Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published A Treatise on the Calculus of Finite Differences in 1860 as a sequel to his Treatise on Differential Equations (1859). Both books became instant classics that were used as textbooks for many years and eventually became the basis for our contemporary digital computer systems. The book discusses direct theories of finite differences and integration, linear equations, variations of a constant, and equations of partial and mixed differences. Boole also includes exercises for daring students to ponder, and also supplies answers. Long a proponent of positioning logic firmly in the camp of mathematics rather than philosophy, Boole was instrumental in developing a notational system that allowed logical statements to be symbolically represented by algebraic equations. One of history's most insightful mathematicians, Boole is compelling reading for today's student of logic and Boolean thinking.
Author(s): George Boole
Series: Cambridge Library Collection - Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 261
Tags: Математика;Вычислительная математика;
Cover......Page 1
A Treatise on the Calculus of Finite Differences......Page 4
Preface......Page 10
Contents......Page 12
CHAPTER I - NATURE OF THE CALCULUS OF FINITE DIFFERENCES......Page 14
CHAPTER II - DIRECT THEOREMS OF FINITE DIFFERENCES......Page 17
Nature of the Problem......Page 41
Given values equidistant......Page 42
Not equidistant......Page 46
Application of Lagrange's Theorem......Page 48
Areas of Curves......Page 49
Application to Statistics......Page 54
Exercises......Page 57
Meaning of Integration......Page 58
Periodical Constants......Page 60
Integrable Forms......Page 61
Summation of Series......Page 69
Connexion of Methods......Page 72
Conditions of extension of direct to inverse forms......Page 74
Exercises......Page 76
Definitions......Page 78
Fundamental Proposition......Page 79
First derived Criterion......Page 82
Supplemental Criteria......Page 84
Exercises......Page 92
Development of [GREEK CAPITAL LETTER SIGMA]ux,......Page 93
Bernoulli's Numbers......Page 96
Applications......Page 97
Limits of the Series for [GREEK CAPITAL LETTER SIGMA]ux......Page 104
Other forms of [GREEK CAPITAL LETTER SIGMA]ux......Page 107
Exercises......Page 110
Genesis......Page 112
Linear Equations of the first orders......Page 114
Linear Equations with constant Coefficients......Page 119
Symbolical Solution......Page 120
Equations reducible to Linear Equations with constant Coefficients......Page 127
Analogy with Differential Equations......Page 131
Fundamental Connexion with Differential Equations......Page 134
Exercises......Page 136
Theory of these Equations......Page 138
Solutions derived from the Variation of a Constant......Page 141
Law of Reciprocity......Page 145
Principle of Continuity......Page 150
Exercises......Page 163
CHAPTER IX - LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS......Page 164
Solution of Linear Equations of Differences in series......Page 171
Finite Solution of Equations of Differences......Page 174
Binomial Equations......Page 176
Exercises......Page 191
CHAPTER X - OF EQUATIONS OF PARTIAL AND OF MIXED DIFFERENCES, AND OF SIMULTANEOUS EQUATIONS OF DIFFERENCES......Page 192
Equations of Partial Differences......Page 195
Method of Generating Functions......Page 204
Equations of Mixed Differences......Page 206
Simultaneous Equations......Page 218
Exercises......Page 219
CHAPTER XI - OF THE CALCULUS OF FUNCTIONS......Page 221
Direct Problems......Page 222
Periodical Functions......Page 228
Functional Equations......Page 231
Exercises......Page 242
CHAPTER XII - GEOMETRICAL APPLICATIONS......Page 245
Answers to the Exercises......Page 258
