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