A Treatise On Universal AlgebraPDF电子书下载

BOOK Ⅰ．PRINCIPLES OP ALGEBRAIC SYMBOLISM 1

CHAPTER Ⅰ．ON THE NATURE OF A CALCULUS 3

1．Signs 3

2．Definition of a Calculus 4

3．Equivalence 5

4．Operations 7

5．Substitutive Schemes 8

6．Conventional Schemes 9

7．Uninterpretable Forms 10

CHAPTER Ⅱ．MANIFOLDS 13

8．Manifolds 13

9．Secondary Properties of Elements 14

10．Definitions 15

11．Special Manifolds 16

CHAPTER Ⅲ．PRINCIPLES OF UNIVERSAL ALGEBRA 18

12．Introductory 18

13．Equivalence 18

14．Principles of Addition 19

15．Addition 21

16．Principles of Subtraction 22

17．The Null Element 24

18．Steps 25

19．Multiplication 25

20．Orders of Algebraic Manifolds 27

21．The Null Element 28

22．Classification of Special Algebras 29

Note 32

BOOK Ⅱ．THE ALGEBRA OF SYMBOLIC LOGIC 33

CHAPTER Ⅰ．THE ALGEBRA OF SYMBOLIC LOGIC 35

23．Formal Laws 35

24．Reciprocity between Addition and Multiplication 37

25．Interpretation 38

26．Elementary Propositions 39

27．Classification 41

28．Incident Regions 42

CHAPTER Ⅱ．THE ALGEBRA OF SYMBOLIC LOGIC(continued) 45

29．Development 45

30．Elimination 47

31．Solution of Equations with One Unknown 55

32．On Limiting and Unlimiting Equations 59

33．On the Fields of Expressions 60

34．Solution of Equations with More than One Unknown 65

35．Symmetrical Solution of Equations with Two Unknowns 67

36．Johnson＇s Method 73

37．Symmetrical Solution of Equations with Three Unknowns 75

38．Subtraction and Division 80

CHAPTER Ⅲ．EXISTENTIAL EXPRESSIONS 83

39．Existential Expressions 83

40．Umbral Letters 86

41．Elimination 89

42．Solutions of Existential Expressions with One Unknown 91

43．Existential Expressions with Two Unknowns 93

44．Equations and Existential Expressions with One Unknown 94

45．Boole＇s General Problem 96

46．Equations and Existential Propositions with Many Unknowns 97

Note 98

CHAPTER Ⅳ．APPLICATION TO LOGIC 99

47．Propositions 99

48．Exclusion of Nugatory Forms 100

49．Syllogism 101

50．Symbolic Equivalents of Syllogisms 103

51．Generalization of Logic 105

CHAPTER Ⅴ．PROPOSITIONAL INTERPRETATION 107

52．Propositional Interpretation 107

53．Equivalent Propositions 108

54．Symbolic Representation of Complexes 108

55．Identification with the Algebra of Symbolic Logic 108

56．Existential Expressions 111

57．Symbolism of the Traditional Propositions 111

58．Primitive Predication 112

59．Existential Symbols and Primitive Predication 113

60．Propositions 114

Historical Note 115

BOOK Ⅲ．POSITIONAL MANIFOLDS 117

CHAPTER Ⅰ．FUNDAMENTAL PROPOSITIONS 119

61．Introductory 119

62．Intensity 119

63．Things representing Different Elements 121

64．Fundamental Propositions 122

65．Subregions 125

66．Loci 128

67．Surface Loci and Curve Loci 130

Note 131

CHAPTER Ⅱ．STRAIGHT LINES AND PLANES 132

68．Introductory 132

69．Anharmonic Ratio 132

70．Homographic Ranges 133

71．Linear Transformations 133

72．Elementary Properties 136

73．Reference-Figures 138

74．Perspective 139

75．Quadrangles 142

CHAPTER Ⅲ．QUADRICS 144

76．Introductory 144

77．Elementary Properties 144

78．Poles and Polars 145

79．Generating Regions 147

80．Conjugate Coordinates 148

81．Quadriquadric Curve Loci 151

82．Closed Quadrics 153

83．Conical Quadric Surfaces 155

84．Reciprocal Equations and Conical quadrics 157

Note 161

CHAPTER Ⅳ．INTENSITY 162

85．Defining Equation of Intensity 162

86．Locus of Zero Intensity 163

87．Plane Locus of Zero Intensity 164

88．Quadric Locus of Zero Intensity 166

89．Antipodal Elements and Opposite Intensities 166

90．The Intercept between Two Elements 167

Note 168

BOOK Ⅳ．CALCULUS OF EXTENSION 169

CHAPTER Ⅰ．COMBINATORIAL MULTIPLICATION 171

91．Introductory 171

92．Invariant Equations of Condition 172

93．Principles of Combinatorial Multiplication 173

94．Derived Manifolds 175

95．Extensive Magnitudes 176

96．Simple and Compound Extensive Magnitudes 177

97．Fundamental Propositions 178

Note 180

CHAPTER Ⅱ．REGRESSIVE MULTIPLICATION 181

98．Progressive and Regressive Multiplication 181

99．Supplements 181

100．Definition of Regressive Multiplication 183

101．Pure and Mixed Products 184

102．Rule of the Middle Factor 185

103．Extended Rule of the Middle Factor 188

104．Regressive Multiplication independent of Reference-Elements 190

105．Proposition 191

106．Müller＇s Theorems 192

107．Applications and Examples 195

Note 198

CHAPTER Ⅲ．SUPPLEMENTS 199

108．Supplementary Regions 199

109．Normal Systems of Points 199

110．Extension of the Definition of Supplements 201

111．Different kinds of Supplements 202

112．Normal Points and Straight Lines 202

113．Mutually normal Regions 203

114．Self-normal Elements 204

115．Self-normal Planes 206

116．Complete Region of Three Dimensions 206

117．Inner Multiplication 207

118．Elementary Transformations 208

119．Rule of the Middle Factor 208

120．Important Formula 208

121．Inner Multiplication of Normal Regions 209

122．General Formula for Inner Multiplication 209

123．Quadrics 210

124．Plane-Equation of a Quadric 212

CHAPTER Ⅳ．DESCRIPTIVE GEOMETRY 214

125．Application to Descriptive Geometry 214

126．Explanation of Procedure 214

127．Illustration of Method 215

128．von Staudt＇s Construction 215

129．Grassmann＇s Constructions 219

130．Projection 224

CHAPTER Ⅴ．DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS 229

131．General Equation of a Conic 229

132．Further Transformations 231

133．Linear Construction of Cubics 233

134．First Type of Linear Construction of the Cubic 233

135．Linear Construction of Cubic through Nine arbitrary Points 235

136．Second Type of Linear Construction of the Cubic 238

137．Third Type of Linear Construction of the Cubic 239

138．Fourth Type of Linear Construction of the Cubic 244

139．Chasles＇ Construction 246

CHAPTER Ⅵ．MATRICES 248

140．Introductory 248

141．Definition of a Matrix 248

142．Sums and Products of Matrices 250

143．Associated Determinant 252

144．Null Spaces of Matrices 252

145．Latent Points 254

146．Semi-Latent Regions 266

147．The Identical Equation 256

148．The Latent Region of a Repeated Latent Root 257

149．The First Species of Semi-Latent Regions 258

150．The Higher Species of Semi-Latent Regions 259

151．The Identical Equation 261

152．The Vacuity of a Matrix 261

153．Symmetrical Matrices 262

154．Symmetrical Matrices and Supplements 265

155．Skew Matrices 267

BOOK Ⅴ．EXTENSIVE MANIFOLDS OF THREE DIMENSIONS 271

CHAPTER Ⅰ．SYSTEMS OF FORCES 273

156．Non-metrical Theory of Forces 273

157．Recapitulation of Formulas 274

158．Inner Multiplication 275

159．Elementary Properties of a Single Force 276

160．Elementary Properties of Systems of Forces 276

161．Condition for a Single Force 277

162．Conjugate Lines 277

163．Null Lines，Planes and Points 278

164．Properties of Null Lines 279

165．Lines in Involution 280

166．Reciprocal Systems 281

167．Formula for Systems of Forces 282

CHAPTER Ⅱ．GROUPS OF SYSTEMS OF FORCES 284

168．Specifications of a Group 284

169．Systems Reciprocal to Groups 285

170．Common Null Lines and Director Forces 286

171．Quintuple Groups 286

172．Quadruple and Dual Groups 287

173．Anharmonic Ratio of Systems 290

174．Self-Supplementary Dual Groups 292

175．Triple Groups 295

176．Conjugate Sets of Systems in a Triple Group 298

CHAPTER Ⅲ．INVARIANTS OF GROUPS 300

177．Definition of an Invariant 300

178．The Null Invariants of a Dual Group 300

179．The Harmonic Invariants of a Dual Group 301

180．Further Properties of Harmonic Invariants 302

181．Formul？ connected with Reciprocal Systems 303

182．Systems Reciprocal to a Dual Group 304

183．The Pole and Polar Invariants of a Triple Group 305

184．Conjugate Sets of Systems and the Pole and Polar Invariants 306

185．Interpretation of P(x) and P(X) 307

186．Relations between Conjugate Sets of Systems 308

187．The Conjugate Invariant of a Triple Group 310

188．Transformations of G(p，p) and G(P，P) 312

CHAPTER Ⅳ．MATRICES AND FORCES 316

189．Linear Transformations in Three Dimensions 316

190．Enumeration of Typos of Latent and Semi-Latent Regions 317

191．Matrices and Forces 322

192．Latent Systems and Semi-Latent Groups 323

193．Enumeration of Types of Latent Systems and Semi-Latent Groups 326

194．Transformation of a Quadric into itself 338

195．Direct Transformation of Quadrics 339

196．Skew Transformation of Quadrics 342

Note 346

BOOK Ⅵ．THEORY OF METRICS 347

CHAPTER Ⅰ．THEORY OF DISTANCE 349

197．Axioms of Distance 349

198．Congruent Ranges of Points 350

199．Cayley＇s Theory of Distance 351

200．Klein＇s Theorem 353

201．Comparison with the Axioms of Distance 354

202．Spatial Manifolds of Many Dimensions 354

203．Division of Space 355

204．Elliptic Space 356

205．Polar Form 356

206．Length of Intercepts in Polar Form 358

207．Antipodal Form 361

208．Hyperbolic Space 362

209．The Space Constant 363

210．Law of Intensity in Elliptic and Hyperbolic Geometry 364

211．Distances of Planes and of Subregions 365

212．Parabolic Geometry 367

213．Law of Intensity in Parabolic Geometry 368

Historical Note 369

CHAPTER Ⅱ．ELLIPTIC GEOMETRY 371

214．Introductory 371

215．Triangles 371

216．Further Formulae for Triangles 374

217．Points inside a Triangle 375

218．Oval Quadrics 376

219．Further Properties of Triangles 378

220．Planes One-sided 379

221．Angles between Planes 382

222．Stereometrical Triangles 382

223．Perpendiculars 383

224．Shortest Distances from Points to Planes 385

225．Common Perpendicular of Planes 386

226．Distances from Points to Subregions 387

227．Shortest Distances between Subregions 388

228．Spheres 391

229．Parallel Subregions 397

CHAPTER Ⅲ．EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY 399

230．Intensities of Forces 399

231．Relations between Two Forces 400

232．Axes of a System of Forces 401

233．Non-Axal Systems of Forces 404

234．Parallel Lines 404

235．Vector Systems 406

236．Vector Systems and Parallel Lines 407

237．Further Properties of Parallel Lines 409

238．Planes and Parallel Lines 411

CHAPTER Ⅳ．HYPERBOLIC GEOMETRY 414

239．Space and Anti-Space 414

240．Intensities of Points and Planes 415

241．Distances of Points 416

242．Distances of Planes 417

243．Spatial and Anti-spatial Lines 418

244．Distances of Subregions 419

246．Geometrical Signification 420

246．Poles and Polars 420

247．Points on the Absolute 422

248．Triangles 422

249．Properties of Angles of a Spatial Triangle 424

250．Stereometrical Triangles 425

251．Perpendiculars 426

252．The Feet of Perpendiculars 427

253．Distance between Planes 428

254．Shortest Distances 429

255．Shortest Distances between Subregions 430

256．Rectangular Rectilinear Figures 433

257．Parallel Lines 436

258．Parallel Planes 439

CHAPTER Ⅴ．HYPERBOLIC GEOMETRY(continued) 441

259．The Sphere 441

260．Intersection of Spheres 444

261．Limit-Surfaces 447

262．Great Circles on Spheres 448

263．Surfaces of Equal Distance from Subregions 451

264．Intensities of Forces 452

265．Relations between Two Spatial Forces 452

266．Central Axis of a System of Forces 454

267．Non-Axal Systems of Forces 455

CHAPTER Ⅵ．KINEMATICS IN THREE DIMENSIONS 456

268．Congruent Transformations 456

269．Elementary Formulae 458

270．Simple Geometrical Properties 459

271．Translations and Rotations 460

272．Locus of Points of Equal Displacement 462

273．Equivalent Sets of Congruent Transformations 463

274．Commutative Law 464

275．Small Displacements 464

276．Small Translations and Rotations 465

277．Associated System of Forces 466

278．Properties deduced from the Associated System 467

279．Work 468

280．Characteristic Lines 470

281．Elliptic Space 470

282．Surfaces of Equal Displacement 472

283．Vector Transformations 472

284．Associated Vector Systems of Forces 473

285．Successive Vector Transformations 473

286．Small Displacements 476

CHAPTER Ⅶ．CURVES AND SURFACES 478

287．Curve Lines 478

288．Curvature and Torsion 479

289．Planar Formul？ 481

290．Velocity and Acceleration 482

291．The Circle 484

292．Motion of a Rigid Body 487

293．Gauss＇ Curvilinear Coordinates 488

294．Curvature of Surfaces 489

295．Lines of Curvature 490

296．Meunier＇s Theorem 493

297．Normals 493

298．Curvilinear Coordinates 494

299．Limit-Surfaces 494

CHAPTER Ⅷ．TRANSITION TO PARABOLIC GEOMETRY 496

300．Parabolic Geometry 496

301．Plane Equation of the Absolute 496

302．Intensities 498

303．Congruent Transformations 500

BOOK Ⅶ．APPLICATION OF THE CALCULUS OF EXTENSION TO GEOMETRY 503

CHAPTER Ⅰ．VECTORS 505

304．Introductory 505

305．Points at Infinity 506

306．Vectors 507

307．Linear Elements 508

308．Vector Areas 509

309．Vector Areas as Carriers 511

310．Planar Elements 512

311．Vector Volumes 513

312．Vector Volumes as Carriers 513

313．Product of Four Points 514

314．Point and Vector Factors 514

315．Interpretation of Formulae 515

316．Vector Formul？ 516

317．Operation of Taking the Vector 516

318．Theory of Forces 518

319．Graphic Statics 520

Note 522

CHAPTER Ⅱ．VECTORS(continued) 523

320．Supplements 523

321．Rectangular Normal Systems 524

322．Imaginary Self-Normal Sphere 524

323．Real Self-Normal Sphere 525

324．Geometrical Formul？ 526

325．Taking the Flux 527

326．Flux Multiplication 528

327．Geometrical Formul？ 529

328．The Central Axis 529

329．Planes containing the Central Axis 530

330．Dual Groups of Systems of Forces 530

331．Invariants of a Dual Group 531

332．Secondary Axes of a Dual Group 531

333．The Cylindroid 532

334．The Harmonic Invariants 533

335．Triple Groups 533

336．The Pole and Polar Invariants 534

337．Equation of the Associated Quadric 535

338．Normals 535

339．Small Displacements of a Rigid Body 536

340．Work 537

CHAPTER Ⅲ．CURVES AND SURFACES 539

341．Curves 539

342．Osculating Plane and Normals 540

343．Acceleration 540

344．Simplified Formul？ 541

345．Spherical Curvature 541

346．Locus of Centre of Curvature 542

347．Gauss＇ Curvilinear Co-ordinates 543

348．Curvature 544

349．Lines of Curvature 545

350．Dupin＇s Theorem 546

351．Eider＇s Theorem 547

352．Meunier＇s Theorem 547

Note 547

CHAPTER Ⅳ．PURE VECTOR FORMUL？ 548

353．Introductory 548

354．Lengths and Areas 549

355．Formul？ 549

356．The Origin 550

357．New Convention 550

358．System of Forces 551

359．Kinematics 551

360．A Continuously Distributed Substance 552

361．Hamilton＇s Differential Operator 554

362．Conventions and Formulas 555

363．Polar Co-ordinates 557

364．Cylindrical Co-ordinates 558

365．Orthogonal Curvilinear Co-ordinates 560

366．Volume，Surface，and Line Integrals 562

367．The Equations of Hydrodynamics 562

368．Moving Origin 563

369．Transformations of Hydrodynamical Equations 565

370．Vector Potential of Velocity 565

371．Curl Filaments of Constant Strength 567

372．Carried Functions 569

373．Clebsch＇s Transformations 570

374．Flow of a Vector 572

Note 573

Note on Grassmann 573

Index 576