 385–386
Peano’s postulates 31, 265,
271, 385, 386
Pearson, E. S. 354
Pearson, Karl 106, 107,
215–216, 386, 386–387,
415
peasant multiplication. See
Russian multiplication
pedal circle, of triangle 387
pedal triangle 387
Pell, John 54
Pell’s equation 54
pencil-turning trick 505–506,
506
pendulums 114–115, 213
Penny Cyclopedia 122
pentagon 285
pentagram (pentacle,
pentalpha, pentangle) 230,
387–388, 388, 424
percentage 388
percentage error 388–389
percentile (centile, quartile)
389
perfect number 18, 174, 338,
389, 423–424
perigon (round angle) 14,
362, 390
perimeter 390, 403, 443
period doubling 151
Index 563