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Ty^ojU.
A
HARVARD COLLEGE LIBRARY
FROM THE
FARRAR FUND
n* i«fMi< 1^ if r*. EUm f»rar m MMawry <^ibr AtMhuwI, /oiM Farrar, EMt PreftMor of MidlimMdiei. Attronoma and NattrJ PkHotojAg, 1807-1836
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A TEEATISE
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UNIVERSAL ALGEBEA.
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lonion: c J. CLAY and sons,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE. •Iwgoto: 268, AB6TLB STREET.
E(tp>ig: F. A. BROCKHAUS.
000 IBorfc: THE MACMILLAN OOMPANT.
ISombaR: E. SEYMOUR HALE.
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A TEEATISE
ON
UNIVEKSAL ALGEBEA
WITH APPLICATIONS.
BY
ALFRED NORTH WHITEHEAD, M.A.
FELLOW AND LECTURER OF TRINITY COLLKOB, CAKBRIDOE.
VOLUME I.
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CAMBRIDGE: AT THE UNIVERSITY PRESS.
1898
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PEEFACE.
TT is the purpose of this work to present a thorough investigation of the various systems of Symbolic Reasoning allied to ordinary Algebra. The chief examples of such systems are Hamilton's Quaternions, Grassmann's Calculus of Extension^ and Boole's Symbolic Logic. Such algebras have an intrinsic value for separate detailed study ; also they are worthy of a comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular.
The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge. Accordingly after the general principles of the whole subject have been discussed in Book I. of this volume, the remaining books of the volume are devoted to the separate study of the Algebra of Symbolic Logic, and of Grassmann's Calculus of Extension, and of the ideas involved in them. The idea of a generalized conception of space has been made prominent, in the belief that the properties and operations involved in it can be made to form a uniform method of interpretation of the various algebras.
Thus it is hoped in this work to exhibit the algebras both as systems of symbolism, and also as engines for the investigation of the possibilities of thought and reasoning connected with the abstract general idea of space. A natural mode of comparison. between the algebras is thus at once provided by the unity of the subject-matters of their interpretation. The detailed comparison of their symbolic structures has been adjourned to the second volume, in which it is intended to deal with Quaternions, Matrices, and the general theory of Linear Algebras. This comparative anatomy of the subject was originated by B. Peirce's paper on Linear Associative Algebra*, and has been carried forward by more recent investigations in Germany.
* Firat read before the National Academy of Soienoes in Washington, 1871, and repabliahed in the American Journal of Mathematics, vol. iv., 1881.
i
VI PREFACE.
The general name to be given to the subject has caused me much thought : that finally adopted, Universal Algebra, has been used somewhat in this signification by Sylvester in a paper, Lectures on the Principles of Universal Algebra, published in the American Journal of Mathematics, vol. vi., 1884. This paper however, apart from the suggestiveness of its title, deals ex- plicitly only with matrices.
Universal Algebra has been looked on with some suspicion by many mathematicians, as being without intrinsic mathematical interest and as being comparatively useless as an engine of investigation. Indeed in this respect Symbolic Logic has been peculiarly unfortunate; for it has been disowned by many logicians on the plea that its interest is mathematical, and by many mathematicians on the plea that its interest is logical. Into the quarrels of logicians I shall not be rash enough to enter. Also the nature of the interest which any individual mathematician may feel in some branch of his subject is not a matter capable of abstract argumentation. But it may be shown, I think, that Universal Algebra has the same claim to be a serious subject of mathematical study as any other branch of mathematics. In order to substantiate this claim for the importance of Universal Algebra, it is necessary to dwell shortly upon the fundamental nature of Mathematics.
Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning.
The reasoning is formal in the sense that the meaning of propositions forms no part of the investigation. The sole concern of mathematics is the inference of proposition from proposition. The justification of the rules of inference in any branch of mathematics is not properly part of mathematics : * it is the business of experience or of philosophy. The business of mathematics is simply to follow the rule. In this sense all mathematical reasoning is necessary, namely, it has followed the rule.
Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart > from any existential import), need only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics. The subject-matter is not necessarily first presented to the mind by definitions : but no idea, which has not been completely defined as far as concerns its relations to other ideas involved in the subject-matter, can be admitted into the reasoning. Mathematical definitions are always to be construed as limitations as well as definitions;
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PREFACE. VU
namely, the properties of the thing defined are to be considered for the purposes of the argument as being merely those involved in the definitions.
Mathematical definitions either possess an existential import or are conventional. A mathematical definition with an existential import is the result of an act of pure abstraction. Such definitions are the starting points of applied mathematical sciences; and in so far as they are given this existential import, they require for verification more than the mere test of self-consistency.
Hence a branch of applied mathematics, in so far as it is applied, is not merely deductive, Unless in some sense the definitions are held to be guaranteed a priori as being true in addition to being self-consistent.
A conventional mathematical definition has no existential import. It sets before the mind by an act of imagination a set of things with fully defined self-consistent types of relation. In order that a mathematical science of any importance may be founded upon conventional definitions, the entities created by them must have properties which bear some afiinity to the properties of existing things. Thus the distinction between a mathematical definition with ibi existential import and a conventional definition is not always very obvious fi*om the form in which they are stated. Though it is possible to make a definition in form unmistakably either conventional or existential, there is often no gain in so doing. In such a case the definitions and resulting propositions can be construed either as refeiiing to a world of ideas created by convention, or as referring exactly or approximately to the world of existing things. The existential import of a mathematical definition attaches to it, if at all, qu& mixed mathematics ; qu& pure mathematics, mathematical defi- nitions must be conventional*.
Historically, mathematics has, till recently, been confined to the theories of Number, of Quantity (strictly so-called), and of the Space of common experience. The limitation was practically justified : for no other large systems of deductive reasoning were in existence, which satisfied our definition of mathematica The introduction of the complex quantity of ordinary algebra, an entity which is evidently based upon conventional definitions, gave rise to the wider mathematical science of to-day. The realization of wider conceptions has been retarded by the habit of mathe- maticians, eminently useful and indeed necessary for its own purposes, of extending ail names to apply to new ideas as they arise. Thus the name
* Cf. Or&ssmann, Ausdehnungslehre von 1S44, Einleitang.
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viii PREFACE.
of quantity was transferred from the quantity, strictly so called, to the generalized entity of ordinary algebra, created by conventional definition, which only includes quantity (in the strict sense) as a special case.
Ordinary algebra in its modem developments is studied as being a large "
body of propositions, inter-related by deductive reasoning, and based upon 'i
conventional definitions which are generalizations of fiindamental conceptions. |
Thus a science is gradually being created, which by reason of its fundamental character has relation to almost every event, phenomenal or intellectual, which can occur. But these reasons for the study of ordinary Algebra apply i
to the study of Universal Algebra ; provided that the newly invented '
algebras can be shown either to exemplify in their sjrmbolism, or to represent in their interpretation interesting generalizations of important systems of i
ideas, and to be useful engines of investigation. Such algebras are j j
mathematical sciences, which are not essentially concerned with number or quantity ; and this bold extension beyond the traditional domain of pure quantity forms their peculiar interest. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive re&soning, or imaginative literature, shall be mathematics developed by means of a calculus.
It is the object of the present work to exhibit the new algebras, in their detail, as being usefiil engines for the deduction of propositions ; and in their several subordination to dominant ideas, as being representative symbolisms of fundamental conceptions. In conformity with this latter object I have not hesitated to compress, or even to omit, developments and applications which are not allied to the dominant interpretation of any algebra. Thus unity of idea, rather than completeness, is the ideal of this book. I am convinced that the comparative neglect of this subject during the last forty years is partially due to the lack of unity of idea in its presentation.
The neglect of the subject is also, I think, partially due to another defect in its presentation, which (for the want of a better word) I will call the lack of independence with which it has been conceived. I will proceed to explain my meaning.
Every method of research creates its own applications : thus Analytical Geometry is a different science from Synthetic Geometry, and both these sciences are diflTerent from modem Projective Geometry. Many propositions
\
PREFACE. IX
are identical in all three sciences, and the general subject-matter, Space, is the same throughout. But it would be a serious mistake in the development of one of the three merely to take a list of the propositions as they occur in the others, and to endeavour to prove them by the methods of the one in hand. Some propositions could only be proved with great difficulty, some could hardly even be stated in the technical language, or symbolism, of the special branch. The same applies to the applications of the algebras in this book. Thus Grassmann's Algebra, the Calculus of Extension, is applied to Descriptive Geometry, Line Geometry, and Metrical Geometry, both non- Euclidean and Euclidean. But these sciences, as here developed, are not the same sciences as developed by other methods, though they apply to the same general subject-matter. Their combination here forms one new and distinct science, as distinct from the other sciences, whose general subject- matters they deal with, as is Analytical Geometry from Pure Geometry. This distinction, or independence, of the application of any new algebra appears to me to have been insufficiently realized, with the result that the developments of the new Algebras have been cramped.
In the use of symbolism I have endeavoured to be very conservative. Strange symbols are apt to be rather an encumbrance than an aid to thought: accordingly I have not ventured to disturb any well-established notation. On the other hand I have not hesitated to introduce fresh symbols when they were required in order to express new ideas.
This volume is divided into seven books. In Book I. the general prin- ciples of the whole subject are considered. Book II. is devoted to the Algebra of Symbolic Logic; the results of this book are not required in any of the succeeding books of this volume. Book III. is devoted to the general principles of addition and to the theory of a Positional manifold, which is a generalized conception of Space of any number of dimensions without the introduction of the idea of distance. The comprehension of this book is essential in reading the succeeding books. Book IV. is devoted to the principles of the Calculus of Extension. Book V. applies the Calculus of Extension to the theory of forces in a Positional manifold of three dimensions. Book VI. applies the Calculus of Extension to Non-Euclidean Geometry, considered, after Cayley, as being the most general theory of distance in a Positional manifold; the comprehension of this book is not necessary in reading the succeeding book. Book VII. applies the Calculus of Extension to ordinary Euclidean Space of three dimensions.
\
I
X PREFACE.
It would have been impossible within reasonable limits of time to have made an exhaustive study of the many subjects, logical and mathematical, on which this volume touches ; and, though the writing of this volume has been continued amidst other avocations since the year 1890, I cannot pretend to have done so. In the subject of pure Logic I am chiefly indebted to Mill, Jevons, Lotze, and Bradley; and in regard to Symbolic Logic to Boole, Schroder and Venn. Also I have not been able in the footnotes to this volume adequately to recognize my obligations to De Morgan's writings, both logical and mathematical. The subject-matter of this volume is not concerned with Quaternions; accordingly it is the more necessary to mention in this preface that Hamilton must be regarded as a founder of the science of Universal Algebra. He and De Morgan (c£ note, p. 131) were the first to express quite clearly the general possibilities of algebraic |
symbolism. j
The greatness of my obligations in this volume to Grassiliaann will be '
understood by those who have mastered his two AusdehnungslehrSs^ The technical development of the subject is inspired chiefly by his work of 1862, but the underlying ideas follow the work of 1844. At the same time I have tried to extend his Calculus of Extension both in its technique and in its ideas. But this work does not profess to be a complete interpretation of Grassmann's investigations, and there is much valuable matter in his Ausdehnungslehres ivhich it has not fallen within my province to touch upon. Other obligations, as far as I am aware of them, are mentioned as they occur. But the book is the product of a long preparatory period- of thought and miscellaneous reading, and it was only gradually that the subject in its full extent . shaped itself in my mind ; since then the various parts of this volume have been systematically deduced* according to the methods appropriate to them here, with hardly any aid from other works. This procedure was necessary, if any unity of idea was to be preserved, owing to the bewildering variety of methods and points of view adopted by writers on the various subjects of this volume. Accordingly there is a possibility of some oversights, which I should very much regret, in the attribution of ideas and methods to their sources. I should like in this connection to mention the names of Arthur Buchheim and of Homersham Cox as the mathematicians whose writings have chiefly aided me in the development of the Calculus of Extension (cf. notes, pp. 248, 346, 370, and 575). In the development of Non-Euclidean Geometry the ideas of Cayley, Klein, and Clifford have been
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PREFACE. XI
chiefly followed ; and in the development of the theory of Systems of Forces I am indebted to Sir R. S. Ball, and to Lindemann.
I have added unsystematieally notes to a few theorems or methods, stating that they are, as far as I know, now enunciated for the first time. These notes are unsystematic in the double sense that I have not made a systematic search in the large literatures of the many branches of mathematics with which this book has to do, and that I have not added notes to every theorem or method which happens to be new to me.
My warmest thanks for their aid in the final revision of this volume are due to Mr Arthur Berry, Fellow of King's College, to Mr W. E. Johnson, of Eling's College, and Lecturer to the University in Moral Science, to Prof Fors3rth, Sadlerian Professor to the University, who read the first three books in manuscript, and to the Hon. B, Russell, Fellow of Trinity College, who has read many of the proofs, especially in the parts connected with ^o Non-Euclidean Geometry.
Mr Johnson not only read the proofs of the first three books, and made many important suggestions and corrections, but also generously placed at my disposal some work of his own on Symbolic Logic, which will be found duly incorporated with acknowledgements.
Mr Berry throughout the printing of this volume has spared himself no trouble in aiding me with criticisms and suggestions. He undertook the extremely laborious task of correcting all the proofs in detail. Every page has been improved either substantially or in expression owing to his suggestions I cannot express too strongly my obligations to him both for his general and detailed criticism.
The high efficiency of the University Press in all that concerns mathe- matical printing, and the courtesy which I have received from its Officials, also deserve grateful acknowledgements.
Gambbidob,
December, 1S97.
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CONTENTS.
The following Books and Chapters are not essential far the comprehension of the subsequent paHs of this volume : Book II, Chapter V of Book IV, Book VI,
BOOK I.
PRINCIPLES OF ALGEBRAIC SYMBOLISM.
CHAPTER I. On the Nature of a Calculus.
ART. PA0E8
1. Signs 3—4
2. Definition of a Calculus 4 — 5
3. Equivalence 6 — 7
4. Operations 7 — 8
5. Substitutive Schemes 8 — 9
6. Conventional Schemes 9 — 10.
7. Uninterpretable Forms 10 — 12
CHAPTER 11. Manifolds.
8. Manifolds 13—14
9. Secondary Properties of Elements 14-15
10. Definitions 15
11. Special Manifolds 16-17
52
xiv CONTENTS.
CHAPTER III. Principles of Universal Alqebra.
ART. I*AOB8
12. Introductory 18
13. Equivalence 18—19
14. Principles of Addition 19 — 21
15. Addition 21—22
16. Principles of Subtraction 22—24
17. The Null Element 24—26
18. Steps 26
19. Multiplication 25—27
20. Orders of Algebraic Manifolds 27—28
21. The Nidi Element 28—29
22. Classification of Special Algebras 29 — 32
Note 32
BOOK 11.
THE ALGEBRA OF SYMBOLIC LOGIC.
CHAPTER I. The Algebra of Symbolic Logic.
23. Formal Laws 35 — 37
24. Reciprocity between Addition and Multiplication .... 37 — 38
25. Interpretation 38—39
26. Elementary Propositions 39 — 41
27. Classification 41-^2
28. Incident Regions 42 — 44
CHAPTER IL The Algebra of Symbolic Logic {c<mixnv/ed),
29. Development 46 — 47
30. Elimination 47—56
31. Solution of Equations with One Unknown 56 — 59
32. On Limiting and Unlimiting Equations 59 — 60
33. On the Fields of Expressions 60—65
34. Solution of Equations with More than One Unknown .... 65 — 67 36. Symmetrical Solution of Equations with Two Unknowns 67 — 73
36. Johnson's Method 73—76
37. Symmetrical Solution of Equations with Three Unknowns . 75 — 80
38. Subtraction and Division 80—82
CONTENTS. XV
CHAPTER m.
Existential Expressions.
ABT. PAGES
39. Existential Expressions 83 — 86
40. Umbral Letters 86-89
41. Elimination 89—91
42. Solutions of Existential Expressions with One Unknown . 91 — 92
43. Existential Expressions with Two Unknowns 93 — 94
44. Equations and Existential Expressions with One Unknown. . 94—96
46. Boole's General Problem 96-97
46. Equations and Existential Propositions with Many Unknowns 97-— 98
NoU 98
CHAPTER IV.
Application to Logic.
47. Propositions 99—100
48. Exclusion of Nugatory Forms 100—101
49. Syllogism 101—103
50. Symbolic Equivalents of Syllogisms 103 — 105
61. Generalization of Logic 106 — 106
CHAPTER V. Propositional Interpretation.
52. Propositional Interpretation 107 — 108
53. Equivalent Propositions 108
54. Symbolic Representation of Complexes 108
55. Identification with the Algebra of Symbolic L<^c .... 108 — 111
56. Existential Expressions Ill
57. Symbolism of the Traditional Propositions Ill — 112
58. Primitive Predication 112—113
59. Existential Symbols and Primitive Predication 113—114
60. Propositions 114—115
Historical Note 115—116
XVI CONTENTS.
BOOK III.
POSITIONAL MANIFOLDS.
CHAPITER I. Fundamental Propositions.
ABT. PAGES
61. Introductory 119
62. Intensity 119—121
63. Things repi-esenting Difierent Elements 121 — 122
64. Fundamental Propositions 122—125
66. Subregions 126 — 128
66. Loci 128—130
67. Surface Loci and Curve Loci 130—131
Note 131
CHAPTER II. Straight Lines and Planes.
68. Introductory 132
69. Anharmonic Ratio 132
70. Homographic Ranges 133
71. Linear Transformations 133 — 136
72. Elementary Properties 136—137
73. Reference-Figiures 138 -139
74. Perspective 139—142
76. Quadrangles 142—143
CHAPTER III.
QUADRICS.
76. Introductory 114
77. Elementary Properties 144 — 146
78. Poles and Polars «... 146—147
79. Generating R^ons 147 — 148
80. Conjugate Coordinates 148 — 151
81. Quadriquadric Curve Loci 151 — 153
82. Closed Quadrics 163-165
83. Conical Quadric Surfaces 156—157
84. Reciprocal Equations and Conical quadrics 167 — 161
Note 161
CONTENTS. XVU
CHAPTER IV. Intensity.
ABT. PA0B8
85. Defining Equation of Intensity 162—163
86. Locus of Zero Intensity 163—164
87. Plane Locus of Zero Intensity 164—166
88. Quadric Locus of Zero Intensity 166
89. Antipodal Elements and Opposite Intensities 166 — 167
90. The Intercept between Two Elements 167—168
Note 168
BOOK IV.
CALCULUS OF EXTENSION.
CHAPTER I. Combinatorial Multiplicahon.
91. Introductory 171—172
92. Invariant Equations of Condition 172 — 173
93. Principles of Combinatorial Multiplication 173 — 17ft
94. Derived Manifolds 175—176
95. Extensive Magnitudes 176—177
96. Simple and Compoimd Extensive Magnitudes 177 — 178
97. Fundamental Propositions 178 — 180
Note 180
CHAPTER II. Regressive Multiplication.
98. Progressive and Regressive Multiplication 181
99. Supplements 181—183
100. Definition of Regressive Multiplication . 183—184
lOL Pure and Mixed Products 184-185
102. Rule of the Middle Factor 185—188
103. Extended Rule of the Middle Factor 188—190
104. Regressive Multiplication independent of Reference-Elements . . 190—191
105. Proposition 191
106. Mtiller's Theorems 192—195
107. Applications and Examples 195—198
Note 198
xviii CONTENTS.
CHAPTER III. Supplements.
ABT. PAGES
108. Supplementary R^ons 199
109. Normal Systems of Points 199—200
110. Extension of the Definition of Supplements 201—202
111. Different kinds of Supplements 202
112. Normal Points and Straight Lines 202—203
113. Mutually normal Regions 203—204
114. Self-normal Elements 204—206
115. Self-normal Planes 206
116. Complete Region of Three Dimensions 206 — 207
117. Inner Multiplication 207
118. Elementary Transformations 208
119. Rule of the Middle Factor 208
120. Important Formula 208—209
121. Inner Multiplication of Normal Regions 209
122. General Formula for Inner Multiplication 209—210
123. Quadrics 210—212
124. Plane-Equation of a Quadric 212—213
CHAPTER IV. Descriptive Geometry.
125. Application to Descriptive Geometry 214
126. Explanation of Procedure 214—215
127. Illustration of Method 215
128. von Staudt's Construction 215 — 219
129. Grassmann's Constructions 219 — 223
130. Projection 224—228
CHAPTER V. Descriptive Geometry of Conics and CuBica
131. General Equation of a Conic 229—231
132. Further Transformations 231—233
133. Linear Construction of Cubics 233
134. First Type of Linear Construction of the Cubic 233—236
135. Linear Construction of Cubic through Nine arbitrary Points . . 236—237
136. Second Type of Linear Construction of the Cubic .... 238—239
137. Third Type of Linear Construction of the Cubic 239—244
138. Fourth Type of Linear Construction of the Cubic .... 244—246
139. ChasW Construction 246—247
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XIX
CHAPTER VI. Matrices.
ABT. PAOB8
140. Introductory 248
141. Definition of a Matrix 248—249
142. Sums and Products of Matrices 260—252
143. Associated Determinant 252
144. Null Spaces of Matrices 252—254
[45. Latent Points 254—255
[46L Semi-Latent Regions 256
147. The Identical Equation 256—257
48. The Latent Region of a Repeated Latent Root 257—258
L49. The First Species of Semi-Latent Regions 258—259
50. The ^igher Species of Semi-Latent Regions 259—261
51. The Identical Equation 261
52. The Vacuity of a Matrix 261—262
153. Symmetrical Matrices 262—265
[54. Symmetrical Matrices and Supplements 265 — 267
[55. Skew Matrices 267—269
BOOK V.
EXTENSIVE MANIFOLDS OF THREE DIMENSIONS.
CHAPTER I.
Systems of Forces.
156. Non-metrical Theory of Forces 273—274
157. Recapitulation of Formula 274—275
158. Inner Multiplication 275—276
159. Elementary Properties of a Single Force 276
160. Elementary Properties of Systems of Forces 276 — 277
161. Condition for a Single Force 277
162. Ck>njugate lines 277—278
163. Null Lines, Planes and Points 278
164. Properties of Null Lines 279—280
165. Lines in Involution 280—281
166. Reciprocal Systems 281—282
167. Formuka for Systems of Forces 282—283
XX CONTENTS.
CHAPTER 11.
Groups of Systems of Forces.
AKT. PAQKS
168. Specifications of a Group 284 — 285
169. Systems Reciprocal to Qroups 285
170. Common Null Lines and Director Forces 286
171. Quintuple Groups 286—287
172. Quadruple and Dual Groups . 287—290
173. Anharmonic Ratio of Systems 290 — 292
174. Self-Supplementary Dual Groups 292—294
176. Triple Groups 295—298
176. Conjugate Sets of Systems in a Triple Group 298 — 299
CHAPTER III. Invariants of Groups.
177. Definition of an Invariant 300
178. The Null Invariants of a Dual Group 300
179. The Harmonic Invariants of a Dual Group 301—302
180. Further Properties of Harmonic Invariants 302 — 303
181. Formulad connected with Reciprocal Systems 303—304
182. Systems Reciprocal to a Dual Group 304
183. The Pole and Polar Invariants of a Triple Group .... 305—306
184. Conjugate Sets of Systems and the Pole and Polar Invariants 306—307
185. Interpretation of P (^) and P (Z) 307—308
186. Relations between Conjugate Sets of Systems 308—310
187. The Conjugate Invariant of a Triple Group 310—312
188. Transformations of Q (p, />) and (? (P, P) 312—315
CHAPTER IV. Matrices and Forces.
189. Linear Transformations in Three Dimensions 316 — 317
190. Enumeration of Types of Latent and Semi-Latent Regions . 317 — 321
191. Matrices and Forces 322—323
192. Latent Systems and Semi-Latent Groups 323—326
193. Enumeration of Types of Latent Systems and Semi-Latent Grou^xs . 326 — 338 194 Transformation of a Quadric into itself 338—339
195. Direct Transformation of Quadrics 339—342
196. Skew Transformation of Quadrics 342—346
Note 346
CONTENTS. Xxi
BOOK VL
THEORY OF METRICS.
CHAPTER I. Thbx)ry of Distance.
A^RT. PAGES
197. Axioms of Distance 349—350
198. Congruent Ranges of Points 350—351
199. Cayle/s Theory of Distance 361 353
200. Klein's Theorem 353 — 354
201. Comparison with the Axioms of Distance 354
202. Spatial Manifolds of Many Dimensions 354 — 355
203. Division of Space . 355 356
204. Elliptic Space 356
205. Polar Form 356 — 368
206. Length of Intercepts in Polar Form 368 — 361
207. Antipodal Form 361—362
208. Hyperbolic Space 362—363
209. The Space Constant 363—364
210. Law of Intensity in Elliptic and Hyperbolic Geometry 364—366
211. Distances of Planes and of Subregions 365—367
212. Parabolic Geometry 367 — 368
213. Law of Intensity in Parabolic Geometry 368 — 369
Historical Note 369—370
CHAPTER IL Elliptic Geometry.
214. Introductory 371
215. Triangles 371—373
216. Further Formulue for Triangles 374—375
217. Points inside a Triangle 375 — 376
218. Oval Quadrics 376—378
219. Pmrther Properties of Triangles 378 — 379
220. Planes One-sided 379—382
221. Angles between Planes 382
222. Stereometrical Triangles 382 — 383
223. Perpendiculars 383—386
224. Shortest Distances from Points to Planes 385—386
226. Common Perpendicular of Planes 386
226. Distances from Points to Subregions 387 — 388
227. Shortest Distances between Subregions 388 — 391
228. Spheres 391—396
229. Pajrallel Subregions 397—398
XXU CONTENTS.
CHAPTER III. Extensive Manifolds and Elliptic Geometry.
ABT. PAOBB
230. Intensities of Forces 399_400
231. Relations between Two Forces 400—401
232. Axes of a System of Forces 401 — 404
233. Non-Axal Systems of Forces 404
234. Parallel Lines 404—406
235. Vector Systems 406 — 407
236. Vector Systems and Parallel Lines 407—408
237. Further Properties of Parallel Lines 409-^11
238. Planes and Parallel Lines 411-^13
CHAPTER IV. Hyperbolic Geometry.
239. Space and Anti-Space 414
240. Intensities of Points and Planes 415—416
241. Distances of Points 416—417
242. Distances of Planes 417—418
243. Spatial and Anti-spatial Lines 418—419
244. Distances of Subregions 419
245. Geometrical Signification 420
246. Poles and Polars 420—422
247. Points on the Absolute 422
248. Triangles 422—424
^ 249. Properties of Angles of a Spatial Triangle 424 — 425
I 250. Stereometrical Triangles 425—426
[ 251. Perpendiculars 426 — 427
I 252. The Feet of Perpendiculars 427—428
253. Distance between Planes 428 — 429
I 254. Shortest Distances 429—430
I 255. Shortest Distances between Subregions 430 — 433
256. Rectangular Rectilinear Figures 433 — 436
^ 257. Parallel Lines 436—438
258. Parallel Planes 439—440
CHAPTER V. Hyperbolic Qeometry (continiied).
269. The Sphere 441—444
260. Intersection of Spheres 444 — 447
261. Limit-Surfaces 447—448
262. Qreat Circles on Spheres 448—451
\
CONTENTS. XXlll
ART. PAQES
263. Surfaces of Equal Distance from Subregions 451
264. Intensities of Forces 452
265. Relations between Two Spatial Forces 452 — 454
266. Central Axis of a System of Forces 454--455
267. Non-Axal Systems of Forces 455
CHAPTER VI. Kinematics in Three Dimensions.
268. Congruent Transformations 456 — 458
269. Elementary Formul® 453^459
270. Simple Geometrical Properties 459 — 460
271. Translations and Rotations 460 — 462
272. Locus of Points of Equal Displacement 462 — 463
273. Equivalent Sets of Congruent Transformations 463
274. Commutative Law 464
275. Small Displacements 464 — 465
276. Small Translations and Rotations 465 — 466
277. Associated System of Forces 466
278. Properties deduced from the Associated System 467 — 468
279. Work 468—469
280. Characteristic Lines 470
281. Elliptic Space 470—471
282. Surfaces of Equal Displacement 472
283. Vector Transformations 472
284. Associated Vector Systems of Forces 473
285. Successive Vector Transformations 473 — 476
286. Small Displacements 476—477
CHAPTER VII. CuRVKs AND Surfaces.
287. Curve Lines 478—479
288. Ciurvature and Torsion 479—481
289. Planar Formulas 481—482
290. Velocity and Acceleration 482 — 484
291. The Circle 484—487
292. Motion of a Rigid Body 487-488
293. Gauss' Curvilinear Coordinates 488-489
294. Curvature of Surfaces 489—490
295. Lines of Curvature 490—493
296. Meunier^s Theorem 493
297. Normals 493—494
298. Curvilinear Coordinates 494
299. Limit-Surfaces 494—495
XXIV CONTENTS.
CHAPTER VIIL
Transition to Parabolic Geometry.
ABT. PAGES
300. Parabolic Geometry 4d6
301. Plane Equation of the Absolute 496 — 498
302. Intensities 498—499
303. Congruent Transformations 500 — 502
%
t
BOOK VII.
APPLICATION OF THE CALCULUS OF EXTENSION TO
GEOMETRY.
CHAPTER I.
Vectors.
304. Introductory 505—606
305. Points at Infinity 506—507
306. Vectors 507—508
307. Linear Elements 508—509
308. Vector Areas 509—511
309. Vector Areas as Carriers 511
310. Planar Elements 512—513
311. Vector Volumes 513
312. Vector Volumes as Carriers 513 — 514
313. Product of Four Points 514
314. Point and Vector Factors 514 — 515
315. Interpretation of Formul® 515 — 516
316. Vector Formul® 516
317. Operation of Taking the Vector 516—518
818. Theory of Foitses 518—520
319. Graphic Statics 520—522
Note 522
CONTENTS.
XXV
CHAPTER II.
Vectors (continued).
f
AET.
320. Supplements ....
321. Rectangular Normal Systems .
322. Imaginaiy Self-Normal Sphere
323. Real Self-Normal Sphere
324. Qeometrical Formulas
325. Taking the Flux
326. Flux Multiplication .
327. Geometrical Formulae
328. The Central Axis .
329. Planes containing the Central Axis
330. Dual Groups of Systems of Forces
331. Invariants of a Dual Group .
332. Secondary Axes of a Dual Group .
333. The Cylindroid ....
334. The Harmonic Invariants
335. Triple Groups
336. The Pole and Polar Invariants
337. Equation of the Associated Quadric
338. Normals
339. Small Displacements of a Rigid Body
340. Work
PAOES
523—524
524 524—525 526—526 526—527 627—528
628
529 529—530
530 530—531
531 531—532 532—533
533 533—534 534—535
535 535 536 636—537 637-538
CHAPTER IIL
Curves and Surfaces.
341. Curves
342. Osculating Plane and Normals
343. Acceleration ....
344. Simplified FormulsB .
345. Spherical Curvature
346. Locus of Centre of Ciurvature
347. Gauss' Curvilinear Co-ordinates
348. Curvature
349. Lines of Curvature
350. Dupin's Theorem 361. Ruler's Theorem 352 Meunier's Theorem
Note
539
540
540
541
641—542
542—643
643—644
644—645
545—646
646—547
647
547
547
r
XXVI CONTENTS.
CHAPTER IV.
\ Pure Vector Formula.
I
ABT. PAQB8
I
363. Introductory 548—549
354. Lengths and Areas 549
355. FormulfiB 549—550
356. The Origin 550
357. New Convention 550 — 551
358. System of Forces 551
359. Kinematics 551—652
360. A Continuously Distributed Substance 552 — 554
361. Hamilton's Differential Operator 564—555
362. Conventions and Formulae 555 — 557
363. Polar Co-ordinates 557—568
364. Cylindrical Co-ordinates 558—560
365. Orthogonal Curvilinear Co-ordinates 560 — 562
366. Volume, Surface, and Line Integrals 562
367. The Equations of Hydrodynamics 562—563
368. Moving Origin 563—565
369. Transformations of Hydrodynamical Equations 565
370. Vector Potential of Velocity 566—566
371. Curl Filaments of Constant Strength 567—569
372. Carried Functions 569—570
373. Clebsch's Transformations 670—572
374. Flow of a Vector 572—573
Note 573
Note on Orastmann 573 — 575
Index 576—586
i
BOOK I.
PRINCIPLES OF ALGEBRAIC SYMBOLISM.
w. 1
i
V
^
r
CHAPTER I. On the nature of a Calculus.
1. Signs. Words, spoken or written, and the symbols of Mathematics ai^ alike signs. Signs have been analysed* into (a) suggestive signs, I (j8) expressive signs, (7) substitutive signs.
A suggestive sign is the most rudimentary possible, and need not be dwelt upon here. An obvious example of one is a knot tied in a band- kerchief to remind the owner of some duty to be performed.
In the use of expressive signs the attention is not fixed on the sign itself but on what it expresses; that is to say, it is fixed on the meaning conveyed by the sign. Ordinary language consists of groups of expressive signs, its primary object being to draw attention to the meaning of the words employed. Language, no doubt, in its secondary uses has some of the characteristics of a system of substitutive signs. It remedies the inability of the imagination to bring readily before the mind the whole extent of complex ideas by associating these ideas with familiar sounds or marks ; and it is not always necessary for the attention to dwell on the complete meaning while using these Sjnoibols. But with all this allowance it remains true that language when challenged by criticism refers us to the meaning and not to the natural or conventional properties of its symbols for an explanation of its processes.
A substitutive, sign is such that in thought it takes the place of that for which it is substituted. A counter in a game may be such a sign : at the end of the game the counters lost or won may be interpreted in the form of money, but till then it may be convenient for attention to be concentrated on the counters and not on their signification. The signs of a Mathematical Calculus are substitutive signs.
The difference between words and substitutive signs has been stated thus, 'a word is an instrument for thinking about the meaning which it
* Cf. Stout, 'Thought and Language,* 3/tit<7, April, 1891, repeated in the same author's Analytic Ptyehology^ (1896), oh. x. § 1: cf. also a more obscure analysis to the same e£Feot by C. S. Peirce, Proe, of the American Academy of, Arts and Scittncet^ 1867, Vol. vii. p. 294.
1—2
I
4 ON THE NATURE OF A CALCULUS. [CHAP. L
expresses ; a substitute sign is a means of not thinking about the meaning which it symbolizes*.' The use of substitutive signs in reasoning is to economize thought.
2. Definition of a Calculus. In order that reasoning may be con- ducted by means of substitutive signs, it is necessary that rules be given for the manipulation of the signs. The rules should be such that the final state of the signs after a series of operations according to rule denotes, when the signs are interpreted in terms of the things for which they are substituted, a proposition true for the things represented by the signs.
The art of the manipulation of substitutive signs according to 6xed rules, and of the deduction therefix^m of true propositions is a Calculua
We may therefore define a sign used in a Calculus as 'an arbitrary mark, having a fixed interpretation, and susceptible of combination with other signs in subjection to fixed laws dependent upon their mutual interpretation f.*
The interpretation of any sign used in a series of operations must be fixed in the sense of being the same throughout, but in a certain sense it may be ambiguous. For instance in ordinary Algebra a letter x may be used in a series of operations, and x may be defined to be any algebraical quantity, without further specification of the special quantity chosen. Such a sign denotes any one of an assigned class with certain un- ambiguously defined characteristics. In the same series of operations the sign must always denote the same member of the class ; but as far as any explicit definitions are concerned any member will do.
When once the rules for the manipulation of the signs of a calculus are known, the art of their practical manipulation can be studied apart from any attention to the meaning to be assigned to the signs. It is obvious that we can take any marks we like and manipulate them according to any rules we choose to assign. It is also equally obvious that in general such occupations must be Mvolous. They possess a serious scientific value when there is a similarity of type of the signs and of the rules of manipulation to those of some calculus in which the marks used axe substitutive signs for things and relations of thinga The comparative study of the various forms produced by variation of rules throws light on the principles of the calculus. Furthermore the knowledge thus gained gives fisMjility in the invention of some significant calculus designed to facilitate^ reasoning with respect to some given subject.
It enters therefore into the definition of a calculus properly so called that the marks used in it are substitutive signs. But when a set of marks and the rules for their arrangements and rearrangements are analogous to
* Of. stout, * Thought and Language,' MUid, April, 1S91. f Boole, Laws of Thought, Ch. ii.
2, 3] DEFINITION OF A CALCULUS. 5
those of a significant calculus so that the study of the allowable forms of their arrangements throws light on that of the calculus, — or when the marks and their rules of arrangement are such as appear likely to receive an interpretation as substitutive signs or to facilitate the invention of a true calculus, then the art of arranging such marks may be called — by an extension of the term — ^an uninterpreted calculus. The study of such a calculus is of scientific value. The marks used in it will be called signs or symbols as are those of a true calculus, thus tacitly suggesting that there is some unknown interpretation which could be given to the calculus.
3. Equivalence. It is necessary to note the form in which propositions occur in a calculu& Such a form may well be highly artificial from some points of view, and may yet state the propositions in a convenient form for the eliciting of deductions. Furthermore it is not necessary to assert that the form is a general form into which all judgments can be put by the aid of some torture. It is sufficient to observe that it is a form of wide appli- cation.
In a calculus of the type here considered propositions take the form of assertions of equivalence. One thing or &ct, which may be complex and involve an inter-related group of things or a succession of facts, is asserted to be equivalent in some sense or other to another thing or fact.
Accordingly the sign = is taken to denote that the signs or groups of signs on either side of it are equivalent, and therefore symbolize things which are so far equivalent. When two groups of symbols are connected by this sign, it is to be understood that one group may be substituted for the other group whenever either occurs in the calculus under conditions for which the assertion of equivalence holds good.
The idea of equivalence requires some explanation. Two things are equivalent when for some purpose they can be used indifferently. Thus the equivalence of distinct things implies a certain defined purpose in view, a certain limitation of thought or of action. Then within this limited field no distinction of property exists between the two things.
As an instance of the limitation of the field of equivalence consider
an ordinary algebraical equation, /(a?, y) = 0. Then in finding ^ by the
formula, ;r^ = — ^ / ^ » ^^ ^^y ^^^ substitute 0 for / on the right-hand
side of the last equation, though the equivalence of the two symbols has been asserted in the first equation, the reason being that the limitations under which /= 0 has been asserted are violated when / undergoes partial dif- ferentiation.
The idea of equivalence must be carefully distinguished from that of
\
6
ON THB NATURE OF A CALCULUS.
[chap. I.
i
mere identity*. No investigations which proceed by the aid of propositions merely asserting identities such as il is ^, can ever result in anything but barren identities^. Equivalence on the other hand implies non-identity as its general case. Identity may be conceived as a special limiting case of equivalence. For instance in arithmetic we write, . 2 -I- 8 » 3 -f 2. This means that, in so far as the total number of objects mentioned, 2 -f 3 and 3 + 2 come to the same number, namely 5. But 2 -f 3 and 3 + 2 are not identical ; the order of the symbols is different in the two combinations, and this difference of order directs different processes of thought. The importance of the equation arises from its assertion that these different processes of thought are identical as far as the total number of things thought of is concerned.
From this arithmetical point of view it is tempting to define equivalent things as being merely different ways of thinking of the same thing as it exists in the external world. Thus there is a certain aggregate, say of 5 things, which is thought of in different ways, as 2 + 3 and as 3 + 2. A sufficient objection to this definition is that the man who shall succeed in stating intelligibly the distinction between himself and the rest of the world will have solved the central problem of philosophy. As there is no universally accepted solution of this problem, it is obviously undesirable to assume this distinction as the basis of mathematical reasoning.
Thus from another point of view all things which for any purpose can be conceived as equivalent form the extension (in the logical sense) of some uni- versal conception. And conversely the collection of objects which together form the extension of some universal conception can for some purpose be treated as equivalent. So 6 = 6^ can be interpreted as symbolizing the fact that the two individual things b and b' are two individual cases of the same general conception B\. For instance if b stand for 2 + 3 and b' for 3 + 2, both b and b' are individual instances of the general conception of a group of five things.
The sign = as used in a calculus must be discriminated from the logical copula ' is.' Two things b and b' are connected in a calculus by the sign =, so that b = b\ when both b and V possess the attribute B. But we may not translate this into the standard logical form, b is b\ On the contrary, we say, b ia By and b' is B; and we may not translate these standard forms of formal logic into the symbolic form, 6 = B, 6' = B ; at least we may not do so, if the sign = is to have the meaning which is assigned to it in a calculua
It is to be observed that the proposition asserted by the equation, b=b\ consists of two elements ; which for the sake of distinctness we will name, and will call respectively the * truism 'and the ' paradox.' The truism is the partial identity of both b and b\ their common J3-nes& The paradox is the
* Cf. Lotze, LogiCf Bk. i. Gh. n. Art. 64.
t Gf. Bradley, PrifieipU$ of Logic, Bk. i. Gh. ▼.
X Ibid, Bk. n. Pt. i. Gh. iv. Art. 3 (p).
4] EQUIVALENCE. 7
distinction between b and b\ so that b is one thing and 6' is another thing : | and these things, as being different, must have in some relation diverse properties. In assertions of equivalence as contained in a calculus the- truism is passed over with the slightest possible attention, the main stress being laid / on the paradox. Thus in the equation 2 + 3 = 3 + 2, the fact that both sides ' represent a common five-ness of number is not even mentioned explicitly. The sole direct statement is that the two different things 3 + 2 and 2 + 3 ^ are in point of number equivalent.
The reason for this unequal distribution of attention is easy to under- stand. In order to discover new propositions asserting equivalence it is ^ requisite to discover easy marks or tests of equivalent things. These tests are discovered by a careful discussion of the truism, of the common ^-ness of b and b'. But when once such tests have been elaborated, we may drop all thought of the essential nature of the attribute B, and simply apply the superficial test to b and 6' in order to verify 6 = 6'. Thus in order to verify that thirty-seven times fifty-six is equal to fifty-six times thirty-seven, we may use the entirely superficial test applicable to this case that the same &ctors ai*e mentioned as multiplied, though in different order.
This discussion leads us at once to comprehend the essence of a calculus of substitutive signs. The signs are by convention to be considered equiva- r lent when certain conditions hold. And these conditions when inter- preted imply the fulfilment of the tests of equivalence.
Thus in the discussion of the laws of a calculus stress is laid on the truism, in the development of the consequences on the paradox.
4 Operations. Judgments of equivalence can be founded on direct perception, as when it is judged by direct perception that two different pieces of stuff match in colour. But the judgment may be founded on a knowledge of the respective derivations of the things judged to be equivalent fix)m other things respectively either identical or equivalent. It is this process of derivation which is the special province of a calculus. The derivation of a thing p from things a, 6, c, ... , can also be conceived as an operation on the things a, 6, c, ... , which produces the thing p. The idea of derivation j^ includes that of a series of phenomenal occurrences. Thus two pieces of stuff J may be judged to match in colour because they were dyed in the same dipping, or were cut from the same piece of stuff. But the idea is more general than that of phenomenal sequence of events: it includes purely logical activities of the mind, as when it is judged that an aggregate of five things has been presented to the mind by two aggregates of three things and of two things respectively. Another example of derivation is that of two propositions a and 6 which are both derived by strict deductive reasoning from the sfime propositions c, d, and e. The two propositions are either both
8 ON THE NATURE OF A CALCULUS. [CHAP. I.
proved or both unproved according as c, d, and e are granted or disputed. Thus a and h are so fai' equivalent. In other words a and 6 may be considered as the equivalent results of two operations on c, d and e.
The words operation, combination, derivation, and synthesis will be used to express the same general idea, of which each word suggests a somewhat specialized form. This general idea may be defined thus : A thing a will be said to result from an operation on other things, c, d, e, etc., when a is presented to the mind as the result of the presentations of c, d and e, etc. under certain conditions; and these conditions are phenomenal events or mcDtal activities which it is convenient to separate in idea into a group by themselves and to consider as defining the nature of the operation which is performed on c, d, e, etc.
Furthermore the fact that c, d, e, etc. are capable of undergoing a certain operation involving them all will be considered as constituting a relation between c, rf, «, etc.
Also the fact that c is capable of undergoing an operation of a certain general kind will be considered as a property of c. Any additional speciali- zation of the kind of operation or of the nature of the result will be considered as a mode of that property.
6. Substitutive Schemes. Let a, a', etc., 6, h\ etc., z, /, etc.,
denote any set of objects considered in relation to some common property <
which is symbolized by the use of the italic alphabet of letters. The
common property may not be possessed in the same mode by different
members of the set. Their equivalence, or identity in relation to this property,
is symbolized by a literal identity. Thus the fiek^t that the things a and m
are both symbolized by letters from the italic alphabet is here a sign that
the things have some property in common, and the fact that the letters
a and m' are different letters is a sign that the two things possess this
common property in different modes. On the other hand the two things
a and a' possess the common property in the same mode, and as far as
this property is concerned they are equivalent. Let the sign = express
equivalence in relation to this property, then a = a\ and m = m\
Let a set of things such as that described above, considered iu relation to their possession of a common property in equivalent or in non-equivalent modes be called a scheme of things ; and let the common property of which the possession by any object marks that object as belonging to the scheme ^
be called the Determining Property of the Scheme. Thus objects belonging ^
to the same scheme are equivalent if they possess the determining property in the same mode.
Now relations must exist between non-equivalent things of the scheme which depend on the differences between the modes in which they possess the determining property of the scheme. In consequence of these relations
<0aummmimmmit^
5, 6] SUBSTITUTIVE SCHEMES. 9
from things a, 6, c, etc. of the scheme another thing m of the scheme can be derived by certain operations. The equivalence, m = m', will exist between m and w!^ if m and w! are derived from other things of the scheme by operations which only differ in certain assigned modes. The modes in which processes of derivation of equivalent things m and w! from other things of the scheme can differ without destro}dng the equivalence of m and m' will be called the Characteristics of the scheme.
Now it may happen that two schemes of things — with of course different determining properties — have the same characteristica Also it may be possible to establish an unambiguous correspondence between the things of the two schemes, so that if a, a\ 6, etc., belong to one scheme and a, a, fi, etc., belong to the other, then a corresponds to a, a' to a\ b to fi and so on. The essential rule of the correspondence is that if in one scheme two things, say a and a\ are equivalent, then in the other scheme their corresponding things a and a! are equivalent. Accordingly to any process of derivation in the italic alphabet by which m is derived from a, 6, eta there must correspond a process of derivation in the Qreek alphabet by which /A is derived from a, fi, etc.
In such a case instead of reasoning with respect to the properties of one scheme in order to deduce equivalences, we may substitute the other scheme, or conversely; and then transpose at the end of the argument. This device of reasoning, which is almost universal in mathematics, we will call the method of substitutive schemes, or more briefly, the method of substitution.
These substituted things belonging to another scheme are nothing else than substitutive signs. For in the use of substituted schemes we cease to think of the original scheme. The rule of reasoning is to confine thought to those properties, previously determined, which are shared in common with the original scheme, and to interpret the results from one set of things into the other at the end of the argument.
An instance of this process of reasoning by substitution is to be found in the theory of quantity. Quantities are measured by their ratio to an arbitrarily assumed quantity of the same kind, called the unit. Any set of quantities of one kind can be represented by a corresponding set of quantities of any other kind merely in so far as their numerical ratios to their unit are concerned. For the representative set have only to bear the same ratios to their unit as do the original set to their unit.
6. Conventional Schemes. The use of a calculus of substitutive signs in reasoning can now be explained.
Besides using substitutive schemes with naturally suitable properties, we may by convention assign to arbitrary marks laws of equivalence which are identical with the laws of equivalence of the originals about which we
I
-*
10 ON THE NATURE OP A CALCULUS. [CHAP. I.
desire to reason. The set of marks may then be considered as a scheme of things with properties assigned by convention. The determining property of the scheme is that the marks are of certain assigned sorts arranged in certain types of sequence. The characteristics of the scheme are the conventional laws by which certain arrangements of the marks in sequence on paper are to be taken as equivalent. As long as the marks are treated as mutually determined by their conventional properties, reasoning concerning the marks will hold good concerning the originals for which the marks are substitutive signs. For instance in the employ- ment of the marks a?, y, -f , the equation, a? + y = y + ar, asserts that a certain union on paper of x and y possesses the conventional quality that the order of x and y is indifferent. Therefore any union of two things with a result independent of any precedence of one thing before the other possesses so far properties identical with those of the union above set down between x and y. Not only can the reasoning be transferred from the originals to the substitutive signs, but the imaginative thought itself can in a large measure be avoided. For whereas combinations of the original things are possible only in thought and by an act of the imagi- nation, the combinations of the conventional substitutive signs of a calculus are physically made on paper. The mind has simply to attend to the rules for transformation and to use its experience and imagination to suggest likely methods of procedure. The rest is merely phjrsical actual inter- change of the signs instead of thought about the originals.
A calculus avoids the necessity of inference and replaces it by an ex- ternal demonstration, where inference and external demonstration are to be taken in the senses assigned to them by F. H. Bradley*. In this connexion a demonstration is to be defined as a process of combining a complex of facts, the data, into a whole so that some new fact is evident. Inference is an ideal combination or construction within the mind of the ' reasoner which results in the intuitive evidence of a new fact or relation between the data. But in the use of a calculus this process of combina- tion is externally performed by the combination of the concrete symbols, with the result of a new fact respecting the symbols which arises for sensuous perception f. When this new fact is treated as k symbol carrying a 'X meaning, it is found to mean the fact which would have been intuitively evident in the process of inference.
7. Uninterpretable Forms. The logical diflScultyJ involved in the use of a calculus only partially interpretable can now be explained. The
♦ Cf. Bradley, Principles of Logic, Bk ii. Pt i. Oh. iii. I t Cf. C. S. Peiroe, Amer, Joum. of Math, VoL vn. p. 1S2 : ' Ab for algebra, the very idea of ; the art is that it presents formolaB which can be manipalated, and that by observing the effects I of BQoh manipulation we find properties not otherwise to be discovered/
X Cf. Boole, Lam of Thought, Ch. v. § 4.
4 <
7] UNINTERPRETABLE FORMS. 11
discussion of this great problem in its application to the special case of
(— 1)* engaged the attention of the leading mathematicians of the first half of this century, and led to the development on the one hand of the Theory of Functions of a Complex Variable, and on the other hand of the science here called Universal Algebra.
The difficulty is this : the symbol (—1)' is absolutely without meaning when it is endeavoured to interpret it as a number; but algebraic trans- formations which involve the use of complex quantities of the form a + 6t, where a and 6 are numbers and i stands for the above symbol, yield pro- positions which do relate purely to number. As a matter of faet the pro- positions thus discovered were found to be true propositions. The method therefore was trusted, before any explanation was forthcoming why algebraic reasoning which had no intelligible interpretation in arithmetic should give true arithmetical results.
The difficulty was solved by observing that Algebra does not depend on \ Arithmetic for the validity of its laws of transformation. If there were » such a dependence, it is obvious that as soon as algebraic expressions are arithmetically unintelligible all laws respecting them must lose their validity. But the laws of Algebra, though suggested by Arithmetic, do not depend on it. They depend entirely on the convention by which it is stated that certain modes of grouping the symbols are to be considered as identical. This assigns certain properties to the marks which form the symbols of Algebra. The laws regulating the manipulation of the algebraic symbols are identical with those of Arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by Arithmetic ; for the reasoning in both cases merely applies the same general laws to diffei-ent classes of things. If an algebraic theorem is interpretable in Arithmetic, the corresponding arithmetical theorem is therefore true. In short when once Algebra is conceived as an independent science dealing with the re- lations of certain marks conditioned by the observance of certain conventional laws, the difficulty vanishes. If the laws be identical, the theorems of the one science can only give results conditioned by the laws which also hold good for the other science ; and therefore these results, when interpretable, are true.
It will be observed that the explanation of the legitimacy of the use of a partially interpretable calculus does not depend upon the fact that in another field of thought the calculus is entirely interpretable. The discovery of an interpretation undoubtedly gave the clue by means of which the true solution was arrived at. For the fact that the processes of the calculus were in- terpretable in a science so independent of Arithmetic as is Geometry at once showed that the laws of the calculus might have been defined in reference to geometrical processes. But it was a paradox to assert that a science like Algebra, which had been studied for centuries without reference to Geometry,
12 ON THE NATURE OF A CALCULUS. [CHAP. I. 7
was after all dependent upon Geometry for its first principles. The step to the true explanation was then easily taken.
But the importance of the assistance given to the study of Algebra by the discovery of a complete interpretation of its processes cannot be over-esti- mated. It is natural to think of the substitutive set of things as assisting the study of the properties of the originals. Especially is this the case with a calculus of which the interest almost entirely depends upon its relation to the originals. But it must be remembered that conversely the originals give immense aid to the study of the substitutive things or symbols.
The wbole of Mathematics consists in the organization of a series of aids
\ to the imagination in the process of reasoning ; and for this purpose device is
1 piled upon device. No sooner has a substitutive scheme been devised to assist
tin the investigation of any originals, than the imagination begins to use the
^originals to assist in the investigation of the substitutive scheme. In some
connexions it would be better to abandon the conception of originals studied
by the aid of substitutive schemes, and to conceive of two sets of inter-related
things studied together, each scheme exemplifying the operation of the same
general laws. The discovery therefore of the geometrical representation of
the algebraical complex quantity, though unessential to the logic of Algebra,
has been quite essential to the modem developments of the science.
V
CHAPTER II.
Manifolds.
8. Manifolds. The idea of a manifold was first explicitly stated by Riemann*; Qrassmannf had still earlier defined and investigated a particular kind of manifold.
Consider any number of things possessing any common property. That property may be possessed by different things in different modes : let each separate mode in which the propeiiiy is possessed be called an element. The aggregate of all such elements is called the manifold of the property.
Any object which is specified as possessing a property in a given mode corresponds t>o an element in the manifold of that property. The element may be spoken of as representing the object or the object as representing the element according to convenience. All such objects may be conceived as equivalent in that they represent the same element of the manifold.
Various relations can be stated between one mode of a property and another mode ; in other words, relations exist between two objects, whatever . other properties they may possess, which possess this property in any two 1 assigned modes. The relations will define how the objects necessarily differ / IB that they possess this property differently : they define the distinction between two sorts of the same property. These relations will be called relations between the various elements of the manifold of the property ; and the axioms from which can be logically deduced the whole aggregate of such relations for all the elements of a given manifold are called the characteristics of the manifold.
The idea of empty space referred to coordinate axes is an example of a manifold. Each point of space represents a special mode of the common property of spatiality. The fundamental properties of space expressed in terms of these coordinates, i.e. all geometiical axioms, form the character- istics of this manifold.
* Ueher die Hypotheten, welche der Qeometrie zu Orunde liegen, QesammeUe Mathematisehe Werke ; a translation of this paper is to be found in Clifford's Collected Mathematical Papers, t Afudefmvngslehre von 1S44.
14 MANIFOLDS. [CUAP. II.
It is the logical deductions from the characteristics of a manifold which are investigated by means of a calculus. The manifolds of separate proper- ties may have the same characteristics. In such a case all theorems which are proved for one manifold can be directly translated so as to apply to the other. This is only another mode of stating the ideas explained in Chapter I. §§ 3, 4, 5.
The relation of a manifold of elements to a scheme of things (cf. § 5), is that of the abstract to the concrete. Consider as explained in § 5 the
scheme of things represented by a, a' etc., 6, V etc., z, / etc. Then
these concrete things are not elements of a manifold. But to such a scheme a manifold always corresponds, and conversely to a manifold a scheme of things corresponds. The abstract property of a common ^-ness which makes the equivalence of a, a\ etc., in the scheme is an element of the manifold which corresponds to this scheme. Thus the relation of a thing in a scheme to the corresponding element of the corresponding manifold is that of a subject of which the element can be predicated. If il be the element corresponding to a, a! etc., then a \& A^ and a' is A. Thus if we write 2-1-3 = 5 at length, the assertion is seen to be
(l + l) + (l-f-l + l) = l-f 1 + 1 + 1 + 1; •
this asserts that two methods of grouping the marks of the type 1 are equivalent as far as the common five-ness of the sum on each side.
The manifold corresponding to a scheme is the manifold of the deter- mining property of the scheme. The cliaiucteristics of the manifold corre- spond to the characteristics of the scheme.
9. Secondary Properties of Elements. In order to state the characteristics of a manifold it may be necessaiy to ascribe to objects coiTe- sponding to the elements the capability of possessing other properties in addition to that definite property in special modes which the elements represent. Thus for the purpose of expressing the relation of an element A of a manifold to the elements B and C it may be necessary to conceive an object corresponding to A which is either Oi or a,, or a,, where the suffix denotes the possession of some other property, in addition to the il-ness of A, in some special mode which is here symbolized by the suffix chosen. Such a property of an object corresponding to A^ which is necessary to define the relation of il to other elements of the manifold, is called a Secondary Property of the element A,
Brevity is gained by considering each element of the manifold, such as A, as containing within itself a whole manifold of its secondary properties. Thus with the above notation A stands for any one of A^, -4„ A^ etc., where the suffix denotes the special mode of the secondary property. Hence the object O], mentioned above, corresponds to A^, and a, to A^, and so on.
9, 10] SECONDARY PROPERTIES OF ELEMENTS. 15
And the statement of the relation between two elements of the original manifold, such as A and B, requires the mention of a special A, say A^ and of a special B, say B4.
For example consider the manifold of musical notes conceived as repre- senting eveiy note so far as it differs in pitch and quality from every other note. Thus each element is a note of given pitch and given quality. The attribute of loudness is not an attribute which this manifold represents; but it is a secondary property of the elements. For consider a tone A and two of its overtones B and C, and consider the relations o{ A, B, C to & note P which is of the same pitch as A and which only involves the overtones B and C. Then P can be described as the pitch and quality of the sound produced by the simultaneous existence of concrete instances o{ A^ B and C with certain relative loudnesses. Hence the relation of P to A, B, C requires the mention of the loudness of each element in order to express it Thus if -dj, -B,, Ci denote A, B, C with the required ratio of their loudnesses, P might be expressed as the combination of A^, B^, C4.
The sole secondary property with which in this work we shall be concerned is that of intensity. Thus in some manifolds each element is to be conceived as the seat of a possible intensity of any arbitrarily assumed value, and this intensity is a secondary property necessary to express the various relations of the elements.
10. Definitions. To partition a manifold is to make a selection of elements possessing a common characteristic : thus if the manifold be a plane, a selection may be made of points at an equal distance from a given point. The selected points then form a circle. The selected elements of a partitioned manifold form another manifold, which may be called a submanifold in reference to the original manifold.
Again the common attribute C, which is shared by the selected elements of the original manifold A, may also be shared by elements of another manifold B. For instance in the above illustration other points in other planes may be at the same distance from the given point. We thus arrive at the conception of the manifold of the attribute C which has common elements with the manifolds A and B, This conception undoubtedly implies that the three manifolds A, B and C have an organic connection, and are in fact parts of a manifold which embraces them all three.
A manifold will be called the complete manifold in reference to its possible submanifolds ; and the complete manifold will be said to contain its submanifolds. The submanifolds will be said to be incident in the complete manifold.
One submanifold may be incident' in more than one manifold. It will then be called a common submanifold of the two manifolds. Manifolds will be said to intersect in their common. submanifolds.
16 MANIFOLDS. [CHAP. II.
11. Special Manifolds. A few definitions of special manifolds will both elucidate the general explanation of a manifold given above and will serve to introduce the special manifolds of which the properties are dis- cussed in this work.
A manifold may be called self-constituted when only the properties which the elements represent are used to define the relations between elements; that is, when there are no secondary properties.
A manifold may be called extrinsically constituted when secondary properties have to be used to define these relations.
The manifold of integral numbers is self-constituted, since all relations of such numbers can be defined in terms of them.
A uniform manifold is a manifold in which each element bears the same relation as any other element to the manifold considered as a whole.
If such a manifold be a submanifold of a complete manifold, it is not necessary that each element of the uniform submanifold bear the same relation to the complete manifold as any other element of that submanifold.
Space, the points being elements, forms a uniform manifold. Again the perimeter of a circle, the points being elements, forms a uniform mani- fold. The area of a circle does not form a uniform manifold.
A simple serial manifold is a manifold such that the elements can be arranged in one series. The meaning of this property is that some determinate process of deriving the elements in order one &om the other exists (as in the case of the successive integral numbers), and that starting from some initial element all the other elements of the manifold are derived in a fixed order by the successive application of this process. Since the process is determinate for a simple serial manifold, there is no ambiguity as to the order of suc- cession of elements. The elements of such a manifold are not necessarily numerable. A test of a simple serial manifold is that, given any three elements of the manifold it may be possible to conceive their mutual relations in such a fashion that one of them can be said to lie between the other two. If a simple serial manifold be uniform it follows that any element can be chosen as the initial element.
A manifold may be called a complex serial manifold when all its elements belong to one or more submanifolds which are simple serial manifolds, but when it is not itself a simple serial manifold. A surface is such a manifold, while a line is a simple serial manifold.
Two manifolds have a one to one correspondence* between their elements if to every element of either manifold one and only one element of the other manifold corresponds, so that the corresponding elements bear a certain defined relation to each other.
* The Bubjeot of the correspondence between the elements of manifolds has been inyestigated by G. Cantor, in a series of memoirs entitled, ' Ueber nnendliche, lineare Panktmannichfaltigkeiten,* Math. Annalen, Bd. 15, 17, 20, 21, 28, and BorehardVt Journal^ Bd. 77, 84.
J
11] SPECIAL MANIFOLDS. 17
A quantitively defined manifold is such that each element is specified by a definite number of measurable entities of which the measures for any element are the algebraic quantities ^, 17, ^, etc., so that the manifold has a one to one correspondence with the aggregate of sets of simultaneous values of these variables.
A quantitively defined manifold is a manifold of an algebraic function when each element represents in some way the value of an algebraic quantity w for a set of simultaneous values of f, rj, (f, etc., where w is a function of f> V> (r> ^^^'> ^^ ^^^ sense that it can be constructed by definite algebraic operations on f, 17, (f, etc., regarded as irresoluble magnitudes, real or imaginary*.
A quantitively defined manifold in which the elements are defined by a single quantity f is a simple serial manifold as far as real values of ^ are concerned. For the elements can be conceived as successively generated in the order in which they occur as f varies from — x to + x .
If an element of the manifold corresponds to each value of ^ as it varies continuously through all its values, then the manifold may be called con- tinuous If some values of f have no elements of the manifold corresponding to them, then the manifold may be called discontinuous.
A quantitively defined manifold depending on more than one quantity is a complex serial manifold. For if the quantities defining it f , 17, (f, eta be put equal to arbitrary functions of any quantity t, so that f =/i (t), rj =/, (t), etc., then a submanifold is formed which is a quantitively deBned manifold depending on the single quantity r. This submanifold is therefore a simple serial manifold. But by properly choosing the arbitrary functions such a submanifold may be made to contain any element of the complete manifold. Hence the complete manifold is a complex serial manifold.
The quantitively defined manifold is continuous if an element corresponds to every set of values of the variables.
A quantitively defined manifold which requires for its definition the absolute values (as distinct from the ratios) of v variables is said to be of 1/ dimensions.
A continuous quantitively defined manifold of v dimensions may also be called a i/-fold extended continuous manifold f.
♦ Cf. Forsyth, Theory of Functiotu, Ch. i. §§ 6, 7. t Cf. Riemann, loo. eit. section i. § 2.
W.
CHAPTER ni. Principles of Universal Algebra.
12. Introductory. Universal Algebra is the name applied to that calculus which symbolizes general operations, defined later, which are called Addition and Multiplication. There are certain general definitions which hold for any process of addition and others which hold for any process of multiplication. These are the general principles of any branch of Universal Algebra. These principles, which are few in number, will be considered in the present chapter. But beyond these general definitions there are other special definitions which define special kinds of addition or of multiplication. The development and comparison of these special kinds of addition or of multipli-
. cation form special branches of Universal Algebra. Each such branch will be called a special algebraic calculus, or more shortly, a special algebra, and the more important branches will be given distinguishing names. Ordinary algebra will, when there is no risk of confusion, be called simply algebra ; but when confusion may arise, the term ordinary will be prefixed.
13. Equivalence. It has been explained in § 3 that the idea of equivalence requires - special definition for any subject-matter to which it is applied. The definitions of the processes of addition and multiplication do carry with them this required definition of equivalence as it occurs in the field of Universal Algebra. One general definition holds both for addition and multiplication, and thus through the whole field of Universal Algebra. This definition may be fi-amed thus: In any algebraic calculus only one recognized type of equivalence exists.
The meaning of this definition is that if two symbols a and a' be equivalent in that sense which is explicitl}' recognized in some algebraic calculus by the use of the symbol =, then either a or a' may be used indifferently in any series of operations of addition or multiplication of the type defined in that calculus.
This definition is so far from being obvious or necessary for any symbolic calculus, that it actually excludes from the scope of Universal Algebra the
12 — 14] EQUIVALENCE. 19
Differential Calculus, excepting limited parts of it. For if /(^, y) be a function of two independent variables x and y, and the equivalence
f(x, y) = 0, be asserted, then ^/{x, y) and ^f(x, y) are not necessarily
7^ 7i
zero, whereas =- 0 and r- 0 are necessarily zero. • Hence the symbols f{x, y)
and 0 which are recognized by the sign of equality as equivalent according to one type of equivalence are not equivalent when submitted to some operations which occur in the calculus.
14. Principles of Addition. The properties of the general operation termed addition will now be gradually defined by successive specifications.
Consider a group of things, cx)ncrete or abstract, material things or merely ideas of relations between other things. Let the individuals of this group be denoted by letters a,b ... z. Let any two of the group of things be capable of a synthesis which results in some third thing.
Let this S3ni thesis be of such a nature that all the properties which are i attributed to any one of the original group of things can also be attributed to this result of the synthesis. Accordingly the resultant thing belongs to the original group.
Let the idea of order between the two things be attributable to their synthesis. Thus if a and b be the two things of which the synthesis is being discussed^ orders as between a first or b first can be attributed to this synthesis. Also let only ttuo possible alternative orders as between a and b be material, so as to be taken into explicit consideration when judging that things are or are not equivalent.
Let the result of the synthesis be unambiguous, in the sense that all possible results of a special synthesis in so &r as the process is varied by the variation of non-apparent details are to be equivalent. It is to be noted in this connection that the properties of the synthesis which are explicitly mentioned cannot be considered as necessarily defining its nature unambiguously. The present assumption therefore amounts to the state- ^ ment that the same words (or symbols) are always to mean the same thing, at leant in eveiy way which can affect equivalence.
This process of forming a synthesis between two things, such as a and 6, and then of considering a and b, thus united, as a third resultant thing, may be symbolized by a /> 6*. Here the order is sjrmbolized by the order in which a and b are mentioned ; accordingly a ^b and b^a symbolize two different things. Then by definition the only question of order as between a and b which can arise in this synthesis is adequately symbolized. Also a ^b whenever it occurs must always mean the same thing, or at least stand for some one of a set of equivalent things.
* Of. Grassniann, Aundehmtngnlehre von 1844, Preface.
2—2
20 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III.
Further a /^ 6 is by assumption a thing capable of the same synthesis with any other of the things a, t, ... u\ Accordingly we may write
p ^ (a ^b) and (a ^b) ^p
to represent the two possible syntheses of the type involving p and a ob. The bracket is to have the usual meaning that the synthesis within the bracket is to be performed first and the resultant thing then to be combined as the symbols indicate.
According to the convention adopted here the symbol a '^ 6 is to be read from left to right in the following manner: a is to be considered as given first, and b as joined on to it according to the manner prescribed by the symbol ^ . Thus (a '^b) ^p means that the result of a /> 6 is first obtained and then p is united to it. But a ^ 6 is obtained by taking a and joining 6 on to it. Thus the total process may equally well be defined by a ^b '^p. Hence, since both its right-hand and left-hand sides have been defined to have the same meaning, we obtain the equation
a ^ b ^ p = {a o b) ^ p.
Definition. Let any one of the symbols, either a single letter or a com- plex of letters, which denotes one of the group of things capable of this synthesis be called a term. Let the symbol ^ be called the sign of the operation of this synthesis.
It will be noticed that this synthesis has essentially been defined as a synthesis between two terms, and that when three terms such as a, 6, j>, are indicated as subjects of the synthesis a sequence or time-order of the opera- tions is also unambiguously defined. Thus in the sjmtheses (a ^ 6) ^ j> there are two separate ideas of order symbolized ; namely^ the determined but unspecified idea of order of synthesis as between the two terms which is involved by hypothesis in the act of synthesis, and further the sequence of the two successive acts of synthesis, and this time-order involves the sequence in which the various terms mentioned are involved in the process. Thus ar\b rsp and jp '^'(a '^ 6) both involve that the synthesis a ^ 6 is to be first performed and then the synthesis of a ^ 6 and p according to the special order of synthesis indicated.
In the case of three successive acts of sjrnthesis an ambiguity may arise. Consider the operations indicated in the symbols
a f^b <^ c^ d, c ^ {a ^b) ^ d.
No ambiguity exists in these two expressions; each of them definitely indicates that the synthesis a ^ 6 is to be made first, then a synthesis with c, and then a S3mthesis of this result with d. Similarly each of the two expressions d '^ (a '^ 6 /> c), and d ^ {c '^ (a /> b)} indicates unambiguously the same sequence of operations, though in the final synthesis of d with the result of the previous syntheses the alternative order of synthesis is adopted to that adopted in the two previous examples.
14, 15] PRINCIPLES OF ADDITION. 21
But consider the expressions
(a ob) o(c ^ d) and (c ^ d) '^ (a '^ b).
Here the two syntheses a ^b and c ^ d are directed to be made and then the resulting terms to be combined together. Accordingly there is an ambiguity as to the sequence in which these sjoitheses a ^^b, c ^d are to be performed. It has been defined however that a ^b and c ^ d are always to be unambiguous and mean the same thing. This definition means that the synthesis ^ depends on no previous history and no varying part of the environment. Accordingly a ^ 6 is independent ot c^d and these operations may take place in any sequence 6f time.
The preceding definitions can be connected with the idea of a manifold. All equivalent things must represent the same element of the manifold. The synthesis a ^ 6 is a definite unambiguous union which by hypothesis it is always possible to construct with any two things representing any two elements of the manifold. This synthesis, when constructed and represented by its result, represents some third element of the manifold. It is also often convenient to express this fact by saying that a ^^b represents a relation between two elements of the manifold by which a third element of the manifold is generated ; or that the term a '^ 6 represents an element of the manifold. An element may be named after a term whicb represents it : thus the element x is the element represented by the term x. The same element might also be named after any term equivalent to x.
It is obvious that any synthesis of the two terms a and b may be conceived as an operation performed on one of them with the help of the other. Accordingly it is a mere change of language without any alteration of real meaning, if we sometimes consider a ^ 6 as representing an operation performed on b or on a.
16. Addition. Conceive now that this synthesis which has been defined above is such that it follows the Commutative and Associative Laws.
The Commutative Law asserts that
a ^b = b ^a.
Hence the two possible orders of synthesis produce equivalent results.
It is to be carefully noticed that it would be erroneous to state the commutative law in the form that, order is not involved in the synthesis a^b. For if order is not predicable of the synthesis, then the equation, a /> 6 = 6 /> a, must be a proposition which makes no assertion at alL Accordingly it is essential to the importance of the commutative law that order should be involved in the synthesis, but that it should be indifferent as £Bur as equi- valence is concerned.
The Associative Law is symbolized by
a ^ b ^ c = a rs (b ^ c)] where a ^ 6 '^ c is defined in § 14.
1
22 PRINCIPLES OF UNIVERSAL ALGEBRA. [cHAP. II f.
The two laws combined give the property that the element of the manifold identified by three given terms in successive synthesis is independent of the order in which the three terms are chosen for the operation, and also of the internal oixler of each synthesis.
Let a synthesis with the above properties be termed addition; and let the manifold of the corresponding type be called an algebraic manifold ; and let a scheme of things representing an algebraic manifold be called an algebraic scheme. Let addition be denoted by the sign +. Accordingly it is to be understood that the symbol a + 6 represents a synthesis in which the above assumptions are satisfied.
The properties of this operation will not be found to vary seriously in the different algebras. The great distinction between these properties turns on the meaning assigned to the addition of a term to itself Ordinary algebra and most special algebras distinguish between a and a + a^ But the algebra of Symbolic Logic identifies a and a+a. The consequences of these assumptions will be discussed subsequently.
16. Principles of Subtraction. Let a and b be terms representing any two given elements of an algebraic manifold. Let us propose the problem, to find an element w of the manifold such that
x + b = a^
There may be no general solution to this problem, where a and b are connected by no special conditions. Also when there is one solution, there may be more than one solution. It is for instance easy to see that in an algebra which identifies a and a'\' a, there will be at least two solutions if there be one. For if a? be one answer, then a? + 6 = a? + 6+6 = a. Hence x + b is another answer.
If there be a solution of the above equation, let it be written in the form, a N^ 6. Then it is assumed that a^ b represents an element of the mani- fold, though it may be ambiguous in its signification.
The definition of a^ b is
a ^ b + b^a : (1).
If c be another element of the manifold let us assume that (a^b)^ c symbolizes the solution of a double problem which has as its solution or solutions one or more elements of the manifold.
Then av^ 6v^ c + (6 + c)=:a ^ 6v/c + (c + 6)
It follows that the problem proposed by the symbol a^(b + c) has one or more solutions, and that the solutions to the problem a v^ 6 ^^ c are included in them.
10] PRINCIPLES OF SUBTKACTION. 23
Conversely suppose that the problem a ^ (6 + c) is solved by one or more elements of the manifold.
Then by hypothesis a ^ (6 -f- c) + (6 + c) == a ; and hence
{a v^ (6 + c)} + c + 6 = a w (6 4- c) + (t + c) = a. But if d + c + 6 = a, then ci + c is one value of a v^ 6 and d is one value of
Accordingly a^b^ c is a problem which by hypothesis must have one or more solutions, and the solutions to a v^ (6 + c) are included in them.
Hence since the solutions of each are included in those of the other, the two problems must have the same solutions. Therefore whatever particular meaning (in the choice of ambiguities) we assign to one may also be assigned to the other. We may therefore write
av^(6 + c) = av/6 ^c (2).
Again we have
a v^ (6 + c) = a v^ (c + 6).
Hence from equation (2),
a'^b^ c = a^ c^h (3).
It may be noted as a consequence of equations (2) and (3), that if a v^ (6 + c) admit of solutions, then also both a^b and a v^ c admit of solutions.
Hence lia^b and 6 v^ c admit of solutions ; then aw6 = av/(6v'C + c); and it follows from the above note that a^ {b^ c) admits of a solution.
Also in this case
av/6-|-c = aw(6wc + c) + c = av/(6v/c)v^c + c, . from equation (2).
Hence a ^ 6 + c = a v^ (6 ^ c) (4).
We cannot prove that a^b-^-c^a-^c^b, and that a + (6 ^^ c) = a + 1 ^^ c, without making the assumption that a v^ 6, if it exists, is unambiguous.
Summing up : for three terms a, b and c there are four equivalent forms symbolized by
(a N^ 6) ^ c = (a ^^ c) v^ 6 = a ^ (6 4- c) = a ^ (c + 6) :
also there are three sets of forms, the forms in each set being equivalent but not so forms taken from different sets, namely
(a>^ 6) + c = a v^(6 ^ c) = c + (av^6) (a),
{c^ 6) + a = c ^ {b^ a)^a + {c^ b) (^8),
(a + c)>^ 6=(c + a) v^ 6 (7).
Subtraction. Let us now make the further assumption that the reverse anal}rtical process is unambiguous, that is to say that only one element of
24 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. lU.
the manifold is represented by a symbol of the type a^b. Let us replace in this case the sign ^ by — , and call the process subtraction.
Now at least one of the solutions of a + b^b is a. Hence in subtraction the solution of o + 1 — 6 is a, or symbolically a + 6 — J = a. But by definition, a — 6 + 6 = a.
Hence, a -f-6 — 6 = a — b + b = a (5).
We may note that the definition, a — b + b = a, assumes that the question a — b has an answer. But equation (5) proves that a manifold may always without any logical contradiction be assumed to exist in which the subtractive question a— 6 has an answer independently of any condition between a and 6, For fix)m the definition, a — b + b, where a — 6 is assumed to have an answer, can then be transformed into the equivalent form a + 6 — 6, which is a question capable of an answer without any condition between a and 6. But it may happen that in special interpretations of an algebra a — 6, though unam- biguous, has no solution unless a and b satisfy certain conditions. The remarks of § 7 apply here.
Again a + 6 — c = a + (6 — c + c) — c
= a + (6 — c) + c — c = a+(6-c) (6).
17. The Null Element. On the assumption that to any question of the type a — b can be assigned an answer, some meaning must be assigned to the term a — a.
Now if c be any other term,
c + a — a^c^c + b — b. Hence it may be assumed that
a — a = 6 — 6. Thus we may put
a-a = 0 (7);
where 0 represents an element of the manifold independent of a. Let the element 0 be called the null element. The fundamental property of the null- element is that the addition of this element and any other element a of the manifold yields the same element a. It would be wrong to think of 0 as I necessarily symbolizing mere nonentity. For in that case, since there can be no differences in nonentities, its equivalent forms a — a and b — b must be not only equivalent, but absolutely identical ; whereas they are palpably different. Let any term, such as a - a, which represents the null element be called a null term.
The fundamental property of 0 is,
a + 0 = a (8).
17 — 19] THE NULL ELEMENT. 25
Other properties of 0 which can be derived from this by the help of the
previous equations are,
0 + 0 = 0;
and a — 0 = a — (6— 6) = a — 6 + 6=a.
Again forms such as 0 — a may have a meaning and be represented by definite elements of the manifold.
The fundamental properties of 0 — a are symbolized by
6 + (0-a) = 6+0-a = 6-a, and 6 — (0 — a) = 6 — 0 + tt = 6+a.
Since in combination with any other element the null element 0 dis- appears, the symbolism may be rendered more convenient by writing — a for 0 — a. Thus — a is to symbolize the element 0 — a.
18. Steps. We notice that, since a = 0 + a, we may in a similar way consider a or + a as a degenerate form of 0 + a. From this point of view every element of the manifold is defined by reference to its relation with the null element. This relation with the null element may be called the step which leads from the null element to the other element. And by fostening the attention rather on the method of reaching the final element than on the element itself when reached, we may call the symbol + a the symbol of the step by which the element a of the manifold is reached.
This idea may be extended to other elements besides the null element. For we may write 6 ==: a + (6 — a) ; and 6 — a may be conceived as the »fep from a to &. The word step has been used* to imply among other things a quantity ; but as defined here there is no necessary implication of quantity. The step + a is simply the process by which any term p is transformed into the term p + a. The two steps + a and — a may be conceived as exactly opposed in the sense that their successive application starting from any term p leads back to that term, thus p + a — a=p. In relation to +a, the step — a will be called a negative step ; and in relation to — a, the step + a will be called a positive step. The frmdamental properties of steps are (1) that they can be taken in any order, which is the commutative law, and (2) that any number of successive steps may be replaced by one definite tesultant step, which is the associative law.
The introduction of the symbols + a and — a involves the equations
+ (+ a) = + (0 + a) = 0 + a = + a = a, -(+a) = -(0 + a) = -0-a = -a,
+ (-.a) = + (0-a) = 0-a = -a, '' ^ ^'
— (- a) = — (0 — a) = — 0 + a = + a = a..
19. Multiplication. A new mode of synthesis, multiplication, is now to be introduced which does not, like addition, necessarily concern terms of a
* Of. Clifford, Elements of Dynumic,
26 _ PRINCIPLES OF UNiyERSAL ALUEBRA. [CHAP. lil,
single algebraic scheme (cf. § 15), nor does it necessarily reproduce as its result a member of one of the algebraic schemes to which the terms S3nithe- SATt I sized belong. Again, the commutative and associated laws do not necessarily hold for multiplication ; but a new law, the distributive law, which defines the relation of multiplication to addition holds. Any mode of synthesis for which this relation to addition holds is here called multiplication. The result of multiplication like that of addition is unambiguous.
Consider two algebraic manifolds; call them the manifolds A and B. Let a, a', a" etc., be terms denoting the various elements oi A, and let 6, b\ V etc., denote the various elements of B. Assume that a mode of synthesis is possible between any two terms, one from each manifold. Let this synthesis result in some third thing, which is the definite unambiguous product under all circumstances of this special synthesis between those two elements.
Also let the idea of order between the two things be attributable to their union in this synthesis. Thus if a and 6 be the two terms of which the synthesis is being discussed, an order as between a first or h first can be attributed to this synthesis. Also let only two possible alternative orders as between a and h exist.
Let this mode of synthesis be, for the moment, expressed by the sign i=^ . Thus between two terms a, h from the respective manifolds can be generated the two things a^h and 6 j=: a.
All the things thus generated may be represented by the elements of a third manifold, call it (7. Also let the symbols a^h and h^a conceived as representing such things be called terms. Now assume that the manifold (7 is an algebraic manifold, according to the definition given above (§ 15). Then its corresponding terms are capable of addition. And we may write (a i=j 6) -f- (6' j=: a") + etc. ; forming thereby another term representing an ele- ment of the manifold (7.
The diefinition of the algebraic nature of G does not exclude the potssi- bility that elements of G exist which cannot be foimed by this synthesis of two elements from A and B respectively. For (a:=:6) + (6"^=^ d) is by definition an element of G ; but it vrill appear that this element cannot in general be formed by a single synthesis of either of the types a^^^ ^ h^^^ or ¥^^ ^ a^^K
Again a + a' + a'' + etc., represents an element of the manifold .^,and 6 + 6' + 6" -f- etc., represents an element of the manifold B, Hence there are elements of the manifold G represented by terms of the form
(a-f-a'-f- a" + etc.);=:(6 + 6' + 6" + ...), and (6 + 6' + 6" + etc.)^(a + a 4-a"H- ...)■
Now let this synthesis be termed Multiplicdtion, when such expreS' sions as the above follow the distributive, law as defined by equations (10) below.
For multiplication let the synthesis be denoted by x or by mere juxta-
20] MULTIPLICATION. 27
position. Then the definition of multiplication yields the following symbolic statements
(a-\-a^)b = ab+a% \
6(a + a') = 6a + 6a', ^ ^^^^•
(b + b')a = ba'^Va,
It will be noticed that the general definition of multiplication does not involve the associative or the commutative law.
20. Orders of Algebraic Manifolds. Consider a single algebraic manifold A, such that its elements can be multiplied together. Call such a manifold a self-multiplicative manifold of the first order. Now the products of the elements, namely cut, aa\ a'a, etc., by hypothesis form another alge- braic manifold ; call it B, Then B will be defined to be a manifold of the second order.
Now let the elements of A and B be capable of multiplication, thus forming another algebraic manifold C, Let C be defined to be a manifold of the third order. Also in the same way the elements of A and C form by multiplication an algebraic manifold, D, of the fourth order ; and so on.
Further let the elements of any two of these manifolds be capable of multiplication, and each manifold be self-multiplicative.
Let the following law hold, which we may call the associative law for manifolds. The elements formed by multiplying elements of the manifold of the mih order with elements of the manifold of the nth order belong to the manifold of the (m + n)th order. • Thus the complete manifold of the mth oi-der is formed by the multiplica- tion of the elements of any two manifolds, of which the sum of the ordere forms m, and also by the elements deduced by the addition of elements thus formed.
For instance <xa, aaW\ aa'a'W, represent elements of the manifolds of the second, third, and fourth oi-ders respectively; also cut represents an element of the manifold of the second order. Also a" (oaf) is an element of the mani- fold of the third order ; and (oa') (a'V) is an element of the manifold of the fourth order; and aa'(aaW") is an element of the manifold of the sixth order; and so on.
Such a system of manifolds will be called a complete algebraic system.
In special algebras it will be found that the manifold of some order, say the mth, is identical with the manifold of the first order. Then the manifold of the m 4- 1th order is identical with that of the second order, and so on.
Such an algebra will be said to be of the m - 1th species. In an algebra of the first species only the manifold of the first order can occur. Such
28 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III.
an algebra is called linear. The Calculus of Exteusion, which is a special algebra invented by Grassmann, can be of any species.
It will save symbols, where no confusion results, to use dots instead of brackets. Thus a" {(w!) is written a'. aa\ and {aa') {a"d") is written aa . a'V, and so on. A dot will be conceived as standing for two opposed bracket signs, thus )(, the other ends of the two brackets being either other dots or the end or beginning of the row of letters. Thus ah . cd stands for {ab) {cd)y and is not (a6) cd, unless in the special algebra considered, the two expressions happen to be identical ; also ab . cde ,fg stands for {ab) (cde) (fg). It will be noticed that in these examples each dot has been replaced by two opposed bracket signs. An ingenious use of dots has been proposed by Mr W. E. Johnson which entirely obviates the necessity for the use of brackets. Thus a {b (cd)] is written a.,b .cdy and a [b {c (de)]] is written a,..b..c,de. The principle of the method is that those multiplications indicated by the fewest dots are the first performed. Thus a {b{cd)} (ef) is written a,.b.cd,.ef, and a {b {cd)] ef is written a.,b,cd.,e ...f, where in the case of equal numbers of dots the left-hand multiplication is first performed.
21. The Null Element. Returning to the original general conception of two algebraic manifolds A and B of which the elements can be multiplied together, and thus form a third algebraic manifold C; let Oi be the null element of -4, Oa the null element of B, and 0, the null element of C,
Then if a and b represent any two elements of the manifolds A and B respectively, we have
a + Oi = a, and 6 + 0, = 6.
Hence (a + Oi) 6 = aft = ab + Oib.
Accordingly, Oib = O3 .
Similarly, 6O1 = 0, = aOj = 0^.
No confusion can arise if we use the same symbol 0 for the null elements of each of the three manifolds.
Accordingly, 0a = aO = O6 = 6O = O (11).
It will be observed that a null element has not as yet been defined for the algebraic manifold in general ; but only for those which allow of the process of subtraction, as defined in § 16. Thus manifolds for which the relation a + a^a holds are excluded from the definition.
In order to include these manifolds let now the null element be defined as that single definite element, if it exist, of the manifold for which the equation
a + 0 = a,
holds, where a is a/ny element of the manifold.
It will be noted that for the definite element a the same property may
21, 22] THE NULL ELEMENT. 29
hold for a as well as for 0; since in some algebras a-\-a = a. But 0 is defined to be the single element which retains this property with all elements. Then in the case of multiplication equations (11) hold.
22. Classification of Special Algebras. The succeeding books of this work vdll be devoted to the discussion and compaiison of the leading special algebraa It remains now to explain the plan on which this in- vestigation will be conducted.
It follows from a consideration of the ideas expounded in Chapter i. that it is desirable to conduct the investigation of a calculus strictly in connection with its interpretations, and that without some such interpretation, however general, no^ great progress is likely to be made. Therefore each special algebra will, as far as possible, be interpreted concurrently with its in- vestigation. The interpretation chosen, where many are available, will be that which is at once most simple and most general ; but the remaining applications will also be mentioned with more or less fulness according as they aid in the development of the calculus. It must be remembered, however, in explanation of certain obvious gaps that the investigation is primarily for the sake of the algebra and not of the interpretation.
No investigation of ordinary algebra will be attempted. This calculus stands by itself in the fundamental importance of the theory of quantity which forms its interpretation. Its formulae will of course be assumed ^
throughout when required.
In the classification of the special algebras the two genera of addition form the first ground for distinction.
For the purpose of our immediate discussion it will be convenient to call the two genera of algebras thus formed the non-numerical genus and the numerical genus.
In the non-numerical genus investigated in Book II. the two symbols a and a'{'a, where a represents any element of the algebraic manifold, are equivalent, thus a = a + a. This definition leads to the simplest and most rudimentary type of algebraic symbolism. No symbols representing number or quantity are required in it. The interpretation of such an algebra may be expected therefore to lead to an equally simple and fundamental science. It will be found that the only species of this genus which at present has been developed is the Algebra of Symbolic Logic, though there seems no reason why other algebras of this genus should not be developed to receive interpretations in fields of science where strict demonstrative reasoning with- out relation to number and quantity is required. The Algebra of Symbolic Logic is the simplest possible species of its genus and has accordingly the simplest interpretation in the field of deductive logic. It is however always desirable while developing the symbolism of a calculus to reduce the inter- pretation to the utmost simplicity consistent with complete generality.
30 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. IIL
Accordingly in discussing the main theory of this algebra the difficulties peculiar to Symbolic Logic will be avoided by adopting the equally general interpretation which considers merely the intersection or non-intersection of regions of space. This interpretation will be developed concurrently with the algebra. After the main theory of the algebra has been developed, the more abstract interpretation of Symbolic Logic will be introduced.
In the numerical genus the two symbols a and a + a are not equivalent. The symbol a + a is shortened into 2a ; and by generalization of this process a symbol of the form fa is created, where f is an ordinary algebraical quantity, real or imaginary. Hence the general type of addition for this genus is symbolized by f a + 176 + 5<^ + etc., where a, b, c, etc. are elements of the algebraic manifold, and f, rf, ^, etc. are any ordinary algebraic quantities (such quantities being always symbolized by Greek letters, c£ Book III. Chapter I. below). There are many species of algebra with im- portant interpretations belonging to this genus; and an important general theory, that of Linear Associative Algebras, connecting and comparing an indefinitely large group of algebras belonging to this genus.
The special manifolds, which respectively form the interpretation of all the special algebras of this genus, have all common properties in that they all admit of a process symbolized by addition of the numerical type. Any manifold with these properties will be called a 'Positional Manifold.' It is therefore necessary in developing the complete theory of Universal Algebra to enter into an investigation of the general properties of a positional manifold, that is, of the properties of the general type of numerical addition. It will be found that the idea of a positional manifold will be made more simple and concrete without any loss of generality by identifying it with the general idea of space of any arbitrarily assigned number of dimensions, but excluding all metrical spatial ideas. In the discussion of the general properties of numerical addition this therefore will be the interpretation adopted as being at once the most simple and the most general. All the properties thus deduced i^ust necessarily hold ^or any special algebra of the genus, though the scale of the relative importance of different properties may vary in different algebras. Positional manifolds are investigated in Book III.
Multiplication in algebras of the numerical genus of course follows all the general laws investigated in this chapter. There is also one other general law which holds throughout this genus. The product of {a and 17& ({ and 17 being numbers) is defined to be equivalent to the product of ^ (ordinary multiplication) into the product of a and b. Thus in symbols
^.7fb = (rfoh, tfb. ^^ ^ba ; where the juxtaposition of f and 17 always means that they are to be multiplied according to the ordinary law of multiplication for numbers.
If this law be combined with equation 10 of § 19, the following general
22] CLASSIFICATION OF SPECIAL ALGEBRAS. 31
equation must hold: let 61, eg, ... e, be elements of the manifold, and let Greek letters denote numbers (i.e. ordinary algebraic quantities, real or imaginary), then
It follows that in the numerical genus of algebras the successive derived manifolds are also positional manifolds, as well as the manifold of the first order.
In the classification of the special algebras of this genus the nature of the process of nmltiplication as it exists in each special algebra is the guide.
The first division must be made between those algebras which involve a complete algebraical system of more than one manifold and those which involve only one manifold, that is, between algebras of an order higher than the first and between linear algebras (cf. § 20). It is indeed possible to consider all algebras as linear. But this simplification, though it has very high authority, is, according to the theory expounded in this work, fallacious. For it involves treating elements for which addition has no mean- ing as elements of one manifold ; for instance in the Calculus of Extension it involves treating a point element and a linear element as elements of one manifold capable of addition, though such addition is necessarily meaningless.
The only known algebra of a species higher than the first is Grassmann's Calculus of Extension ; that is to say, this is the only algebra for which this objection to its simplification into a linear algebra holds good. The Calculus of Extension will accordingly be investigated first among the special algebras of the numerical genus. It can be of any species (cf. § 20). The general type of manifold of the first algebraic order in which the algebra finds its interpretation will be called an Extensive Manifold. Thus an extensive manifold is also a positional manifold.
In Book IV. the fundamental definitions and formulae of the Calculus of Extension will be stated and proved. The calculus will also be applied in this book to an investigation of simple properties of extensive manifolds which, though deduced by the aid of this calculus, belong equally to the more general type of positional manifolds. One type of formulae of the algebra will thus receive investigation. Other types of formulae of the same algebra are developed in Books V., VI. and VII., each type being developed in conjunction with its peculiar interpretation. The series of interpretations will form, as they ought to do, a connected investigation of the general theory of spatial ideas of which the foundation has been laid in the discussion of positional manifolds in Book III.
This spatial interpretation, which also applies to the algebra of Symbolic Logic, will in some form or other apply to every special algebra, in so far as interpretation is possible. This fact is interesting and deserves investigation.
32 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III. 22
The result of it is that a treatise on Universal Algebra is also to some extent a treatise on certain generalized ideas of space.
In order to complete this .subsidiary investigation an appendix on a mode of arrangement of the axioms of geometry is given at the end of this volume.
The second volume of this work will deal with Linear Algebras. In addition to the general theory of their classification and comparison, the special algebras of quaternions and matrices will need detailed development.
Note. The discussions of this chapter are largely based on the ' Ueber- sicht der allgemeinen Formenlehre* which forms the introductory chapter to Grassmann's Ausdehnungslehre von 1844.
Other discussions of the same subject are to be found in Hamilton's Lectures on Quaternions, Preface; in Hankel's Vorlesungen uber Complexe ZahUn (1867) ; and in De Morgan's Trigonometiy and Double Algebra, also in a series of four papers by De Morgan, * On the Foundation of Algebra/ Transactions of the Cambridge Philosophical Society, vols. vii. and VIIL, (1839. 1841, 1843, 1844).
BOOK II.
THE ALGEBRA OF SYMBOLIC LOGIC.
w.
CHAPTER 1. The Algebra of Symbolic Logic.
23. Formal Laws. The Algebra of Symbolic Logic* is the only known member of the non-numerical genus of IJDiversal Algebra (cf. Bk. I., Ch. ill., § 22).
It will be convenient to collect the formal laws which define this special algebra before considering the interpretations which can be assigned to the symbols. The algebra is a linear algebra (cf. § 20), so that all the terms used belong to the same algebraic scheme and are capable of addition.
Let a, b, c, etc. be terms representing elements of the algebraic manifold of this algebra. Then the following symbolic laws hold.
(1) The general laws of addition (cf. Bk. I. Ch. in., g 14, 15) :
a + 6 = 6 -f a, a + 6 + c = (a + 6) + c = a + (6 + c).
(2) The special law of addition (cf § 22) :
a^ a = a,
(3) The definition of the null element (cf. § 21) :
a + 0 = a.
(4) The general laws of multiplication (cf § 19) :
c(a-\'b)^ca + cb, (a-\-b)c^ac + be,
(5) The special laws of multiplication :
ab=^ba, ahc = ab.c=^a.bc,
* This algebra in all essential partioulars was invented and perfected bj Boole, cf. his work entitled, An Investigation of the Laws of Thought^ London, 1854.
3—2
36 THE ALGEBRA OF STMBOUC LOGIC. [CHAP. 1.
(6) The law of ' absorption ' :
a + a& = a. This law includes the special law (2) of addition.
(7) The definition of the ' Universe.* This is a special element of the manifold, which will be always denoted in future by i, with the following property :
ai = a,
(8) Supplementary elements. An element b will be called supple- mentary to an element a if both a + 6 = t, and a6 = 0. It will be proved that only one element supplementary to a given element can exist ; and it will be assumed that one such element always does exist. If a denote the given element, a will denote the supplementary element. Then a will be called the supplement of a. The supplement of an expression in a bracket, such as (a + 6), will be denoted by " (a + 6).
The theorem that any element a has only one supplement follows from the succeeding fundamental proposition which develops a method of pi'oof of the equivalence of two terms.
Proposition I. If the equations xy = xz, and a? -f y = a? + ^, hold simul- taneously, then y = z.
Multiply the second equation by x, where x is one of the supplements of X which by hypothesis exists.
Then ^ (a? + y) = ac (a? + z).
Hence by (4) xx -{-xy = xx -hxz,
hence by (8) and (3) xy = xz.
Add this to the first equation, then by (4)
{x-hx)y=={x + x)z, hence by (8) iy =-t5,
hence by (7) y = z.
Corollary I. There is only one supplement of any element x. For if possible let x and x' be two supplements of x.
Then a^ = 0 = xx\ and a? + al = i = a? -f a?'.
Hence by the proposition, x « x\
Corollary II. I{x = y, then S = y. Corollary III. i; = 0, and 6 = i.
Corollary IV. i = a? ;
where x means the supplement of the supplement of x. The prooiEs of these corollaries can be left to the reader.
24] FORMAL LAWS. 37
Proposition II. (a? + y) (a? + ^r) as a; + yz.
For (x •\' y){x -^^ z) ^ XX •\' xy -\- xz •\' yz — X -^^ a:{y -{• z) •\' yz
^x-\-yz, by (6).
Proposition III. a^O = 0 = Oa?, and a? + i = i = i + a;.
The first is proved in § 21. The proof of the second follows at once from (6) and (7).
24. Reciprocity between Addition and Multiplication. A reci- procity between addition and multiplication obtains throughout this algebra ; so that corresponding to every proposition respecting the addition and multiplication of terms there is another proposition respecting the multi- plication and addition of terms. The discovery of this reciprocity was first made by C. S. Peirce*; and later independently by Schroder f.
The mutual relations between addition and multiplication will be more easily understood if we employ the sign x to represent multiplication. The definitions and fundamental propositions of this calculus can now be arranged thus.
The Conmiutative Laws are (cf. (1) and (5))
xxy = yxx,) ^ ^'
The Distributive Laws are (cf. (4) and Prop. II.)
X x(y-h z) ^{x X y) -^ (x X z),) ^
X -h {y X z)=^ (x -\- y) X (x + z),) ^ ^'
The Associative Laws are (cf. (1) and (5))
x + (y + z)^x-hy + z,] ^ .^
X x(ifx z) = xxy xz.) ■ ^'
The Laws of Absorption are (cf. (6))
x-\'{xxy) = x^x-\-x,) .j^.
a?x(a:-hy)= a? = a?xa?.J
The properties of the Null element and of the Universe are (cf. (3), (7), and Prop. III.),
::oro] ■ <«■
a? + 0 = a?,l a? X i = X,)
in
The definition of the supplement of a term gives (cf (8) and Prop. I.)
(G).
x-\- x = %,) a; X « = 0. 1
* Proe, of the American Academy of Arts and Seiencei, 1867. t Der Operatiomkreit des Lo0ikkalk(iU, 1877.
S8 THE ALGEBRA OF SYMBOUC LOGIC. [CHAP. 1.
There can therefore be no distinction in properties between addition and multiplication. All propositions in this calculus are necessarily divisible into pairs of reciprocal propositions; and given one proposition the reciprocal proposition can be immediately deduced from it by interchanging the signs + and X , and the terms % and 0. An independent proof can of course always be found : it will in general be left to the reader.
Also any interpretation of which the calculus admits can always be inverted so that the interpretation of addition is assigned to multiplication, and that of multiplication to addition, also that of t to 0 and that of 0 to i,
25. Interpretation. It is desirable before developing the algebraic formulae to possess a simple and general form of interpretation (cf. § 7 and §22>
Let the elements of this algebraic manifold be regions in space, each region not being necessarily a continuous portion of space. Let any term symbolize the mental act of determining and apprehending the region which it represents. Terms are equivalent when they place the same region before the mind for apprehension.
Let the operation of addition be conceived as the act of apprehending in the mind the complete region which comprises and is foimed by all the regions represented by the terms added. Thus in addition the symbols represent firstly the act of the mind in apprehending the component regions represented by the added terms and then its act in apprehending the complete region. This last act of apprehension determines the region which the resultant term represents. This interpretation of terms and of addition both satisfies and requires the formal laws (1) and (2) of § 23. For the complete region does not depend on the order of apprehension of the com- ponent regions ; nor does it depend on the formation of subsidiary complete regions out of a selection of the added terms. Hence the commutative and associative laws of addition are required. The law, a + a=a, is satisfied since a region is in no sense reduplicated by being placed before the mind repeatedly for apprehension. The complete region represented by a + a re- mains the region represented by a. This is called the Law of Unity by Jevons (cf. Pure Logic, ch. vi).
The null element must be interpreted as denoting the non-existence of a region. Thus if a term represent the null element, it symbolizes that the mind after apprehending the component regions (if there be such) symbolized by the term, further apprehends that the region placed by the term before the mind for apprehension does not exist. It may be noted that the addition of terms which are not null cannot result in a null term. A null teim can however arise in the multiplication of terms which are not null.
Let the multiplication of terms result in a term which represents the entire region common to the terms multiplied. Thus xyz represents the
25, 26] INtERPBEtATIOK. 39
entire region which is at once incident in the regions x and y and z. Hence the term xy symbolizes the mental acts first of apprehending the regions symbolized by x and y, and then of apprehending the region which is their complete intersection. This final act of apprehension determines the region which «y represents
This interpretation of multiplication both satisfies and requires the distributive law, numbered (4) in § 23, and the commutative and associative laws marked (5) in § 23. The law, aa = a, which also occurs in (5) of § 23 is satisfied ; for the region which is the complete intersection of the region a with itself is again the region a. This is called the Law of Simplicity by Jevons (cf. loc. cit).
The Law of Absorption (cf (6) § 23) is also required and satisfied. For the complete region both formed by and comprising the regions a and ah is the region a, and the final act of apprehension symbolized by a + a6 is that of the region a. Hence
a 4- aft = a.
This interpretation also requires that i( x-^y^x, then y^xy. And this proposition can be shown to follow from the formal laws (cf § 26, Prop. viii.).
The element called the Universe (cf § 23 (7)), must be identified with all space ; or if discourse is limited to an assigned portion of space which is to comprise all the regions mentioned, then tlie Universe is to be that complete region of space.
The term supplementary (cf § 23 (8)) to any term a represents that region which includes all the Universe with the exception of the region a. The two regions together make up the Universe ; but they do not overlap, so that their region of intersection is non-existent.
It follows that the supplement of the Universe is a non-existent region, and that the supplement of a non-existent region is the Universe (cf Prop. I. Cor. 3).
26. Elementary Propositions. The following propositions of which the interpretation is obvious can be deduced from the formal laws and from the propositions already stated.
Proposition IV. If a? -h y = 0, then a? = 0, y = 0.
For multipljdng by x, x{x-\-y)^ 0.
But a;(ar + y) = a?, by (6) § 23.
Hence a? = 0. Similarly, y = 0.
The reciprocal theorem is, if ajy = i, then a? = i, y = i.
Proposition V. x + y = x + yx, and xy^x{y-\-x). For X + y ^ X -\' y {x -hx) = X + yx + yx ^ X + yx.
The second part is the reciprocal proposition to the first part.
40 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. I.
Proposition VI. - {xy) = x+y,
and ■■(fl? + y) = «y-
For by Prop, v., x-^y-x + xy^x-^-xy.
Hence ay +(x + y) = xy'hay + x=^x(if + y)+x
•*• X<^+^f i^^ =flj + ^ = i. — — v)
Also {vy(x-\-y)-(cxy + xyy = 0. - - — --(?-)
ZtnUa^A^^iS) o-^z.) . Hence by § 23 (8) x + y = - (xy).
The second part is the reciprocal of the first part. Also it can be deduced from the first part thus :
-(xy) = x + y = x + y.
Taking the supplements of both sides
"(^ + y) = "(^)=^.
CoROiJ^ARY. The supplement of any complex expression is found by replacing each component term by its supplement and by interchanging + and X throughout.
Proposition VII. If xy = xz, then
osy=:xz, and x+y^x-\-z.
For taking the supplement of both sides of the given equation, by Prop. VI.,
x-^-y^x + z.
Multiplying by a?, xiy = xz.
Again taking the supplement of this equation, then
x + y =^x + z.
The reciprocal proposition is, if x-\-y = x + z, then x + y = x + z, and xy = xz.
Proposition VIII. The following equations are equivalent, so that from any one the remainder can be derived :
y = xy, x-\-y = x, xy-0, xi-y^i.
Firstly : assume y = xy.
Then X'\-y^x + xy — x,
And xy =^ xxy = 0.
And a? + j^ = a? + " (xy) = a? + ^ + y = i.
Secondly: assume x + y=ix.
m
Then «y=(«H"y)y *y.
Hence the other two equations can be derived as in the first case. Thirdly: assume ^ = 0.
1
i
27] ELEMENTARY PROPOSITIONS. 41
Then y^(x-\-x)y^0Dy ^-xy^xy.
Hence the other two equations can be derived. Fourthly: assume x-\-y = i.
Then taking the supplements of both sides
Hence by the third case the other equations are true.
Corollary. By taking the supplements of the first and second equations two other forms equivalent to the preceding can be derived, namely
y^x-hy, ^ = x.
Proposition IX. If x^xyz, then x^xy = xz, and if x = x-{-y-\' z, then x = x + y = x + z.
For osy^xy{z-{-z)=^ xyz + xyz = a? + xyz = x, from (6) § 23.
The second part of the proposition is the reciprocal theorem to the first ^
part. ('X^^4-i)(T^^4i) ^7-^^ ^ ii-\rX(H^iy- X Pn^.JL c^< ^1 f^IJ^')-
Corollary. A similar proof shews that if z=z{xu+yv), then z^z(x'{-y); and that if ir = 2:+(a? + w)(y-f v), then z = z + xy.
27. Classification. The expression a? + y + £r + , . . , which we can denote by u for the sake of brevity, is formed by the addition of the regions x, y, etc. Now these regions may be overlapping regions: we re- quire to express u as a sum of regions which have no common part. To this problem there exists the reciprocal problem, given that u stands for the product xyz.,., to express t^ as a product of regions such that the sum of any two completes the universe. These problems may be enunciated and proved sjmibolically as follows.
Proposition X. To express t^ (= a? + y + 2: + . . .), in the form
X+T+Z+...;
where X, F, Z have the property that for any two of them, Y and Z say, the condition FZ = 0, holds.
Also to express u (= xyz...) in the form XYZ,.. ; where X, F, Z have the property that for any two of them, F and Z say, the condition F+Z = i, holds.
Now from Prop. IV., i{ x(y + z)^ 0, then an/ — 0,xz^ 0. Hence for the first part of the proposition the conditions that Z, F, Z, etc. must satisfy can be expressed thus,
Z(F+Z + r-|-...)=cO, F(Z+y+...) = 0, Z(r+...) = 0,etc.
Now by Prop. V., u^x-{'y-\-z-\- ...
42 THE ALGEBRA OB* SYMBOLIC LOGIC. [CHAP. L
and y-f ir + ^+... =y + y(2r + ^+ ...);
and z + t + ,..=z +z(t'\' .,,).
Proceeding in this way, we find
u = x-\-xy-^ xyz + xyzt + . . . .
Hence we may write
X^x, Y=xy, Z^xyZy etc.
It is obvious that there is more than one solution of the problem.
Again for the second part of the proposition, consider
S = ^ + y + ^ + . . . .
By the fii-st part of the proposition,
S = iic + a^ + xyz + xyzi +
Here any two terms, 1" and Z, satisfy the condition YZ = 0. Taking the supplements of these equations,
= xyz. . .
= '-(^'\-xy ^-xyz-^- .„)
Hence we may write XYZ.., for xyz...y where X = a?, Y=x-\-y, Z^x + y + Zj etc. and any two of X, F, Z, etc., for instance Fand Z, satisfy the condition F+ Z= i.
It is obvious that there is more than one solution of this problem.
These problems are of some importance in the logical applications of the algebra.
28. Incident Regions. The symbolic study of regions incident (cf. § 10) in other regions has some analogies to the theory of inequalities in ordinary algebra. These relationships also partly possess the properties of algebraic equations. Two mixed symbols have therefore been adopted to express them, namely 4 ^^'^ ^ (cf Schroder, Algebra der Logik). Then, y^Xy expresses that y is incident in x ; and x^ y expresses that x contains y. Expressions of this kind will be called, borrowing a term from Logic, subsumptions. Then a subsumption has analogous properties to an inequality. The Theoty of Symbolic Logic has been deduced by C. S. Peirce from the type of relation symbolized by ^, cf. American Journal of MaJlk&nuitics, Vols, ill and Vll (1880, 1885). His investigations are incorporated in Schroder's Algebra der Logik.
In order to deduce the properties of a subsumption as far as possible purely symbolically by the methods of this algebra, it is necessary to start from a proposition connecting subsumptions with equations. Such an initial proposition must be established by considering the meaning of a subsumption.
28] INCIDENT REGIONS. 43
Proposition XI. liy^x, then y^xy\ and conversely.
For if y be incident in x, then y and ayy denote the same region.
The converse of this proposition is also obvious.
It is obvious that any one of the equations proved in § 26, Prop. VIII., to be equivalent to y = xy is equivalent to y ^x. In fact the subsumptiou y^x may be considered as the general expression for that relation between X and y which is implied by any one of the equations of Prop. VIII. It follows that an equation is a particular case of a subsumption.
Corollary, xz^x^x-k-z.
Proposition XII. If a; :): y, and y^z\ then x^z.
For by Prop. XI. and by § 26, Prop. IX.
z — zy = zxy = zx. Hence x^z,
^ Proposition XIII. \i x^y, and y^x\ then x = y. For since ^^V^ then y — xy.
And since y^^> iki^xa y^^x-^-y.
xHence y = xy = x{x + y) = x.
Proposition XIV. U x^y, and u^v\ then iix^vy, and u-hx^v + y.
For y = yx, and v = vu ; hence vy = yxvu = vy . xu.
Therefore ux^vy.
Also a? = a? + y, and u = u + v;
hence a7 + w = iP + y + M + t; = (a:+M)H-(y + v).
Therefore x-j-u^y + v.
Corollary. U x^y, and u — v\ then iix ^ vy, and u-{-x^v-\-y. For v = vu, and u = w + 1; ; hence the proof can proceed exactly as in the proposition.
The proofe of the following propositions may be lefb to the reader.
■ Proposition XV. If x^yytheny^x.
Proposition XVI. 1{ z^xy, then z^x, z^y, z^x + y.
Proposition XVII. If z 4 ^y, then xy^z, x + y^z.
Proposition XVIII. Uz^x + y, then z^x, z^y, z^xy.
Proposition XIX. Itz^x + y, then xy^z, x + y^z.
Proposition XX. If xz 4 y, and x^y -^z, then x^y.
44 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. I. 28.
Proposition XXI. It z^wu-^yv, then s^x + y.
The importance of Prop. XXI. demands that its proof be given. By Prop. IX. Cor., z = z (am + yv) = 5 (a? + y). Therefore -a 4 ^ + y-
Corollary. If z^xu-^-yv, then z^x-\-y; that is, xu-k-yv^x-^y.
Prop.* XXII. If z^{x-\- u) (y + v), then z ^ xy. Corollary, (x -\-u)(y + v)^ xy,
* This proposition, which I had overlooked, was pointed out by Mr W. E. Johnson.
CHAPTER II. The Algebra of Symbolic Logic {continiLed).
29. Development. (1) The expression for any region whatsoever may be written in the form ox + Imc] where x represents any region. For let z be any region. Now x + x = %.
Hence z^^x-^x
^ZX'\'ZX,
Now Xeta^zx-^- van, and b — zx-^ vx, where u and v are restricted by no conditions.
Then ax + bx^ {zx + vai) x + {zx + vx) x^zx + zx^z.
Hence by properly choosing a and b, ax-^-lw can be made to represent any region z without imposing any condition on x.
Again the expression for any region can be written in the form
(a + a?) (6 + x),
where x represents any other region. For by multiplication
(a H- a?) (6 + a) = oft -f oS + 6a? = (a + ab) S + (6 H- ah) x = a^ + bx.
This last expression has just been proved to represent the most general region as &r as its relation to the term x is concerned.
(2) Binomial expressions of the form cuc-^bx have many important properties which must be studied. It is well to notice at once the follow- ing transformations :
cur + te = (6 + a?) (a + «) ;
" {ax + Iko) = (a + x) (6 +a?)
(ax -h bx) (ex -h cte) = acx + bdx ;
aa7 + te + c=s(a + c)aj-|-(6 + c)«; ax'^bx^ax-\'bx + ah.
46 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
(3) Let f{x) stand for any complex expression formed according to the processes of this algebra by successive multiplications and additions of x and x and other terms denoting other regions. Then /(a?) denotes some region with a specified relation to x. But by (1) of this article f{x) can also be written in the form (ix-\-bx. Furthermore a and h can be regarded as specified by multiplications and additions of the other terms involved in the formation of f{x) without mention of x. For if a be a complex expression, it must be
expressible, by a continual use of the distributive law, as a sum of products of which each product either involves ar or S or neither. Since a only appears when multiplied by a?, any of these products involving « as a factor can be rejected, since xx = 0\ also any of these products involving a; as a factor can be written with the omission of x, since xx = x. Hence a can be written in a form not containing x or x. Similarly h can be written in such a form.
(4) Boole has shown how to deduce immediately from /(a?) appropriate forms for a and h. For write /(a?) = ax-\-bx. Let i be substituted for a?, then
f(t) = at + 6i; = ai H- 0 = a.
Again let 0 be substituted for x, then
/(O) = aO + 60 = 0 + W = 6.
Hence /(a?) =r/(») x +/(0) x.
For complicated expressions the rule expressed by this equation shortens the process of simplification. This process is called by Boole the develop- ment of/ (a?) with respect to x.
The reciprocity between multiplication and addition gives the reciprocal
rule
/(^) = {/(0) + «){/(t') + S}.
(5) The expressions /(»') and /(O) may involve other letters y, z, etc. They may be developed in respect to these letters also.
Consider for example the expre8sion/(a:, y, z) involving three letters.
/(<», y, i)=f(i, y, «)a! +/(0, y, z)x,
f(i, y, z)=/(i, t, z)y+fii, Q, g)y, f(i, i. *)=/(». i, i)z+f{x. i, 0)i, f(i, 0, z) ^/(i, 0, i) z +f{i, 0, 0) z, /(O. y, z) =/(0. i. z) y +/((), 0, z) y, f(0, i, z) =/(0, i, i) z +/(0, i, 0) z. /(O, 0, z) =/(0, 0, t) z +/(0. 0, 0) z.
Hence by substitution
/(«, y, z) =f{%, i, i) asyz +/(0, i, i) xyz +f(i, 0, t) asyz +/(t, t, 0) xyz +/(i, 0, 0) xyz +/(0, i, 0) xyz +/(0, 0. t) ^z +/(0, 0, 0) xyz.
30] DEVELOPMENT. 47
The reciprocal fonnula, owing to the brackets necessary, becomes too complicated to be written down here.
Let any term in the above developed expression for f{x, y, z), say /(O, i, 0) xyz, be called a constituent term of the type xyz in the develop- ment.
(6) The rule for the supplement of a binomial expression given in subsection (1) of this article, namely ~" {ax + bx) = aa? + fe, can be extended to an expression developed with respect to any number of terms x^y, z, .... The extended rule is that if
/(a?, y, Zy ..,)^axyz... + ... H-gf^^...,
then ""/(^> y> ^> ...) = dxyz .., + ... +^^^....
In applying this proposition any absent constituent term must be replaced with 0 as its coefficient and any constituent term with the form xyz... must be written ixyz.., so that % is its coefficient.
For assume that the rule is true for n terms x, y, z... and let ^ be an (n + l)th term.
Then developing with respect to the n terms a?, y, j&, . . .
f(^,y*^, ...t)-Axyz.:.^',..-\-Owgz...y
where the products such as xyz ..., ..., xyz ... do not involve t, and
Then the letters a, a\ ..., g, g' are the coefficients of constituent terms of the expression as developed with respect to the n + 1 terms Xy y, z,,,,t.
Hence by the assumption
"/(a?, y, Zy.,.t)-Axyz .,. + ... + G«p.... But by the rule already proved for one term,
A -dt-^ctiy ..., 0=gt-\-gt.
Hence the rule holds for {n + l)th terms. But the rule has been proved for one term. Thus it is true always.
30. Elimination. (1) The object of elimination may be stated thus : Given an equation or a subsumption involving certain terms among others, to find what equations or subsumptions can be deduced not involving those terms.
The leading propositions in elimination are Propositions XXI. and XXII. of the last chapter, namely that, if z^xu-^yv, then z^x-j-y; and if jer :^ (a? + w) (y + v), then z^xy; and their Corollaries that, xu-^yv^x + y, and, (a: + w)(yH-v)a^ajy.
(2) To prove that if aa? + te = c, then a + b^c^ah.
Eliminating x and x by the above-mentioned proposition from the equation
c=*aa?-»-6«, c^a + fc.
48 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP, XL
Also takiug the supplementary equation,
Hence from above
c^a + b.
Taking the supplementary subsumption,
oft = *" (a + 6)
Therefore finally
a + b^c^ab.
The second part can also be proved* by taking the reciprocal equation, c = (tt •+ ^) (6 + x), and by using Prop. XXII. Corollary.
The same subsumptions, written in the supplementary form, are
a + 6 a^ c :^ ofc.
(3) By Prop. XI. each of these subsumptions is equivalent to an equation, which by Prop. VIII. can be put into many equivalent forms.
Thus a + 6 s^ c, can be written
c = c(a-\'b)',
and this is equivalent to
a6c = 0. And c^ab, can be written
ab = abc; and this is equivalent to abc = 0.
(4) Conversely, if
c^^a-^b, ^ab
f6,) > J
then it has to be shown that we can write dx + bx^c; where we have to determine the conditions that x must fulfil. This problem amounts to proving that the equation dx + b^^c has a solution, when the requisite conditions between a, b, c are fulfilled. The solution of the problem is given in the next article (cf. § 31 (9)).
The equation ax-j-l^ = c includes a number of subsidiary equations.
For instance ax^cx; thence a-^x^c-^x, and thence ax = cx. Similarly i^^txc, and ^ = c^. The solution of the given equation will satisfy identi- cally all these subsidiary equations.
(5) Particular Cases, There are two important particular cases of this equation, when c = t, and when c = 0.
Firstly, c = i. Then cw? + te = i.
* Pointed ont to me by Mr W. E. Johnson.
30] ELIMINATION. 49
Hence a-^b^i.
But the only possible case of this subsumption is
a H- 6 = i.
Also ab 4 i, which is necessarily true.
Therefore finally, a + 6 = i, is the sole deduction independent of x. Secondly, c = 0. Then cue + 65c = 0.
Hence a-^b^O, which is necessarily true.
Also ab 4 0. But the only possible case of this subsumption is aft = 0. Therefore finally, a6 = 0, is the only deduction independent of w. If the equation be written f(x) = 0, the result of the elimination becomes
/(»)/(0)=o.
These particular cases include each other. For if
cuV'{-bx = i, then ~ {ax + b^) = 0,
that is ax + 6^ = 0.
And a + 6 = i is equivalent to ofc = 0.
(6) General Equoition. The general form ^{x) —i^{x), where ^ (a?) and '^ (x) are defined in the same way as f(x) in § 29 (3), can be reduced to these cases. For this equation is equivalent to
<l>(x)^(x) + ^(x)^fr{x) = 0.
This is easily proved by noticing that the derived equation implies
that is ^ (^) 4 ^ (^) » '^ (^) 4 ^ (^)>
that is <^ (a?) = ^ (x).
Hence the equation ^ (^) = '^ (^) can be written by § 29 (4) in the form
{* (*)^ « + * » ^ (i)} ^ + {<^ (0) ^(0) + ^ (0) ^ (0)} S = 0. Hence the result of eliminating x from the general equation is
[<!> (t) ?(») + * (t) t (»)} {"^ (0) t (0) + * (0) t (0)1 = 0-
This equation includes the four equations
The reduction of the general equation to the form with the right-hand side null is however often very cumbrous. It is best to take as the standard form
ax + bx = cX'{-daD (1).
This form reduces to the form, ax + bx — c^ when c = d. For
cx + c^ = c(x-^x) = c. w. 4
50 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
The equation is equivalent to the two simultaneous equations
cue = ex, 6S = d£ ;
as may be seen by multiplying the given equation respectively by x and x.
Let the equation ax = ex he called the positive constituent equation, and the equation bx = c^ be called the negative constituent equation of equation (1).
Taking the supplements
a + x^c-i-x, b-^x = d-^x. Hence multiplying by x and x respectively
dx = cx, and bx=cbb.
So equation (1) can also be written
ax-^bx = cx-^dxi and the two supplementary forms give
aa? + fe = ca? H- dx, ax-^bx = cx-^ dx.
The elimination of x can also be conducted thus. Put each side of equation (1) equal to z.
Then (ix-j-bx = z,
cx-^-dx^z.
Hence the following subsumptions hold,
a + b^z^db; c-^-d^z^cd.
Therefore a-hb^cd,
c + d^ah.
Also similarly from the form, ax-hbx=^cx-\-dx,we find the subsumptions
d-^b^cd, o-^d^ab.
These four subsumptions contain (cf § 31 (9), below) the complete result of eliminating x from the given equation (1). The two supplementary forms give the same subsumptions, only in their supplementary forms, but in- volving no fi'esh information.
From these four subsumptions it follows that,
abed = abed = abed = abed = 0.
These are obviously the four equations found by the other method, only written in a different notation.
30] ELIMINATION. 61
From these equations the original subsumptions can be deduced. For abed = 0 can be written
oft 4 "" (^^), and therefore ab^c-\-d.
Similarly for the other subsumptions.
Also it can easily be seen that the four subsumptions can be replaced by the more symmetrical subsumptions, which can be expressed thus,
(The sum of any two coeflScients, one from each constituent equation) ^ (The product of the other two).
(7) IHscrimiruints. All these conditions and (as it will be shown) the solution of the equation can be expressed compendiously by means of certain functions of the coefficients which will be called the Discriminants of the equation.
The discriminant o{ax = cx is dc + ac. Let it be denoted by A. The discriminant of b^ = dx is bd + bd. Let it be denoted by B, Then A and B will respectively be called the positive and negative discriminants of the equation
ax + bx = cX'\'C^ (1).
Now il = ■" (ac + ac) = ac + ac,
and B = - (bd + bd) = bd + bd.
Therefore, remembering that y + ^ = 0 involves y = 0, z = 0, it follows that all the conditions between the coefficients a, b, c, d can be expressed in the form
This equation will be called the resultant* of the equation
aw + bx=:cx + dx. It can be put into the following forms,
A^B, B^A, AB=^A, BA^B, A+B = B, B+A=A.
It is shown below in § 31 (9) that the resultant includes every equation between the coefficients and not containing x which can be deduced fix)m equation (1).
The equation ax-i-bx^cx + dx when written with its right-hand side null takes the form
(8) Again let there be n simultaneous equations aiX=CiX,a^==c^,.,., On^^Cn^, and let Ai, A^, ... An be the discriminants of the successive equations respectively; then their product AiA^... An is called the resultant
* Of. Schrdder, Algebra der Logik, § 21.
4—2
52 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
discriminant of the n equations. It will be denoted by 11(^1^), or more shortly A. Similarly let there be n simultaneous equations biX = diX b^=d^,...bnCb='dnX; and let Bi, B^-.-Bn be the discriminants. Then B1B2 . . . £n is the resultant discriminant of the n equations. It will be called n (Br) or B.
The n equations diX + 61^ = CiX + diX,
On^ + bnX =CnX-\' dnX,
involve the 2n equations just mentioned and conversely.
The functions A and B are called the positive and negative resultant discriminants of these equations.
Now A = ill H- ila + . . . H- Any
5 = -Bi + JSj + . . . H- B^,
Hence AB = 1 ArB,.
Now any equation a,a! = CfX may be joined with any equation b^ = d^ to form the equation OfX + 6^ = c^a? + d^. Hence all the relations between the coefficients are included in all the equations of the type,
ArB, = 0.
But these equations are all expressed by the equation
25 = 0.
This equation may therefore conveniently be called the resultant of the n equations.
This is the complete solution of the problem of the elimination of a single letter which satisfies any number of equations.
The single equation
Ax-^Bx = 0,
is equivalent to the 71 given equations.
It must be carefully noticed that in this algebra the distinctions of
procedure, which exist in ordinary algebra according to the number of equations given, do not exist. For here one equation can always be found
which is equivalent to a set of equations, and conversely a set of equations
can be found which are equivalent to one equation.
(9) More than one Unknown, The general equation involving two un- knowns, X and y, is of the type
axy + bay + c% + d^ = exy •{-fa^ -f gxy + Aay .
This equation is equivalent to the separate constituent equations, axy = ftry, bay —foffy, etc. Let a constituent equation involving x (as distinct from ^) be called a constituent positive with respect to x^ and
30] ELIMINATION. 53
let a constituent equation involving x (as distinct ftom x) be called a constituent negative with respect to x. Thus, axy = exy^ is positive with respect both to x and y ; bay ^^fxy^ is positive Avith respect to x, negative with respect to y^ and so on.
Let A, B, Cy 1) stand for the discriminants of these constituents. Thus
A^ae-^ae, B = hf-v If, C — cg + ^, D^^dh-^dh, Then the discriminant A is called the discriminant positive with respect to x and y, B ia the discriminant positive with respect to x and negative with respect to y, and so on.
The equation can be written in the form
(ay '\-by)x + (cy + dy) x=^{ey +/y)a? 4 {gy + hy) x.
If we regard x as the only unknown, the positive discriminant is
(ay + Wi {ey -^fy) + (ay + by) (ey +/y),
that is Ay + By.
The negative discriminant is Oy + Dy.
The resultant is (Ay-^ By) ( Cy + Dy) = 0 ;
that is AGy-^BDy = 0.
This is the equation satisfied by y when x is eliminated. It will be noticed that A and C are the discriminants of the given equation positive with respect to y, and B and D are the discriminants negative with respect toy.
Similarly the equation satisfied by x when y is eliminated is
ABx-^ CDx = 0. The resultant of either of these two equations is
ABCD = 0.
This is therefore the resultant of the original equation. The original equation when written with its right-hand side null takes the form
Axy'^Bxy-\-Cxy + Ds^ = 0 (1).
Again suppose there are n simultaneous equations of the above type the coefEcients of which are distinguished by suffixes 1,2, ... n.
Then it may be shown just as in the case of a single unknown x, that all
equations of the type, ApBqCrDg = 0, hold.
Hence if A stand for AiA^ ... A^, and B for BiB^ ... B^, C for (7i(7, ... Cm
and D for AA ••• A, the resultant of the equations is ABCD = 0.
The n equations can be replaced by a single equation of the same form as (1) above.
Also the equation satisfied by x, after eliminating y only is
ABx-^CDx=^0,
64 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
where A and B are the positive discriminants with respect to x, and C and D are the negative discriminants. The equation satisfied by ^ is
ACy + BDy = 0, where a similar remark holds.
(10) This formula can be extended by induction to equations involving any number of unknowns. For the sake of conciseness of statement we will only give the extension from two unknowns to three unknowns, though the reasoning is perfectly general.
The general equation for three unknowns can be written in the form {az H- a'z) xy + {bz 4- Vz) aiy + (cz + dz) xy + {dz + d!z) xy
= (ez + ez) xy + (fz -\-fz) xy-^(gz + g'z) xy-^{hz + Kz) s^.
Then, if A =a€ + ae, A' = ae + a'e, and so on, A is the discriminant positive with respect to x, y, and z, and A' is the discriminant positive with respect to x and y, but negative with respect to z] and so on.
If X and y be regarded as the only unknowns, then the two discriminants positive with respect to x are
(az + az) {ez + e'z) + (az + a'z) (ez + e'z\
and (bz + b'z) (fz -{-fz) -h (Iz + b'z) (fz ^fz\
that is, Az->r A% and Bz + Rz.
Similarly the two discriminants negative with respect to x are
Cz + (7z, and Dz + Uz.
Hence the equation for x after eliminating y is
(Az-{-A'z)(Bz + Wz)x'\'(Cz-{-G'z)(I)z-\'D'z)x = Q,
that is (ABz -h A'Ez) x + ( CDz + CD'z) x = 0.
The result of eliminating z from this equation is
AA'BB'x + CCDD'x = 0.
Hence the equation for x after eliminating the other unknowns is of the form, Pa? + Qx = 0, where P is the product of the supplements of the discri- minants positive with respect to a?, and Q is the product of the supplements of the discriminants negative with respect to. x.
The resultant of the whole equation is
AA'MGCDD' = 0,
that is the product of the supplements of the discriminants is zero.
The given equation when written with its right-hand side null takes the form
AxyZ'\-A'xyz-\-Ba^Z'\'Fayz-\- Cxyz + Cxyz + l>c^z + Us^=^0. The same formulae hold for any number of equations with any number of variables, if resultant discriminants are substituted for the discriminants of a single equation.
31] ELIMINATION. 60
(11) It is often convenient to notice that if
^ (a?, y, z, ...) ^-^{x, y, z, ...),
be an equation involving any number of variables, then any discriminant is of the form
*lo,o,o,..J^Uo,o,.J + *Uo,o,..J^U,o,o,..J'
where % is substituted for each of the unknowns with respect to which the discriminant is positive and 0 is substituted for each of the unknowns with respect to which the discriminant is negative.
(12) The formula for the elimination of some of the unknowns, say, UjV,w,..., from an equation involving any number of unknowns, a?, y, xr, ... u,v,w,.,,, can easily be given. For example, consider only four unknowns, ^> y, ^> t, and let it be desired to eliminate z and t from this equation, so that
a resultant involving only x and y is left. Let any discriminant of the
• • ■ • .
equation be written D Q' l" *' * J , where either i or 0 is to be written ac- cording to the rule of subsection (11). The equation can be written {D(i t, I, %)xyz-hn(i i 0, i)xyz + I){i 0, i, i)xyz'^n(0, i, i, %)xyz
+ D{i 0, 0, %)xyZ'\-D (0, i, 0, %) xyz + D (0, 0, t, i) xyz + D (0, 0, 0, t) xyz] t
+ [D (%, %, i 0)xyz + 5(i, t, 0, 0)xyz-j-...]~t = 0.
Hence eliminating t, the resultant is
B(iiii)D{i^,iO)xyz + n(i,iO,{)n{iiO,0)xyz
+ 5(i,0,i,t)5(t,0,i,0)a:y^+...+5(0,0,0,i)5(0,0,0,0)Sp = 0.
Again eliminating z by the same method, the resultant is
5(i, i i %)D(i i i 0)5(t, i, 0, i)S(t, i, 0, 0)xy
4-5(i,0,t,i)5(i,0,i,0)5(i,0,0,t)B(i,0,0,0)a^
+ D (0, t/i, i) D (0, i, i, 0) D (0, t, 0, t) D (0, i, 0, 0) xy
+ 5(0, 0,i,i)5(0,0,i, 0)5(0, 0,0,t)2>(0,0,0,0)^y = 0.
It is evident from the mode of deduction that the same type of formula holds for any number of unknowns.
31. Solution of equations with one unknown. (1) The solutions of equations will be found to be of the form of sums of definite regions together vrith sums of undetermined portions of other definite regions ; for example to be of the form aH- Vift-h VjC, where a, 6, c are defined regions and Vi, Vj are entirely arbitrary, including % or 0.
Now it is to be remarked that u(b-{- o), where u is arbitrary, is as
86 1*HE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. 11.
general as vjb + VjC. For writing w = Vji + v, (6 f 6c), which is allowable since u is entirely arbitrary, then
u {h •\- c) = [vjb -{-v^^h -\-hc)] {6 + c} = Vih + Vj (6c H- 6c)
= Vj) + VjC.
Hence it will always be sufEcient to use the form t^ (6 + c), unless v^ and v^ are connected by some condition in which case vji> + v^ may be less general than w (6 H- c).
(2) ax = ca?.
Then by § 30, (7) Ax = 0.
Hence by § 26, Prop. VIII. x = Ax.
But instead of x on the right-hand side of this last equation, (x + vA) may be substituted, where v is subject to no restriction. But the only restriction to which x is subjected by this equation is that it must be
incident in A. Hence x-\-vA \& perfectly arbitrary.
|
Thus finally |
x=-vA\ |
|
jre V is arbitrary. |
|
|
(3) |
bx^d^. |
|
From subsection (2) ; |
x = uB. |
|
Hence |
x=u-^B, |
(4) Gw? + 6ac = ca; + cte ; where AB = 0.
From the equation (ix==cx,ii follows that x = uA ;
and from 6^ = d«, that x = v-^ B,
Hence uA = » + A
Therefore vA = 0.
Hence v = wA.
Finally, x = B + wA ;
where w is arbitrary.
This solution can be put into a more symmetrical form, remembering that
B-\'A=A. For x = B{w-^w)'{-wA^Bw + w(A ■^B)^wA +wB. Hence the solution can bo written
x = B-^wAA
x = A -j-wBj
31] SOLUTION OF EQUATIONS WITH ONE UNKNOWN. 57
Or a; = wA + wBy^
x = wA+wB.)
The first form of solution has the advantage of showing at a glance the terms definitely given and those only given with an undetermined factor.
(5) To sum up the preceding results in another form: the condition that the equations ax^cx, bx^da may be treated as simultaneous is
15 = 0.
The solution which satisfies both equations is
x = B + uA.
The solution which satisfies the first and not necessarily the second is
x^uA.
The solution which satisfies the second and not necessarily the first is
x = uB, that is x==u-{-B.
In all these cases u is quite undetermined and subject to no limitation.
(6) The case cw? + te = c, is deduced from the preceding by putting d = c.
Then A — ac-\-ac, B = bc-\-bc.
The solutions retain the same form as in. the general case.
The relations between a, b, c are all included in the two subsumptions
a-\-b^c^ab. The case oo? + 6S = 0 is found by putting c == d = 0.
The equation can be written
(M? H- te = Oa? + 0^.
The positive discriminant is aO + aO, that is a, the negative is 6. The resultant is oi = 0. The solution is
a? = 6 -I- lid,
(7) The solution for n simultaneous equations can be found with equal ease.
Let X satisfy the n equations
OiX + biX = CiX + d^, Oja? + ftjS = Cgfl? + d^,
ttnX+bnX = Cf^-i-dnX,
Then x satisfies the two groups of n equations each, namely
diX^CiXy a^=^c^ ... Of^^CfiiX] and biX = diX, b^ = d^ . . . 6„a = d^ac.
= wA + wB,\ = wA + wB J
58 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. H.
From the first group
Hence x = uA ; where u is not conditioned. Similarly from the second group
X = ViBi = v A = . . . = vJBn . Hence x = vB, a; = » + jB ; where v is not conditioned. Therefore uA = v + B.
Hence vA = 0, that is v = wA.
So finally the solution of the n equations is
x = B-{- wA =
x = A-{- wB
The group Oja? = CiX, eye = c^, etc. can always be treated as simultaneous, and so can the group of typical form 6^ = d^^.
The condition that the two groups can be treated as simultaneous is
25 = 0.
(8) It has been proved that the solution B + uA satisfies the equation, AxA- Bx=sO, without imposing any restriction on u. It has now to be
proved that any solution of the equation can be represented hy B-\-uA, when u has some definite value assigned to it.
For if some solution cannot be written in this form, it must be capable
of being expressed in the form mB + wA •{-nAB.
But Ax^O, and AB = 0, hence, by substituting for x its assumed form, nAB = 0. Thus the last term can be omitted.
Again, Bx=0; and AB = 0, hence B{m-{-B)(w + A) = 0; that is mwB = 0. Hence m=p(w-\-B)y and therefore m=p + wB. Therefore the solution becomes
x = mB-^wA = (p-j-wB)B-\-w{A-^B),
-pB-^-B-^-wA^B-^wA,
Thus the original form contains all the solutions.
(9) To prove that the resultant AB = 0, includes all the equations to be found by eliminating x from
Ax + Bx = 0.
For a? = £ H- wA satisfies the equation on the assumption that AB = 0, and without any other condition.
Hence AB is the complete resultant.
It easily follows that for more than one unknown the resultants found in § 30 are the complete resultants.
32] SOLUTION OF EQUATIONS WITH ONE UNKNOWN. 69
(10) Subsumptions of the general type
aa + bx^cx + d^
can be treated as particular cases of equations.
For the subsumption is equivalent to the equation
cfl? + d» = (ca? + dS) (ax + 6^) = dcx + bdx.
Hence A^ac + c" (ac) = ac + c==a + c,
-B=6d + d-(6d) = 6d + d = 6 + d,
A ^c{a + c) = ac,
B = d(b-\-d)^bd.
Therefore the resultant -4-8 = 0 is equivalent to fl^crf = 0. This is the only relation between the coefficients to be found by eliminating x. The given subsumption is equivalent to the two subsumptions
ax ^ ex J bx^dx;
that is, to the two equations
ex = acx, dac = bdx.
The solution of aa^cx is x = uA = w(a + c).
The solution of bx^dx \b x = U'\-B = u-\-bd, The solution of ax + bx^cX'\-dx
is a? = -B + t^il =• Si + w (a H- c)
= uB + uA = M (a + c) + ubd.
The case of n subsumptions of the general type with any number of unknowns can be treated in exactly the same way as a special type of equation.
32. On Limiting and Unlimiting Equations. (1) An equation
^(^> y» ^> ••• 0 — V^C^* y> ^> ••• 0 involving the n unknowns x, y, z, ... t is called unlimiting with respect to any of its unknowns (x say), if any arbitrarily assigned value of x can be substituted in it and the equation can be satisfied by solving for the remaining unknowns j/y z, ... t; otherwise the equation is called limiting with respect to x. The equation is. unlimiting with respect to a set of its variables x, y, z, ..., i{ the above property is true for each one of the unknowns of the set. The equation is unlimiting with respect to all its unknowns, if the above is true for each one of its unknowns. Such an equation is called an unlimiting equation.
The equation is unlimiting with respect to a set of its unknowns simultaneously f if arbitrary values of each of the set of unknowns can be simultaneously substituted in the equation.
60 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
It is obvious that an equation cannot be unlimiting with respect to all its unknowns simultaneously, unless it be an identity.
(2) The condition that any equation is unlimiting with respect to an unknown x is found from § 30 (10). For let P be the product of the supple- ments of the discriminants positive with respect to x and Q be the product of the supplements of the discriminants negative with respect to x. Then the equation limiting the arbitrary choice of x is, Px -f (2^ = 0. Hence if the given equation be unlimiting with respect to x, the equation just found must be an identity. Hence P= 0, Q = 0.
(3) The condition that the equation be unlimiting with respect to a set of its unknowns is that the corresponding condition hold for each variable.
(4) The condition that the equation is unlimiting with respect to a set x,y,z, ... of its unknowns simultaneously is that the equation found after eliminating the remaining unknowns t^u^v, ... should be an identity. The conditions are found by reference to § 30 (12) to be that each product of supplements of all the discriminants of the same denomination (positive or negative) with respect to each unknown of the set, but not necessarily of the same denomination for different unknowns of the set, vanishes.
(6) Every equation can be transformed into an unlimiting equation. For let the equation involve the unknowns x,y,Zj ... t: and let the resultant of the elimination of all the unknowns except x be, Px + Qaj = 0.
Then a? = Q + uP, and if u be assigned any value without restriction, then X will assume a suitable value which may be substituted in the equation previous to solving for the other unknowns. Thus if all the equations of the type Px+Qx = 0, be solved, and the original equation be transformed
by substitution of, x=^Q-\-uP, y=8 + vR, etc., then the new equation between t^, v, ... is unlimiting.
(6) The field of an unknown which appears in an equation is the collection of values, any one of which can be assigned to the unknown consistently vrith the solution of the equation. If the equation be un- limiting with respect to an unknown, the field of that unknown is said to be unlimited ; otherwise the field is said to be limited.
Let the unknown be x, and with the notation of subsection (5), let the resultant after eliminating the other unknowns be Px + Qx = 0. Then
a? = Q + uP. Hence the field of x is the collection of values found by substituting all possible values for u, including % and 0. Thus every member
of the field of x contains Q; and P contains every member of the field, since PQ = 0. The field of x will be said to have the minimum extension Q
and the maximum extension P.
33. On the Fields of Expressions. (1) Definition. The 'field of the expression if){x, y, z, ... t)* will be used to denote the collection of
33] ON THE FIELDS OF EXPRESSIONS. 61
values ^hich the expression ^ (x, j/y Zy ... t) can be made to assume by different choices of the unknowns Xyj/.z, ... t If ^ (x, y, z, ... t) can be made to assume any assigned value by a proper choice of x, y, z, ... t, then the field o{ <f>{x, y, Zy...t) will be said to be unlimited. But if ^(a;, y,Zy...t) cannot by any choice of a?, y, Zy ...t, be made to assume some values, then the field of <l> (xy y, z, ... t) will be said to be limited.
(2) To prove that
axyz ... t + bxyz ... i + ...kxyz ... t,
is capable of assuming the value a + 6 + ... + A. This problem is the same as proving that the equation
axyz ...t-\-bxyz ...t-\-... •\-kxyz ...? = a + 6 + ...+&,
is always possible.
The discriminants (cf. § 30 (11)) are
il=a + a6...&, 5=6 + d6...^, ...K^k-\-ah ,..Tc. Hence
il =a(6+c + ...&), -B = 6(a + c + ... +A?), ...Z^=i(a + 6 + c + ..•)•
Hence the resultant AB ... if = 0 is satisfied identically.
It is obvious that each member of the field of the expression must be incident in the region a + 6 + c+...+i: for a + 6 + c + ... + A is the value assumed by the expression when i is substituted for each product xyz...t,
xyz ... ty ... xyz ...t. But this value certainly contains each member of the field.
(3) To prove that any member of the field of
axyz ... t-\-bxyz ...i + ... +J(^cyz ...t
contains the region abc ... A?.
For let <f>(Xy y, z,... t), stand for the given expression. Then the region
containing any member of the field of 0 (Xy yyZy...t) by the previous subsection
is a + E + c+... + fc. Hence the region contained by any member of the field of <l>(XyyyZ, ...t) is abc.k. Hence combining the results of the previous and present subsections
aH-6 + c + ... + A^^(a?, y, z, ...t)^abc ...k.
The field of (f>(x,yyZ... t) will be said to be contained between the maximum extension a + 6 + . . . + A;, and the minimum extension ah ... k. '
(4) The most general form of p, where
a + 6 + c + ...■\-k^p^ahc...ky is p^ahc... A; + w(a + 6 + c + ... +fc).
In order to prove that the fields of
^{Xy y, Zy...t) and abc ... A; -I- w(a + 6 + C + ... A;),
62 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
are identical, it is necessary to prove that the equation
<l>{x, y, z, ... t) = abc ... A? + w(a+ 6 + c + ... +i),
is unlimiting as regards u.
The equation can be written
axyz ...t-\- hxyz ... it + ... •\-kxyz ... t=ahc ...fci6 + (a + 6 + c + ...i)w.
The discriminants positive with resjpect to u are (cf. § 30 (11))
a(a + 6 + c.i.+A;) + a. ahc ...k, that is, a + abc... k,
and b-^dbc...k, c + dbc ...k, ...k + dbc ...k.
Their supplements are ^
a(6 + c + ...+A;), b(a + c-\-... + k), c(a-f 6 + .,. +i), ... i(a + 6 + c + ...).
Hence the product of the supplements is identically zero. Similarly the discriminants negative with respect to u are
abc...k + d(a + b + ...-¥k\ abc ...k + b{d i-b+ ... +i),
and so on. Their supplements are a(b'\-c + ... +k), and so on. The product of the supplements is identically zero. Hence (cf. § 32 (2)) the equation is unlimiting with respect to u.
Thus* the fields of ^ (a?, y,^,...^) and of a6c ...A; + w (a + 6 + c+... + A:) are identical and therefore without imposing any restriction on u we may write
^(^> y> z,...t) = dbc ...k + u(a-\-b + c+ ...-{-k).
(5) The conditions that the field of ^ (x, y,z,...t) may be unlimited are obviously a6c... A; = 0, a + 6 + c + ... -f A:=t,
The two conditions may also be written
ahc ...k = 0 = dbc . . . i\
(6) Consider the two expressions
axyz ... f + bxyz ...i+ ... -{-kxyz ...i,
and c^uvw ...p'\-biuvw ...p+ ... hiuvw ...p,
not necessarily involving the same number of unknowns. Call them ^ (^, y, ^ . . . 0 8^d "^(^j v,w...p). The conditions that the field o{ <l>{x,y,z... t) may contain the field of -^(w, v,w...p), i.e. that all the values which -^ may assume shall be among those which 0 may assume, are ahc... k 4 (hbiCi ... Aj, and a + 6 + c...+i^ai + 6i + Ci+...+Ai. The two conditions may also be written
abc ...k^OfibiCi ... ^,
dbc ... k^dibiCi ...hi.
* Cf. SchnSder, Algehra def Lo^k^ Lecture 10, § 19, where this theoren) is dedaoed b^ another proof.
33] ON THE FIELDS OF EXPRESSIONS. 63
(7) The conditions that the fields of <l>(x,y,z ...t) and -^(m, v, u;...|)) may be identical are obviously
abc ... k = (iibiCi ... Ai, cbbc ... k^OribiCi ... A].
(8) To find the field o{f{x, y,z»..t), when the unknowns are conditioned by any number of equations of the general type
<l>r(^, y, ^ ... t)^^lrr{x, y, z... t).
Write p = /(a?, y,z ...t); and eliminate x,y,z ..,t from this equation and the equations of condition. Let the discriminant of the typical equation of condition positive with respect to all the variables be Ar, let the dis- criminant positive with respect to all except t be Br, and so on, till all the discriminants are expressed. Then the resultant discriminants (cf. § 30 (8) and (9)) of these equations are A=^H (Ar), -8 = 11 (Br), etc
Also let f(x, y, z .,.t) be developed with respect to all its unknowns, so that we may write
p=^axyz ...t + bxyz ... i-\- ... .
The discriminants of this equation are pa+p^, pb-hpb, etc. Hence the resultant after eliminating x,y, z ...tis
'{(pa+pa)A}-{(pb-\-pb)B}...=^0,
that is, {|)(d + Z) + p(a + Z)}{p(6 + 5)+j5(6 + 5)}...=0.
Hence |>(a + Z)(6 + £) ...+p(a + Z)(6 + £) ... = 0.
Thus (cf. § 32 (6)) the field of |> is comprised between
(a + Z)(6 + 5)... and aA-¥bB+.... But apart firom the conditioning equations the field of p is comprised be- tween abc . . . and a + 6 -h c + . . . . Thus the effect of the equations in limiting the field of j9 is exhibited.
The problem of this subsection is Boole's general problem of this algebra, which is stated by him as follows (cf. Laws of Thought, Chapter ix. § 8) : 'Given any equation connecting the symbols x,y,...w, z,..., required to determine the logical expression of any class expressed in any way by the symbols a?, y... in terms of the remaining symbols, w, z, etc.' His mode of solution is in essence followed here, w, z, ... being replaced by the coefficients and discriminants. Boole however did not notice the distinction between expressions with limited and unlimited fields, so that he does not point out that the problem may also have a solution where no equation of condition is given.
A particular case of this general problem is as follows :
Given n equations of the type ayX + bfX==CrX + d^, to determine z, where z is given hy z^ex +fx.
Let the discriminants of the n equations be A and B, those of the equation which defines z are ez + ez, fz -{-fz.
64 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
Hence the resultant is "" (eAz + eAz) ~ {/Bz -\-fBz) = 0, that is (ef+fA '^eB)z'^ (?/"+/4 + cJS) 0 = 0; where AB = 0.
Hence z = (ef+fA ■\-eB)-\-u (eA -h/B).
Another mode of solution, useful later, of this particular case is as follows :
The solution for x of the equations is a? = £ -f vA, x=^A + vB.
Substitute this value of ^ in the expression for z.
Then z^eB 4-/2 + veA + lfB^{eA +fA) v + (fB + eS) v.
It is easy to verify by the use of subsection (7) that this solution is equivalent to the previous solution.
(9) An example of the general problem of subsection (8), which leads to important results later (cf. § 36 (2) and (3)), is as follows.
Given the equation Aon/ + Bxy k- Gxy + Dxy = 0, to determine ay, ay,
Put z = ay, then by comparison with subsection (8) a = i, 6 = 0 = c = d. Hence (a + A)(b + B)(c-\-C)(d-\-D) becomes BCD, and aA-\-bB + cC+ dD becomes A.
Thus, remembering that ABCD^ 0,
xy = BOD-^-uA = A (BGD + u).
Similarly a!y = ACD + uB = B{ACD'^ u),
ay^ABD-\-uC^C{ABDJfu), xy=-ABC + uD^D{ABd-hu).
Also xy + xy^BC +u{A+D) = (A-¥D){BC ■{-u],
ay + xy:r=A~D-\-u(B + C)^(B-\-C){AD + u}.
It is to be noticed that the arbitrary term u of one equation is not identical with the arbitrary term u of any other equation. But relations between the various w's must exist, since xy ■{- xy ■{■xy -{-xy = i.
(10) It is possible that the dependence of the value of an expression f{x, y,z ...t) on the value of any one of the unknowns may be only apparent. For instance if f{x) stand for x + x, then /(a?) is always % for all values of x.
It is required to find the condition that, when the values of y, ^, ... ^ are given, the value of f(x, y, z ...t)iB also given.
For letf(x, y, z .., 0 = ^i + ^f%* where /i and /, are functions of y, z ... t only. Then on the right-hand side either i or 0 may by hypothesis be put for X without altering the value of the function.
Hence /i =/(a?, y...t)=fi.
Thus /i =/j is the requisite condition.
34] ON THE FIELDS OF EXPRESSIONS. 65
Let /(a?, y, z ...t)he written in the form
w{ayz ...t + byz.., i + ...) + x(a'yz ...t -\-Vyz ..A-^ ...), then the required condition is a = a', 6 = V, etc.
34 Solution of Equations with more than one unknown. (1) Any equation involving n unknowns, x, y, z ,..r,s,t can always be transformed into an equation simultaneously unlimiting with respect to a
set of any number of its unknowns, say with respect to x,y, z For let
Pi be the product of the supplements of the discriminants positive with respect to a?, and Qi the product of the supplements of those negative with respect to x. Then (cf § 30 (11)) the resultant after the elimination of all unknowns except x is,
Pl30 + QiX = 0.
•Hence we may write, x ^ Q^^ P^Xi^ Q^-\- P^Xi, where x^ is perfectly arbitrary. Substitute this value of x in the given equation, then the transformed equation is unlimiting with respect to its new unknown Xi.
Again, in the original equation treat x a& known, and eliminate all the other unknowns except y.
Then the resultant is an equation of the form
{Rx + 8x)y + {Tx +Ux)y=- 0,
where ii, S, T, U can easily be expressed in terms of the products of the supplements of discriminants of the original equation. The discriminants in each product are to be selected according to the following scheme (cf. §30(12)):
J2, s, r, u
+, -, +, - +, +, -, -
X
y
Now substitute for x in terms of a?i, and the resultant becomes
P^y + Q^y = 0, where P, and Q, are functions of x^.
Solving, y = Qs + P>ys = Qiy^ + P^y^ ;
where y, is an arbitrary unknown.
If this value for y be substituted in the transformed equation, then an equation between a^i, y,, j? ... r, », t is found which is unlimiting with respect to Xy and y, simultaneously.
Similarly in the original equation treat x,yaa known, and eliminate all the remaining unknowns except z : a resultant equation is found of the type
{Vjxy^ V^y + V^xy -\-V^y) z + (Wixy -\- W^ + W^y ■\- W^y)z = 0; W. 5
66 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
where the F's and W'& are products of the supplements of discriminants selected according to an extension of the above scheme. Now substitute for X and y in terms of Xi and y,, and there results an equation of the
where P, and Q, contain Xi,y^,
Solving, g^Q^ + P,^, = Q,e, + P, j,,
where 5, is an arbitrary unknown. Then substituting for z, the transformed equation involving Xi, y^, ^, ... r^Syt is unlimiting with regard to x^, y^, z^ simultaneously.
Thus by successive substitutions, proceeding according to this rule, any set of the unknowns can be replaced by a corresponding set with respect to which the transformed equation is simultaneously unlimiting.
(2) If this process has been carried on so as to include the n — 1 un- knowns x,y,z,..8, then the remaining unknown t is conditioned by the
equation Pn^ + Qn^ = 0; where Pn and Q^ involve a?!, y, ...««-! which are unlimited simultaneously.
Solving for f, ^ = Qn + PJn = Qnin + ?n^ ;
where t^ is an arbitrary unknown.
Thus the general equation ia solved by the following system of values,
where a?i, y, ... *„ are arbitrary unknowns.
(3) The generality of the solution, namely the &ct that the field of the solution for any variable is identical with the field of that variable as implicitly defined by the original equation, is proved by noting that each step of the process of solution is either a process of forming the resultant of an equation or of solving an equation for one unknown. But since the resultant thus formed is known to be the complete resultant (cf § 31 (9)), and the solution of the equation for one unknown is known to be the complete solution (cf. § 31 (8)), it follows that the solutions found are the general solutions.
It follows from this method of solution that the general solution of the general equation involving n unknowns requires n arbitrary unknowns.
(4) Consider, as an example*, the general equation involving two un- knowns,
axy + bay + (% + ds^ = exy -¥ fxy ■{• gxy + h^.
Let A, B,C,Dhe its discriminants.
Then x = CD + -{lB)x, = ~(AB)x^-\- CDx,,
* Cf. Schroder, Algebra der T^flik, $ 22.
35] SOLUTION OF EQUATIONS WITH MORE THAN ONE UNKNOWN. 67
Also {Ax'\'Cx)y-\- (Bx + Dx) y = 0.
Hence y=Bx + lJx-\- (Ax + Cx) y^^(Ax'^Cx)y^'\-(Bx + 3x)y^ = {(A+ABC)x,-^(G-i-ACD)x,]y,+ l(AB'\-AB~D)x,+(GD + WD^^
As a verification it may be noticed that the field of y as thus expressed is contained between A-\-C and BD. This is easily seen to be true, re- membering that ABCD = 0,
(5) The equation' involving two unknowns maybe more symmetrically solved by substituting (cf. § 32 (5))
x = CD + -{AB)u = CDu'^'(AB)u,
y-=BD+-(AC)v=-BDv'\--(AC)v. Then u and v are connected by the equation*,
A BGuv -f ABDuv + A CDuv + BCDuv = 0.
This is an unlimiting equation: thus either t^ or v may be assumed arbitrarily and the other found by solving the equation.
Thus V = ABDu + BCDu + -(ABCu + ACDu)p,
or u = AGDv + BCDv + -(ABCv + ABDv) q ;
where p and q are arbitrary.
36. Symmetrical Solution of Equations with two unknowns. (1) Schroderf has given a general symmetrical solution of the general equation involving two unknowns in a form involving three arbitrary un- knowns.
The following method of solution includes his results but in a more general form.
(2) Consider any unlimiting equation involving two unknowns. Let Ay By 0, D be its four discriminants. Then the equation can be written in the form
Axy -\- Bxy^ Cxy -\- Dxy-=0 .....(a).
Now put X = a^uv + bjuv + Ciuv + diuv (13),
y^a^uv + biUv + c^uv + diXlv (7).
Since the equation (a) is unlimiting (cf. § 32 (2)),
* This equation was pointed oat to me by Mr W. £. Johnson and formed the starting-point for my inyestigations into limiting and unlimiting equations and into expressions with limited and unlimited fields. As far as I am aware these ideas have not previously been developed, nor have the general symmetrical solutions for equations involving three or more unknowns been previously given, of. §§ 35 — 37.
t Alffihra der Logik, Lecture xn. § 24.
5—2
68 THE ALQEBBA OF SYMBOLIC LOGIC. [CHAP. II.
Also since the fields both of x and y are unlimited, then (cf. § 33 (5))
aJbiCidi = 0 = OibjCidi = CLJb^c^ = a^biC^di. Substitute for x and y from (fi) and (7) in (a), and write ^ (p, q) for the
«
expression
Apq + Bpq + Cpq + Dp^.
Then the equation between u and v is found to be
^(oi, (ii)uv'^(l>{bi, b^)uv-i-<l>(ci, Ci)uv + <f>(di, di)uv = 0 (S).
Equation (S) is the result of a general transformation from unknowns x and y to unknowns u and v,
(3) If the forms (J3) and (7) satisfy equation (a) identically for any two simultaneous values of u and v, then
Thus if the pairs (a,, Oj), (61, 6j), (Ci, Cj), (di, dj) be any pairs of simul- taneous particular solutions of the original equation, then (/3) and (7) are also solutions.
(4s) Assuming that (oj, a,) ... (di, d,) are pairs of simultaneous particular solutions of (a), it remains to discover the condition that the expressions (/3) and (7) for x and y give the general form of the solution.
This condition is discovered by noting that the solution is general, if when X has any arbitrarily assigned value, the field of y as defined by equation (a) is the same as the field of y as defined by (7) when u and v are conditioned by equation (fi).
Now equation (a) can be written
(Ja?+ Cx)y-\-(Bx+^x)y-0.
Hence the field of y as defined by (a) is contained between the maximum extension (cf. § 32 (6)) Ax + Cx and the minimum extension Bx + Dx.
Now let Axy Bx, Gxy Dx be the discriminants of (/8) considered as an equation between u and v. Then
ila5 = aia; + aiS, Bx = b^x-\-ZxX, Oaj = <a« + c,^, Dx^^diX + diX.
But by § 33 (8) the field of y as defined by (7), where u and v are conditioned by (fi) is contained between the maximum extension
diAx + b^Bg + c^Gx + d^Dx, and the minimum extension
(a, + Ax)(b,'\-Bx){c, + Cx)(d,^-I>x): that is, between the maximum extension
(oiOj + bibi + 0108 + didj) x + (OiOj + bib^ + Cip, + did,) x, and the minimum extension
{oi + a,) (61 + 6s) (c, + Cj) (di + dj) a? + (oj + a,) (61 + 6,) (Ci -^ c») (^i + d,) ag.
35] SYMMETRICAL SOLUTION OF EQUATIONS WITH TWO UNKNOWNS. 69
If the field of y be the same according to both definitions, then
ctiaj + 6i6a + (hPt + diCk^ A (e),
aia, + 5A + CiC + di(/, = 0 (?)»
(ai + a,)(6i'f6i)(ci + c,)(^ + cZ,) = -B (i;),
(ai + a,)(ti + 6a)(C| + c)(d, + cZ,) = B (0).
These equations can be rewritten in the form
ctiO, + ti6, + CiCa + did^ = il ( €i ) ,
OiO, + 6i6j + CiCa + didj = 0 ((:,),
Mj + 6i6a + CjCj + dida = J8 (17,),
OiOj + ftiSj + CiCj + didj^Z) (^1).
It follows from their symmetry that if y be given, the field of a; as defined by (0) and conditioned by (7) is the same as the field of ^ as defined by (a).
By adding €1 and rju di+hi + Oi-^di — A +B. Hence Oiii^dli = AB,
By adding (CO and (0,),
ai + 6i + Ci + di = (7 + Z).
Hence OibjCidi = CD,
Similarly, d,6jCa^ = -Z(7, ajb^o^^^BD.
Thus if the conditions between A, B,G, D o{ subsection (2) are fulfilled, then the conditions between Oj, b^, C], di and a,, ft,, c,, c2, of subsection (2) are also fulfilled.
Hence finally if (Oi, Oj), (61, 6,), (ci, c,), (dj, d,) be any pairs of simul- taneous solutions of (a) which satisfy equations (€]), (fi), (971), (0i), then the expressions (fi) and (7) for x and y form the general solution of equation (a).
(5) Now take one pair of coefficients, say Oi and a,, to be any pair of particular simultaneous solutions of the equations
Axy + Bxy -^ Cxy + Dxy =^0 (/c),
and xy^A (X).
These two equations can be treated as simultaneous ; for the discriminants of (X) are A^ A, A, A. Hence the complete resultant of the two equations
is _____
4(J8 + il)(a+il)(2) + il) = 0,
that is ABCD^Q]
and this equation is satisfied by hypothesis. Thus {k) and (X) can be combined into the single equation
70 THE ATX3EBBA OF SYMBOLIC LOGIC. [CHAP. II.
that is, since AB = AG = 0,
ixy + Aary + Axy ■\-{D-\- A)xy = Q.
Any solution of this equation gives xy = A^ osy^B, xy^C, ay:^D', and hence any solution is consistent with equations (cj), (fi), (rfi\ (tfj).
This equation is a limiting equation. By § 34 (5) it can be transformed into an unlimiting equation.
Put x = A-\-k, y^A'\-l.
Then the equation becomes
Akl + ADkl-=0.
Let another pair of the coefficients, say bi and 6,, be choseo to be any particular solutions of the equations
Aay + Bay -f Cxy + Dxy = 0, a/y^B,
These equations can be treated as simultaneous; and are equivalent to
the single equation _ _
Bxy-^Bxy + {C'\-B)xy + Bxy = 0.
Any solutions of this equation give xy ^ A, aiy = B, xy ^ C, xy ^ D,
To transform into an unlimiting equation, put a?=JB + m, jr = J8 + n. Then the equation becomes
Bmn + BCmn = 0.
Let another pair of the coefficients, say Ci and c,, be chosen to be any particular solutions of the equations
Axy + Bxy + Cxy + Dxy = 0, xy = G.
These equations can be treated as simultaneous; and are equivalent to the single equation
Gxy-\-(B-\- G)aJy-\-Cxy + Gxy = 0.
Any solutions of this equation give xy ^ A, ay ^ B, xy = G, xy ^ D.
To transform into an unlimiting equation, put x = G+p, y^G+q, Then the equation becomes
BCpq + Gpq = 0.
Let the last pair of coefficients, namely di and d,, be chosen to be any particular solutions of the equations
Axy + Bay + Gxy + Dxy = 0, xy = D,
These equations can be treated as simultaneous; and are equivalent to the single equation
(A + D)xy + Dxy + Dxy + Dxy = 6.
/
35] SYMMETRICAL SOLUTION OF EQUATIONS WITH TWO UNKNOWNS. 71
Any solutions of this equation give
ay^A, xy:^B, xy^G, xy^D.
To transform into an unlimiting equation, put x^^D-^Vy y = D-h8, Then the equation becomes
ADr8'hDr8 = 0.
If the coefficients Oi, as..<(2,, have these values, then the equations (e), (?)» (v)j (^) are necessarily satisfied.
Hence finally we have the result that the most general solution of the unlimiting equation
Axy-\- Bay + Cxy + Dxy = 0, can be written
x = {A-\-k)uv + (B + m)uv + Cpav + Drav,
y'={A-¥l)uv + Bnuv + (C-\-q)uv -\- Dsuv ;
where u, v are arbitrary unknowns, and k and I, m and n, p and q, r and 8, are any particular pairs of simultaneous solutions of
Akl +15101 = 0^ 5mn + 50mr? =0, Cpq-hBCpq^O, T)r8 + ADf8 = 0.)
Let these equations be called the auxiliary equations. The auxiliary equations can also be written,
A<l>{kj)^0, 5^(m, n) = 0, G<l>(p,q)^0, 5<^(r, 5) = 0.
(6) As an example, we may determine k, I, m, n, p, q, r, 8 so that the general solution has a kind of skew symmetry; namely so that x has the same relation to il as ^ has to D,
Thus put A: = 0, i = Z; m = JB, n = (7; q = G,p = B; « = 0, r = 5. These satisfy the auxiliary equations. Hence the general solution can be written,
remembering that BG = JB, GB = C,
X =s Auv + Buv + Guv, x=^Auv + Buv + Guv + uv,
y = uv + Buv + Guv + Duv, y = Buv + Guv -f Duv.
Again, put A; = t, 1 = 0] m=0, w = i; |) = 0, 5^ = t; r = i, « = 0. The solution takes the skew symmetrical form
a? = tiv + Buv + Cttv, T = Buv + Cut; + uv ;
y = ilttt; + UV + Duv, y = ilw + uv + Diiw.
As another example, notice that the auxiliary equations are satisfied by k ^ Wy I ^w, m ^ Wy n =sWy p = w, q = w, r = w, 8 = w.
72 THE ALQEfiRA Olf SYMBOLIC LOGIC. [CHAP. IL
Hence the general solution can be written
x — {A 4 w)uv + {B + w)uv + Cwuv + Dviniv,
y=^{A-\-w)uv-\- Bwuv + (0 + w) uv + Dwuv \
where u, v and w are unrestricted, and any special value can be given to w without limiting the generality of the solution.
(7) The general symmetrical solution of the limiting equation can now be given. Let Aay + Bivy + Gxy + Dxy = 0 be the given equation.
By§34(5),put x = GD-\-{A '^B)X, y = BD-\-(A + C)Y;
where X and Y are conditioned by
ABGXY+A~BDXY+ ACDXY+ BCDXY^ 0.
The general symmetrical solution for X and Y is therefore by (5) of this section,
X ^ {A -^-B + 0 '\- k)uv -\- {A + B -\-I) + m)uv + ACDpav + BCDfuv,
F=(il + 5 + C+Z)wi; + ABDnuv + (A + C + D + q)uv + BCDsuv ;
where k,l; m,n; p, q; r, 8 are any simultaneous particular solutions of the auxiliary equations
ABCkl + ABCDkl^O,\ ABDmn + ABGDmh = 0, ACDpq^-ABGDpq^O, BGDrs + ABGDfs = 0. j
(8) As a particular example, adapt the first solution of subsection (6) of this section. Then a general solution of the equation is
a? = C5 -f (il + BG) uv -\-{B + AD)uv-\- AGD uw,
y = B5 + (4 + C) w + ABDm + {G + AD)uv + BGDuv.
(9) If a number of equations of the type,
V^i (^» y) = xi (^> y\ V^> (^» y) = x» (^> y)» etc.,
be given, then (assuming that they satisfy the condition for their possibility) their solution can be found by substituting their resultant discriminants (cf. § 30, (8), (9)) for the discriminants of the single equation which has been considered in the previous subsections of this article,
(10) The symmetiical solution of an equation with two unknowns has been obtained in terms of two arbitrary unknowns, and of one or more unknowns to which any arbitrary particular values can be assigned without loss of the generality of the solution. It was proved in § 34 (3) that no solution with less than two unknowns could be general. It is of im- portance in the following articles to obtain the general symmetrical
■(^)-
36] Johnson's method. 73
solution with more than two arbitrary unknowns. For instance take three unknowns, u, v, w (though the reasoning will apply equally well to any number). Let the given unlimiting equation be
Aon/ + Bay -^ Cxy -\- Dxy = 0 (a).
Put
a? = tti uvw + bi uvw + Ci uvw + di uvw ^
+ a^uvw + hlmw + c^uvw + diuvw,
y = a2UVW'\- +d^uvw
'\-(iiuvw-\- •\-d^umv.
Consider a? as a known, then the maximum extension of the field of y as
defined by (a) is -4 a? + Cfe, and its minimum extension is Sr + Dx,,
Also the maximum extension of the field of y as defined by {fi) is
SaiOs.ar+SaiOs.^c, and its minimum extension is n(«i + ai)a?4-n(ai + aa).;».
Hence, if {fi) is the general solution of (a), the following four conditions
must hold
SaiOasil, SoiOjssJS, XdjOi^^C, Soidi^D.
Also Oi, a,; 6i, 6a ••• d^\ d^\ must be pairs of simultaneous solutions of the given equation (a).
36. Johnson's Method. (1) The following interesting method of solving symmetrically equations, limiting or unlimiting, involving any number of unknowns is due to Mr W. E. Johnson.
(2) Lemma, To divide a + 6 into two mutually exclusive parts x and y, such that x^a and y^h.
The required conditions are
These can be written xy + ctary + hxy + (a + 6) aiy = 0.
Hence by § 34 (5), a? = a6 + aw = a (6 + w),) x, v
y=a6 + 6t; = 6(a + t;);j ^ ^'
where a6 (t^t; + uv) = 0 (2).
Solving (2) for v in terms of w, by § 31 (5), v = a&u + (a + 6 + u)t(;.
Substituting for v in (1) and simplifying,
a? = a (6 + w), y = 6 (a + u).
(3) Let the equation, limiting or unlimiting, be
Axy •}- Bay ■{- Cxy ■{- D^ ^ 0 (3).
The resultant of elimination can be written A + B + C + D^^i. Also xy -^r^ and ay + xy are mutually exclusive, their sum
= t = 4 + J8 + (7 + A and obviously fi'om the given equation xy-{-xy^A+D,a!y + xy:^B + C,
74 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
Hence by the lemma
xy + c^==(A-{D)(BG'^u\ xy + ccy = {B + C) (AD + u) (4).
The course of the proof has obviously secured that u does not have to satisfy some further condition in order that equation (4) may express the full knowledge concerning xy-\-xy and xy -f %, which can be extracted from equation (3).
Also, as an alternative proof of this point, § 33 (9) secures that equation (4) represents the complete solution for these expressions.
Again, by equations (4) xy + xy^^ BC + u, hence xy 4 BG + u.
Also by equation (3), xy 4 A. Hence by § 28, Prop. XIV., xy^A (BC + u).
Similarly ^^D (BC! + u).
Therefore by the lemma and equations (4) and simplifying
xy^A(BG^-u)(D-^p\\
c^ = D(BC-\'u)(A+p).\ ^^^•
Also, as before, it follows that equations (5) are a complete expression of the information respecting xy and sty to he extracted from equation (3).
Similarly ay=-B (15 + u)(d+ g),
xy=^C(AD+u)(B + q)
Adding appropriate equations out of (5) and (6),
x^A(BO + u)(5+p)+B (AD + u)(C + q\) y = A(BC^u)(D+p) + C(AD + u)(B + q).\
This symmetrical solution with three arbitraries is the symmetrical solution first obtained by Schroder (cf. loc. cit.).
(4) A simplified form of this expression has also been given by Johnson. For A (BC + u)(D + p)=^A (BCu + u) (D+p\
and B (AD -\-u)(^ + q) = B (ADu + u) (C' + g).
Hence
a; = u {AD + Ap + BAD(G + q)] + u {ABG(D +p) + JSO + Bq] =^u{AD + Ap+BD(G + q)]'\'u{AD(D + p) + BG+Bq\ = A(d+u)(D + p) + B(D + u)(0'\'q).
Similarly y^A(B + u)(B+p) + C(D'hu)(B + q).
(5) This method of solution can be applied to equations involving any number of unknowns. The proof is the. same as for two unknowns, and the headings of the argument will now be stated for three unknowns.
Consider the equation
Axyz + Bxyz + Gxyz + Dxyz + A'xyz + B'afyz + G'xyz + Ds^z = 0. . .(1).
ary^B(AD + u)(G-^q\\
xy=^C(AD+u)(B + q)i ^^^'
37] SYMMETRICAL SOLUTION OF EQUATIONS WITH THREE UNKNOWNS. 75
The resultant isA+B-hC + D + A' + R + CT + D'^i.
Also from (1) xyz + ayz + xi/z + xyz ^A+D-\-B-{-Cy
xyz + (tyz + xyz + xyz :^B + C+ A' + D'.
Hence by the lemma, cf. subsection (2)
xyz-\-aiyz + xyz + 3^z=^(A'{'D+ B' + C') {BCA^U + 8)A
xyz + xyz + xyzi-£yz^(B+C + A'-{-iy)(ADB'G' + 8).) ^ ^'
Again, from (2) and (1),
xyz + xyz 4 (B' + C) {BCA'U + «), xyz 4- iry-j 4 (il + D) (BCA'S^ + a).
Hence from the lemma, cf. subsection (2), and simplifying,
xyz + xyz=^{R + C) {BCAU + a) {AD + m)\ xyz+xyz^{A+D){BCA'D''\-8)(BV'+w).\ ^'^^^
Similarly
xyz ^-xyz^iB+G) {ADWG' + s) (A'& + n),) xyz+xyz^(A' + iy){ADBV' + -8)(BC-^v)] ^ ^'
Again, fix)m equations (3) and (1),
xyz^A {BGA'D' + a) {B'C + w), a^z^D {BCA'U + a) (S'C + w). Hence by