A Treatise on Universal Algebra: With Applications

Cover......Page 1
Frontmatter......Page 6
PREFACE......Page 10
Contents......Page 18
BOOK I - PRINCIPLES OF ALGEBRAIC SYMBOLISM......Page 32
1. Signs......Page 34
2. Definition of a Calculus......Page 35
3. Equivalence......Page 36
4. Operations......Page 38
5. Substitutive Schemes......Page 39
6. Conventional Schemes......Page 40
7. Uninterpretable Forms......Page 41
8. Manifolds......Page 44
9. Secondary Properties of Elements......Page 45
10. Definitions......Page 46
11. Special Manifolds......Page 47
13. Equivalence......Page 49
14. Principles of Addition......Page 50
15. Addition......Page 52
16. Principles of Subtraction......Page 53
17. The Null Element......Page 55
19. Multiplication......Page 56
20. Orders of Algebraic Manifolds......Page 58
21. The Null Element......Page 59
22. Classification of Special Algebras......Page 60
Note......Page 63
BOOK II - THE ALGEBRA OF SYMBOLIC LOGIC......Page 64
23. Formal Laws......Page 66
24. Reciprocity between Addition and Multiplication......Page 68
25. Interpretation......Page 69
26. Elementary Propositions......Page 70
27. Classification......Page 72
28. Incident Regions......Page 73
29. Development......Page 76
30. Elimination......Page 78
31. Solution of Equations with One Unknown......Page 86
32. On Limiting and Unlimiting Equations......Page 90
33. On the Fields of Expressions......Page 91
34. Solution of Equations with More than One Unknown......Page 96
35. Symmetrical Solution of Equations with Two Unknowns......Page 98
36. Johnson\'s Method......Page 104
37. Symmetrical Solution of Equations with Three Unknowns......Page 106
38. Subtraction and Division......Page 111
39. Existential Expressions......Page 114
40. Umbral Letters......Page 117
41. Elimination......Page 120
42. Solutions of Existential Expressions with One Unknown......Page 122
43. Existential Expressions with Two Unknowns......Page 124
44. Equations and Existential Expressions with One Unknown......Page 125
45. Boole\'s General Problem......Page 127
46. Equations and Existential Propositions with Many Unknowns......Page 128
Note......Page 129
47. Propositions......Page 130
48. Exclusion of Nugatory Forms......Page 131
49. Syllogism......Page 132
50. Symbolic Equivalents of Syllogisms......Page 134
51. Generalization of Logic......Page 136
52. Propositional Interpretation......Page 138
55. Identification with the Algebra of Symbolic Logic......Page 139
57. Symbolism of the Traditional Propositions......Page 142
58. Primitive Predication......Page 143
59. Existential Symbols and Primitive Predication......Page 144
60. Propositions......Page 145
Historical Note......Page 146
BOOK III - POSITIONAL MANIFOLDS......Page 148
62. Intensity......Page 150
63. Things representing Different Elements......Page 152
64. Fundamental Propositions......Page 153
65. Subregions......Page 156
66. Loci......Page 159
67. Surface Loci and Curve Loci......Page 161
Note......Page 162
69. Anharmonic Ratio......Page 163
71. Linear Transformations......Page 164
72. Elementary Properties......Page 167
73. Reference-Figures......Page 169
74. Perspective......Page 170
75. Quadrangles......Page 173
77. Elementary Properties......Page 175
78. Poles and Polars......Page 176
79. Generating Regions......Page 178
80. Conjugate Coordinates......Page 179
81. Quadriquadric Curve Loci......Page 182
82. Closed Quadrics......Page 184
83. Conical Quadric Surfaces......Page 186
84. Reciprocal Equations and Conical quadrics......Page 188
Note......Page 192
85. Defining Equation of Intensity......Page 193
86. Locus of Zero Intensity......Page 194
87. Plane Locus of Zero Intensity......Page 195
89. Antipodal Elements and Opposite Intensities......Page 197
90. The Intercept between Two Elements......Page 198
Note......Page 199
BOOK IV - CALCULUS OF EXTENSION......Page 200
91. Introductory......Page 202
92. Invariant Equations of Condition......Page 203
93. Principles of Combinatorial Multiplication......Page 204
94. Derived Manifolds......Page 206
95. Extensive Magnitudes......Page 207
96. Simple and Compound Extensive Magnitudes......Page 208
97. Fundamental Propositions......Page 209
Note......Page 211
99. Supplements......Page 212
100. Definition of Regressive Multiplication......Page 214
101. Pure and Mixed Products......Page 215
102. Rule of the Middle Factor......Page 216
103. Extended Rule of the Middle Factor......Page 219
104. Regressive Multiplication independent of Reference-Elements......Page 221
105. Proposition......Page 222
106. Mülle\'s Theorems......Page 223
107. Applications and Examples......Page 226
Note......Page 229
109. Normal Systems of Points......Page 230
110. Extension of t h e Definition of Supplements......Page 232
112. Normal Points and Straight Lines......Page 233
113. Mutually normal Regions......Page 234
114. Self-normal Elements......Page 235
116. Complete Region of Three Dimensions......Page 237
117. Inner Multiplication......Page 238
120. Important Formula......Page 239
122. General Formula for Inner Multiplication......Page 240
123. Quadrics......Page 241
124. Plane-Equation of a Quadric......Page 243
126. Explanation of Procedure......Page 245
128. von Staudt\'s Construction......Page 246
129. Grassmann\'s Constructions......Page 250
130. Projection......Page 255
131. General Equation of a Conic......Page 260
132. Further Transformations......Page 262
134. First Type of Linear Construction of the Cubic......Page 264
135. Linear Construction of Cubic through Nine arbitrary Points......Page 266
136. Second Type of Linear Construction of the Cubic......Page 269
137. Third Type of Linear Construction of the Cubic......Page 270
138. Fourth Type of Linear Construction of the Cubic......Page 275
139. Chasles\' Construction......Page 277
141. Definition of a Matrix......Page 279
142. Sums and Products of Matrices......Page 281
144. Null Spaces of Matrices......Page 283
145. Latent Points......Page 285
147. The Identical Equation......Page 287
148. The Latent Region of a Repeated Latent Root......Page 288
149. The First Species of Semi-Latent Regions......Page 289
150. The Higher Species of Semi-Latent Regions......Page 290
152. The Vacuity of a Matrix......Page 292
153. Symmetrical Matrices......Page 293
154. Symmetrical Matrices and Supplements......Page 296
155. Skew Matrices......Page 298
BOOK V - EXTENSIVE MANIFOLDS OF THREE DIMENSIONS......Page 302
156. Non-metrical Theory of Forces......Page 304
157. Recapitulation of Formulæ......Page 305
158. Inner Multiplication......Page 306
160. Elementary Properties of Systems of Forces......Page 307
162. Conjugate Lines......Page 308
163. Null Lines, Planes and Points......Page 309
164. Properties of Null Lines......Page 310
165. Lines in Involution......Page 311
166. Reciprocal Systems......Page 312
167. Formulae for Systems of Forces......Page 313
168. Specifications of a Group......Page 315
169. Systems Reciprocal to Groups......Page 316
171. Quintuple Groups......Page 317
172. Quadruple and Dual Groups......Page 318
173. Anharmonic Ratio of Systems......Page 321
174. Self-Supplementary Dual Groups......Page 323
175. Triple Groups......Page 326
176. Conjugate Sets of Systems in a Triple Group......Page 329
178. The Null Invariants of a Dual Group......Page 331
179. The Harmonic Invariants of a Dual Group......Page 332
180. Further Properties of Harmonic Invariants3......Page 333
181. Formulae connected with Reciprocal Systems......Page 334
182. Systems Reciprocal to a Dual Group......Page 335
183. The Pole and Polar Invariants of a Triple Group......Page 336
184. Conjugate Sets of Systems and t h e Pole and Polar Invariants......Page 337
185. Interpretation of P(x) and P(X)......Page 338
186. Relations between Conjugate Sets of Systems......Page 339
187. The Conjugate Invariant of a Triple Group......Page 341
188. Transformations of G (p, p) and G (P, P)......Page 343
189. Linear Transformations in Three Dimensions......Page 347
190. Enumeration of Types of Latent and Semi-Latent Regions......Page 348
191. Matrices and Forces......Page 353
192. Latent Systems and Semi-Latent Groups......Page 354
193. Enumeration of Types of Latent Systems and Semi-Latent Groups......Page 357
194. Transformation of a Quadric into itself......Page 369
195. Direct Transformation of Quadrics......Page 370
196. Skew Transformation of Quadrics......Page 373
Note......Page 377
BOOK VI - THEORY OF METRICS......Page 378
197. Axioms of Distance......Page 380
198. Congruent Ranges of Points......Page 381
199. Cayley\'s Theory of Distance......Page 382
200. Klein\'s Theorem......Page 384
202. Spatial Manifolds of Many Dimensions......Page 385
203. Division of Space......Page 386
205. Polar Form......Page 387
206. Length of Intercepts in Polar Form......Page 389
207. Antipodal Form......Page 392
208. HyperMic Space......Page 393
209. The Space Constant......Page 394
210. Law of Intensity in Elliptic and Hyperbolic Geometry......Page 395
211. Distances of Planes and of Subregions......Page 396
212. Parabolic Geometry......Page 398
213. Law of Intensity in Parabolic Geometry......Page 399
Historical Note......Page 400
215. Triangles......Page 402
216. Further Formulæ for Triangles......Page 405
217. Points inside a Triangle......Page 406
218. Oval Quadrics......Page 407
219. Further Properties of Triangles......Page 409
220. Planes One-sided......Page 410
222. Stereometrical Triangles......Page 413
223. Perpendiculars......Page 414
224. Shortest Distances from Points to Planes......Page 416
225. Common Perpendicular of Planes......Page 417
226. Distances from Points to Subregions......Page 418
227. Shortest Distances between Subregions......Page 419
228. Spheres......Page 422
229. Parallel Subregions......Page 428
230. Intensities of Forces......Page 430
231. Relations between Two Forces......Page 431
232. Axes of a System of Forces......Page 432
234 Parallel Lines......Page 435
235. Vector Systems......Page 437
236. Vector Systems and Parallel Lines......Page 438
237. Further Properties of Parallel Lines......Page 440
238. Planes and Parallel Lines......Page 442
239. Space and Anti-Space......Page 445
240. Intensities of Points and Planes......Page 446
241. Distances of Points......Page 447
242. Distances of Planes......Page 448
243. Spatial and Anti-spatial Lines......Page 449
244. Distances of Subregions......Page 450
246. Poles and Polars......Page 451
248. Triangles......Page 453
249. Properties of Angles of a Spatial Triangle......Page 455
250. Stereoinetrical Triangles......Page 456
251. Perpendiculars......Page 457
252. The Feet of Perpendiculars......Page 458
253. Distance between Planes......Page 459
254. Shortest Distances......Page 460
255. Shortest Distances between Subregions......Page 461
256. Rectangular Rectilinear Figures......Page 464
257. Parallel Lines......Page 467
258. Parallel Planes......Page 470
259. The Sphere......Page 472
260. Intersection of Spheres......Page 475
261. Limit-Surfaces......Page 478
262. Great Circles on Spheres......Page 479
263. Surfaces of Equal Distance from Subregions......Page 482
265. Relations between Two Spatial Forces......Page 483
266. Central Axis of a System of Forces......Page 485
267. Non-Axal Systems of Forces......Page 486
268. Congruent Transformations......Page 487
269. Elementary Formulæ......Page 489
270. Simple Geometrical Properties......Page 490
271. Translations and Rotations......Page 491
272. Locus of Points of Equal Displacement......Page 493
273. Equivalent Sets of Congruent Transformations......Page 494
275. Small Displacements......Page 495
276. Small Translations and Rotations......Page 496
277. Associated System of Forces......Page 497
278. Properties deduced from the Associated System......Page 498
279. Work......Page 499
281. Elliptic Space......Page 501
283. Vector Transformations......Page 503
285. Successive Vector Transformations......Page 504
286. Small Displacements......Page 507
287. Curve Lines......Page 509
288. Curvature and Torsion......Page 510
289. Planar Formulæ......Page 512
290. Velocity and Acceleration......Page 513
291. The Circle......Page 515
292. Motion of a Rigid Body......Page 518
293. Gauss\' Curvilinear Coordinates......Page 519
294. Curvature of Surfaces......Page 520
295. Lines of Curvature......Page 521
297. Normals......Page 524
299. Limit-Surfaces......Page 525
301. Plane Equation of the Absolute......Page 527
302. Intensities......Page 529
303. Congruent Transformations......Page 531
BOOK VII - APPLICATION OF THE CALCULUS OF EXTENSION TO GEOMETRY......Page 534
304. Introductory......Page 536
305. Points at Infinity......Page 537
306. Vectors......Page 538
307. Linear Elements......Page 539
308. Vector Areas......Page 540
309. Vector Areas as Carriers......Page 542
310. Planar Elements......Page 543
312. Vector Volumes as Carriers......Page 544
314. Point and Vector Factors......Page 545
315. Interpretation of Formulæ......Page 546
317. Operation of Taking the Vector......Page 547
318. Theory of Forces......Page 549
319. Graphic Statics......Page 551
Note......Page 553
320. Supplements......Page 554
322. Imaginary Self-Normal Sphere......Page 555
323. Real Self-Normal Sphere......Page 556
324. Geometrical Formula)......Page 557
325. Taking the Flux......Page 558
326. Flux Multiplication......Page 559
328. The Central Axis......Page 560
330. Dual Groups of Systems of Forces......Page 561
332. Secondary Axes of a Dual Group......Page 562
333. The Cylindroid......Page 563
335. Triple Groups......Page 564
336. The Pole and Polar Invariants......Page 565
338. Normals......Page 566
339. Small Displacements of a Rigid Body......Page 567
340. Work......Page 568
341. Curves......Page 570
343. Acceleration......Page 571
345. Spherical Curvature......Page 572
346. Locus of Centre of Curvature......Page 573
347. Gauss\' Curvilinear Co-ordinates......Page 574
348. Curvature......Page 575
349. Lines of Curvature......Page 576
350. Dupin\'s Theorem......Page 577
Note......Page 578
353. Introductory......Page 579
355. Formulæ......Page 580
357. New Convention......Page 581
359. Kinematics......Page 582
360. A Continuously Distributed Substance......Page 583
361. Hamilton\'s Differential Operator......Page 585
362. Conventions and Formulæ......Page 586
363. Polar Co-ordinates......Page 588
364. Cylindrical Co-ordinates......Page 589
365. Orthogonal Curvilinear Co-ordinates......Page 591
367. The Equations of Hydrodynamics......Page 593
368. Moving Origin......Page 594
370. Vector Potential of Velocity......Page 596
371. Curl Filaments of Constant Strength......Page 598
372. Carried Functions......Page 600
373. Clebsch\'s Transformations......Page 601
374. Flow of a Vector......Page 603
Note on Grassmann......Page 604
Index......Page 608