## Principia Mathematica, Volume 1Principia Mathematica was first published in 1910-13; this is the ninth impression of the second edition of 1925-7. The Principia has long been recognised as one of the intellectual landmarks of the century. It was the first book to show clearly the close relationship between mathematics and formal logic. Starting from a minimal number of axioms, Whitehead and Russell display the structure of both kinds of thought. No other book has had such an influence on the subsequent history of mathematical philosophy. |

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### Contents

1AGE | 1 |

SERIES continued 250 Elementary properties of wellordered series | 4 |

251 Ordinal numbers | 18 |

252 Segments of wellordered aeries | 27 |

253 Sectional relations of wellordered series | 32 |

254 Qreater and less among wellordered series | 44 |

255 Greater and less among ordinal numbers | 58 |

256 The series of ordinals | 73 |

306 Addition of simple ratios | 289 |

307 Generalized ratios | 296 |

308 Addition of generalized ratios | 299 |

309 Multiplication of generalized ratios | 309 |

310 The series of real numbers 316 | 316 |

311 Addition of concordant real numbers | 320 |

312 Algebraic addition of real number | 327 |

313 Multiplication of real numbers | 333 |

257 The transfinite ancestnvl relation | 81 |

258 Zermelos theorem | 96 |

259 Inductively defined correlations | 102 |

SECTION E FINITE AND INFINITE SERIES AND ORDINALS | 108 |

260 On finite intervals in a series | 109 |

261 Finite and infinite series | 118 |

262 Finite ordinals | 131 |

263 Progressions | 143 |

264 Derivatives of wellordered series | 156 |

265 The series of alephs | 169 |

SECTION F COMPACT SERIES RATIONAL SERIES AND CONTINUOUS SERIES | 179 |

270 Compact series | 180 |

271 Median classes in series _ | 186 |

272 Similarity of position | 191 |

273 Rational series | 199 |

274 On series of finite subclasses of a series | 207 |

275 Continuous series | 218 |

276 On series of infinite subulasses of a series | 221 |

QUANTITY | 231 |

Summary of Part VI | 233 |

SECTION A GENERALIZATION OF NUMBER | 234 |

300 Positive and negative integers and numerical relations 235 | 235 |

301 Numerically defined powers of relations | 244 |

302 On relative primes | 251 |

303 Ratios | 260 |

304 The series of ratios | 278 |

305 Multiplication of simple ratios | 283 |

314 Real numbers as relations | 336 |

SECTION B VECTORFAMILIES | 339 |

330 Elementary properties of vectorfamilies | 350 |

331 Connected families | 360 |

332 On the representative of relation in a family | 367 |

333 Open families | 376 |

334 Serial families | 383 |

335 Initial families | 390 |

336 The seriea of vectors | 393 |

337 Multiples and aubroultiplca of vectors | 403 |

MEASUREMENT | 407 |

350 Ratios of members of a family | 412 |

351 Submultipliable families | 419 |

352 Rational multiples of a given vector | 423 |

353 Rational families | 431 |

354 Rational nets | 436 |

356 Measurement by real numbers | 443 |

359 Existencetheorems for vectorfamilies | 452 |

CYCLIC FAMILIES | 457 |

370 Elementary properties of cyclic families | 462 |

371 The series of vectors | 466 |

372 Integral sections of the series of vectors | 470 |

373 Submultiplns of identity | 475 |

374 Principal submultiples 435 | 485 |

375 Principal ration | 487 |

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### Common terms and phrases

a e NC axiom of infinity Bord Cantor Cl induct'C'P Cls induct comp connected family correlator Dedekindian defined definition DK Prop e Cls induct existence-theorems existent sub-class field finite ordinals FM ap conx FM conx FM cycl following propositions ft infin Hence hypothesis inductive cardinals Infin ax infinite KfFM last term less Q mathematical induction median class multiplicative axiom NC induct Nr'ft Nr'P Nr'Q ordinal number ordinally similar present number Prog proof proper fractions proper section properties prove Q less rational series ratios real numbers Rel num id relation Rl'P Rl'Q sect'P serial Similarly h smof smor Q submultiple Transp typically indefinite v e NC ind v e NC induct vector vector-family well-ordered series whence Ye Rat