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BOOK Ⅰ．PRINCIPLES OP ALGEBRAIC SYMBOLISM1

CHAPTER Ⅰ．ON THE NATURE OF A CALCULUS3

1．Signs3

2．Definition of a Calculus4

3．Equivalence5

4．Operations7

5．Substitutive Schemes8

6．Conventional Schemes9

7．Uninterpretable Forms10

CHAPTER Ⅱ．MANIFOLDS13

8．Manifolds13

9．Secondary Properties of Elements14

10．Definitions15

11．Special Manifolds16

CHAPTER Ⅲ．PRINCIPLES OF UNIVERSAL ALGEBRA18

12．Introductory18

13．Equivalence18

14．Principles of Addition19

15．Addition21

16．Principles of Subtraction22

17．The Null Element24

18．Steps25

19．Multiplication25

20．Orders of Algebraic Manifolds27

21．The Null Element28

22．Classification of Special Algebras29

Note32

BOOK Ⅱ．THE ALGEBRA OF SYMBOLIC LOGIC33

CHAPTER Ⅰ．THE ALGEBRA OF SYMBOLIC LOGIC35

23．Formal Laws35

24．Reciprocity between Addition and Multiplication37

25．Interpretation38

26．Elementary Propositions39

27．Classification41

28．Incident Regions42

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

29．Development45

30．Elimination47

31．Solution of Equations with One Unknown55

32．On Limiting and Unlimiting Equations59

33．On the Fields of Expressions60

34．Solution of Equations with More than One Unknown65

35．Symmetrical Solution of Equations with Two Unknowns67

36．Johnson＇s Method73

37．Symmetrical Solution of Equations with Three Unknowns75

38．Subtraction and Division80

CHAPTER Ⅲ．EXISTENTIAL EXPRESSIONS83

39．Existential Expressions83

40．Umbral Letters86

41．Elimination89

42．Solutions of Existential Expressions with One Unknown91

43．Existential Expressions with Two Unknowns93

44．Equations and Existential Expressions with One Unknown94

45．Boole＇s General Problem96

46．Equations and Existential Propositions with Many Unknowns97

Note98

CHAPTER Ⅳ．APPLICATION TO LOGIC99

47．Propositions99

48．Exclusion of Nugatory Forms100

49．Syllogism101

50．Symbolic Equivalents of Syllogisms103

51．Generalization of Logic105

CHAPTER Ⅴ．PROPOSITIONAL INTERPRETATION107

52．Propositional Interpretation107

53．Equivalent Propositions108

54．Symbolic Representation of Complexes108

55．Identification with the Algebra of Symbolic Logic108

56．Existential Expressions111

57．Symbolism of the Traditional Propositions111

58．Primitive Predication112

59．Existential Symbols and Primitive Predication113

60．Propositions114

Historical Note115

BOOK Ⅲ．POSITIONAL MANIFOLDS117

CHAPTER Ⅰ．FUNDAMENTAL PROPOSITIONS119

61．Introductory119

62．Intensity119

63．Things representing Different Elements121

64．Fundamental Propositions122

65．Subregions125

66．Loci128

67．Surface Loci and Curve Loci130

Note131

CHAPTER Ⅱ．STRAIGHT LINES AND PLANES132

68．Introductory132

69．Anharmonic Ratio132

70．Homographic Ranges133

71．Linear Transformations133

72．Elementary Properties136

73．Reference-Figures138

74．Perspective139

75．Quadrangles142

CHAPTER Ⅲ．QUADRICS144

76．Introductory144

77．Elementary Properties144

78．Poles and Polars145

79．Generating Regions147

80．Conjugate Coordinates148

81．Quadriquadric Curve Loci151

82．Closed Quadrics153

83．Conical Quadric Surfaces155

84．Reciprocal Equations and Conical quadrics157

Note161

CHAPTER Ⅳ．INTENSITY162

85．Defining Equation of Intensity162

86．Locus of Zero Intensity163

87．Plane Locus of Zero Intensity164

88．Quadric Locus of Zero Intensity166

89．Antipodal Elements and Opposite Intensities166

90．The Intercept between Two Elements167

Note168

BOOK Ⅳ．CALCULUS OF EXTENSION169

CHAPTER Ⅰ．COMBINATORIAL MULTIPLICATION171

91．Introductory171

92．Invariant Equations of Condition172

93．Principles of Combinatorial Multiplication173

94．Derived Manifolds175

95．Extensive Magnitudes176

96．Simple and Compound Extensive Magnitudes177

97．Fundamental Propositions178

Note180

CHAPTER Ⅱ．REGRESSIVE MULTIPLICATION181

98．Progressive and Regressive Multiplication181

99．Supplements181

100．Definition of Regressive Multiplication183

101．Pure and Mixed Products184

102．Rule of the Middle Factor185

103．Extended Rule of the Middle Factor188

104．Regressive Multiplication independent of Reference-Elements190

105．Proposition191

106．Müller＇s Theorems192

107．Applications and Examples195

Note198

CHAPTER Ⅲ．SUPPLEMENTS199

108．Supplementary Regions199

109．Normal Systems of Points199

110．Extension of the Definition of Supplements201

111．Different kinds of Supplements202

112．Normal Points and Straight Lines202

113．Mutually normal Regions203

114．Self-normal Elements204

115．Self-normal Planes206

116．Complete Region of Three Dimensions206

117．Inner Multiplication207

118．Elementary Transformations208

119．Rule of the Middle Factor208

120．Important Formula208

121．Inner Multiplication of Normal Regions209

122．General Formula for Inner Multiplication209

123．Quadrics210

124．Plane-Equation of a Quadric212

CHAPTER Ⅳ．DESCRIPTIVE GEOMETRY214

125．Application to Descriptive Geometry214

126．Explanation of Procedure214

127．Illustration of Method215

128．von Staudt＇s Construction215

129．Grassmann＇s Constructions219

130．Projection224

CHAPTER Ⅴ．DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS229

131．General Equation of a Conic229

132．Further Transformations231

133．Linear Construction of Cubics233

134．First Type of Linear Construction of the Cubic233

135．Linear Construction of Cubic through Nine arbitrary Points235

136．Second Type of Linear Construction of the Cubic238

137．Third Type of Linear Construction of the Cubic239

138．Fourth Type of Linear Construction of the Cubic244

139．Chasles＇ Construction246

CHAPTER Ⅵ．MATRICES248

140．Introductory248

141．Definition of a Matrix248

142．Sums and Products of Matrices250

143．Associated Determinant252

144．Null Spaces of Matrices252

145．Latent Points254

146．Semi-Latent Regions266

147．The Identical Equation256

148．The Latent Region of a Repeated Latent Root257

149．The First Species of Semi-Latent Regions258

150．The Higher Species of Semi-Latent Regions259

151．The Identical Equation261

152．The Vacuity of a Matrix261

153．Symmetrical Matrices262

154．Symmetrical Matrices and Supplements265

155．Skew Matrices267

BOOK Ⅴ．EXTENSIVE MANIFOLDS OF THREE DIMENSIONS271

CHAPTER Ⅰ．SYSTEMS OF FORCES273

156．Non-metrical Theory of Forces273

157．Recapitulation of Formulas274

158．Inner Multiplication275

159．Elementary Properties of a Single Force276

160．Elementary Properties of Systems of Forces276

161．Condition for a Single Force277

162．Conjugate Lines277

163．Null Lines，Planes and Points278

164．Properties of Null Lines279

165．Lines in Involution280

166．Reciprocal Systems281

167．Formula for Systems of Forces282

CHAPTER Ⅱ．GROUPS OF SYSTEMS OF FORCES284

168．Specifications of a Group284

169．Systems Reciprocal to Groups285

170．Common Null Lines and Director Forces286

171．Quintuple Groups286

172．Quadruple and Dual Groups287

173．Anharmonic Ratio of Systems290

174．Self-Supplementary Dual Groups292

175．Triple Groups295

176．Conjugate Sets of Systems in a Triple Group298

CHAPTER Ⅲ．INVARIANTS OF GROUPS300

177．Definition of an Invariant300

178．The Null Invariants of a Dual Group300

179．The Harmonic Invariants of a Dual Group301

180．Further Properties of Harmonic Invariants302

181．Formul？ connected with Reciprocal Systems303

182．Systems Reciprocal to a Dual Group304

183．The Pole and Polar Invariants of a Triple Group305

184．Conjugate Sets of Systems and the Pole and Polar Invariants306

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

186．Relations between Conjugate Sets of Systems308

187．The Conjugate Invariant of a Triple Group310

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

CHAPTER Ⅳ．MATRICES AND FORCES316

189．Linear Transformations in Three Dimensions316

190．Enumeration of Typos of Latent and Semi-Latent Regions317

191．Matrices and Forces322

192．Latent Systems and Semi-Latent Groups323

193．Enumeration of Types of Latent Systems and Semi-Latent Groups326

194．Transformation of a Quadric into itself338

195．Direct Transformation of Quadrics339

196．Skew Transformation of Quadrics342

Note346

BOOK Ⅵ．THEORY OF METRICS347

CHAPTER Ⅰ．THEORY OF DISTANCE349

197．Axioms of Distance349

198．Congruent Ranges of Points350

199．Cayley＇s Theory of Distance351

200．Klein＇s Theorem353

201．Comparison with the Axioms of Distance354

202．Spatial Manifolds of Many Dimensions354

203．Division of Space355

204．Elliptic Space356

205．Polar Form356

206．Length of Intercepts in Polar Form358

207．Antipodal Form361

208．Hyperbolic Space362

209．The Space Constant363

210．Law of Intensity in Elliptic and Hyperbolic Geometry364

211．Distances of Planes and of Subregions365

212．Parabolic Geometry367

213．Law of Intensity in Parabolic Geometry368

Historical Note369

CHAPTER Ⅱ．ELLIPTIC GEOMETRY371

214．Introductory371

215．Triangles371

216．Further Formulae for Triangles374

217．Points inside a Triangle375

218．Oval Quadrics376

219．Further Properties of Triangles378

220．Planes One-sided379

221．Angles between Planes382

222．Stereometrical Triangles382

223．Perpendiculars383

224．Shortest Distances from Points to Planes385

225．Common Perpendicular of Planes386

226．Distances from Points to Subregions387

227．Shortest Distances between Subregions388

228．Spheres391

229．Parallel Subregions397

CHAPTER Ⅲ．EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY399

230．Intensities of Forces399

231．Relations between Two Forces400

232．Axes of a System of Forces401

233．Non-Axal Systems of Forces404

234．Parallel Lines404

235．Vector Systems406

236．Vector Systems and Parallel Lines407

237．Further Properties of Parallel Lines409

238．Planes and Parallel Lines411

CHAPTER Ⅳ．HYPERBOLIC GEOMETRY414

239．Space and Anti-Space414

240．Intensities of Points and Planes415

241．Distances of Points416

242．Distances of Planes417

243．Spatial and Anti-spatial Lines418

244．Distances of Subregions419

246．Geometrical Signification420

246．Poles and Polars420

247．Points on the Absolute422

248．Triangles422

249．Properties of Angles of a Spatial Triangle424

250．Stereometrical Triangles425

251．Perpendiculars426

252．The Feet of Perpendiculars427

253．Distance between Planes428

254．Shortest Distances429

255．Shortest Distances between Subregions430

256．Rectangular Rectilinear Figures433

257．Parallel Lines436

258．Parallel Planes439

CHAPTER Ⅴ．HYPERBOLIC GEOMETRY(continued)441

259．The Sphere441

260．Intersection of Spheres444

261．Limit-Surfaces447

262．Great Circles on Spheres448

263．Surfaces of Equal Distance from Subregions451

264．Intensities of Forces452

265．Relations between Two Spatial Forces452

266．Central Axis of a System of Forces454

267．Non-Axal Systems of Forces455

CHAPTER Ⅵ．KINEMATICS IN THREE DIMENSIONS456

268．Congruent Transformations456

269．Elementary Formulae458

270．Simple Geometrical Properties459

271．Translations and Rotations460

272．Locus of Points of Equal Displacement462

273．Equivalent Sets of Congruent Transformations463

274．Commutative Law464

275．Small Displacements464

276．Small Translations and Rotations465

277．Associated System of Forces466

278．Properties deduced from the Associated System467

279．Work468

280．Characteristic Lines470

281．Elliptic Space470

282．Surfaces of Equal Displacement472

283．Vector Transformations472

284．Associated Vector Systems of Forces473

285．Successive Vector Transformations473

286．Small Displacements476

CHAPTER Ⅶ．CURVES AND SURFACES478

287．Curve Lines478

288．Curvature and Torsion479

289．Planar Formul？481

290．Velocity and Acceleration482

291．The Circle484

292．Motion of a Rigid Body487

293．Gauss＇ Curvilinear Coordinates488

294．Curvature of Surfaces489

295．Lines of Curvature490

296．Meunier＇s Theorem493

297．Normals493

298．Curvilinear Coordinates494

299．Limit-Surfaces494

CHAPTER Ⅷ．TRANSITION TO PARABOLIC GEOMETRY496

300．Parabolic Geometry496

301．Plane Equation of the Absolute496

302．Intensities498

303．Congruent Transformations500

BOOK Ⅶ．APPLICATION OF THE CALCULUS OF EXTENSION TO GEOMETRY503

CHAPTER Ⅰ．VECTORS505

304．Introductory505

305．Points at Infinity506

306．Vectors507

307．Linear Elements508

308．Vector Areas509

309．Vector Areas as Carriers511

310．Planar Elements512

311．Vector Volumes513

312．Vector Volumes as Carriers513

313．Product of Four Points514

314．Point and Vector Factors514

315．Interpretation of Formulae515

316．Vector Formul？516

317．Operation of Taking the Vector516

318．Theory of Forces518

319．Graphic Statics520

Note522

CHAPTER Ⅱ．VECTORS(continued)523

320．Supplements523

321．Rectangular Normal Systems524

322．Imaginary Self-Normal Sphere524

323．Real Self-Normal Sphere525

324．Geometrical Formul？526

325．Taking the Flux527

326．Flux Multiplication528

327．Geometrical Formul？529

328．The Central Axis529

329．Planes containing the Central Axis530

330．Dual Groups of Systems of Forces530

331．Invariants of a Dual Group531

332．Secondary Axes of a Dual Group531

333．The Cylindroid532

334．The Harmonic Invariants533

335．Triple Groups533

336．The Pole and Polar Invariants534

337．Equation of the Associated Quadric535

338．Normals535

339．Small Displacements of a Rigid Body536

340．Work537

CHAPTER Ⅲ．CURVES AND SURFACES539

341．Curves539

342．Osculating Plane and Normals540

343．Acceleration540

344．Simplified Formul？541

345．Spherical Curvature541

346．Locus of Centre of Curvature542

347．Gauss＇ Curvilinear Co-ordinates543

348．Curvature544

349．Lines of Curvature545

350．Dupin＇s Theorem546

351．Eider＇s Theorem547

352．Meunier＇s Theorem547

Note547

CHAPTER Ⅳ．PURE VECTOR FORMUL？548

353．Introductory548

354．Lengths and Areas549

355．Formul？549

356．The Origin550

357．New Convention550

358．System of Forces551

359．Kinematics551

360．A Continuously Distributed Substance552

361．Hamilton＇s Differential Operator554

362．Conventions and Formulas555

363．Polar Co-ordinates557

364．Cylindrical Co-ordinates558

365．Orthogonal Curvilinear Co-ordinates560

366．Volume，Surface，and Line Integrals562

367．The Equations of Hydrodynamics562

368．Moving Origin563

369．Transformations of Hydrodynamical Equations565

370．Vector Potential of Velocity565

371．Curl Filaments of Constant Strength567

372．Carried Functions569

373．Clebsch＇s Transformations570

374．Flow of a Vector572

Note573

Note on Grassmann573

Index576