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An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities - an unabridged edition. ... Read more

Customer Reviews (7)

Lucid reading
Descartes made a revolutionary move when he symbolized geometry with algebra and used the methods of algebra to solve geometrical problems.For example, he defined a line with y=mx+b.Similarly, Boole made a revolutionary move here when he symbolized logic with algebra and used the methods of algebra to solve logical problems.That is the whole gist of this book.In addition, Boole has an eloquent writing style.

Boole was the founder of probabilistic interval estimates
The most important part of this very important book is chapters 16-21 on the application of Boole's unique approach to standard algebra to probability(Boolean algebra is NOT the algebra of George Boole).Boole developed a method that allows one to calculate probability intervals.J M Keynes discovered Boole's approach and adapted it into a formalmathematical technique for the estimation of probability estimates.Theodore Hailperin ,in 1965,showed how Boole's approach could be translated into a linear programming problem formulation using integer and mixed integer solution approaches based on the standard simplex algorithm.The linear programming approach is the modern version ,then,of Boole's original technique used by him in 1854 and in two later journal articles published after 1854.The failure(ignorance) of modern logicians and philosophers to acknowledge(recognize) the priority of Boole in probability theory is a major shortcoming in this profession.Koopman and Good,rather than Boole and Keynes,are credited with developing the interval estimate approach technically.All one has to do is read Boole and Keynes to see that they had a fully operational mathematical technique.An Investigation of the Laws of Thought

This Edition Introduces Boole to a New Generation
Instead of writing an original review, I decided to quote excerpts from a review by Prof. James Van Evra, a noted authority on Boole and on the history of logic since 1800. The entire review can be found in the journal PHILOSOPHY IN REVIEW; Volume 24 (2004) pages 167-169. The words below are all by Prof. Van Evra.

The body of this book is a replica of the 1854 edition of George Boole's great work in logic. While it has been widely available in this form for over a century, what sets this edition apart is the inclusion of John Corcoran's extensive and penetrating introduction both to the text and to Boole's logical thought more generally. The result is a valuable addition to Boole scholarship conveniently bound with Boole's major work.
Corcoran's commentary is valuable to those already familiar with Boole's work, but is especially helpful to those approaching it for the first time. Many existing commentaries approach Boole from a present day perspective, i.e. as anticipating, however imperfectly, things to come (W. V. O. Quine's review of Desmond MacHale's biography of Boole ("In the Logical Vestibule") is an excellent example of this approach). There is some justification for doing this--Boole, after all, tended to be forward looking and had little positive to say about the tradition which preceded him. The effect of such an approach, however, is a tendency to stress what is lacking in Boole, rather than his positive contribution. Corcoran, by contrast, uses Aristotle's theory of logic as a baseline for his analysis. Starting with simple sentences and immediate inference, Corcoran clearly and accurately shows how Boole's logic covers the same ground. As he puts it, `Boole was one of the last logicians to take [the subject-connector-predicate view of simple propositions] seriously' (xiii). The result of Corcoran's approach is a view in which Boole's logic is seen to be simpler than Aristotle's in one respect (i.e. as a unified system), and more complicated in another (extending the range of propositions covered within it). By beginning with Aristotle, Corcoran's analysis provides an exceptionally clear account of Boole's positive contributions to logic.
At the same time, Corcoran also describes things that Boole's system lacks. Thus he points out that Boole never recognized indirect inference, and he notes problems that arise when Boole attempts to use algebraic devices (such as solving equations) as a warrant for logical inference (not all algebraic operations result in logically valid inferences). By detailing both the strengths and weaknesses in Boole's theory, Corcoran provides a balanced and accurate account Boole's proper place in the modern development of logic.
Another welcome feature of Corcoran's introduction is the inclusion of references, often to recent encyclopedia articles, at just those points at which readers with relatively little technical background encounter concepts that require some further explanation. Such an addition makes it easier for those with modest backgrounds in logic and algebra to work through Laws of Thought.
This year is the sesquicentennial anniversary of the publication of Laws of Thought. So much of what has happened in the meantime bears the mark of Boole's influence that it is appropriate to mark the occasion with a fresh look at the work. Corcoran's excellent introduction does this with clarity and rigor.

Boole's LT Breaks Ancient Mold , Founds Math Logic
.Aristotle and Boole are the two most original logicians before the era of modern logic. Aristotle presented the world's first system of logic.His system involves the standard three parts: first, a limited formalized predicational language; second, a formal method of step-by-step deductions for establishing validity of arguments having unlimited numbers of premisses; and third, an equally general method of countermodels for establishing invalidity. Boole's LAWS OF THOUGHT showed that logic is mathematical. Its stated aims were to refine, systematize, and complete the project started by Aristotle and, more ambitiously, to demonstrate the mathematical character of logic.His two-part system involves, first, a limited formalized equational language capable of expressing tautologies or "laws of thought", a breakthrough dramatically altering Aristotle's plan, and, second, a semi-formal method of derivation using equational reasoning totally absent from previous systematic logic.Boole's primary goals included construction of a method for generating solutions to sets of equations regarded as conditions on "unknowns", an unprecedented innovation with radical implications for the future development of logic. As for the third part of a system of logic, a method of establishing invalidity, surprisingly, Boole's book contains no systematic discussion of independence nor does it contain anything like a method of countermodels. Boole's LAWS OF THOUGHT set in motion forces that would lead to the ultimate fulfillment many of his goals including the establishment of mathematical logic.

NOW IS A GOOD TIME TO STUDY BOOLE.
The publication of The Laws of Thought in 1854 launched modern mathematical logic. The author George Boole (1815-1864) was already a celebrated mathematician specializing in what is known as analysis. If, as Aristotle (384-322 B.C.E.) tells us, we do not understand a thing until we see it growing from its beginning, then those who want to understand modern mathematical logic should begin with The Laws of Thought.There are many wonderful things about this book besides its historical importance. For one thing, the reader does not need to know any mathematical logic. There was none available to the audience for which it was written-even today a little basic algebra and a semester's worth of beginning logic is all that is required. For another thing, the book is exciting reading. The reader comes to feel through Boole's intense, serious, and sometimes labored writing that the birth of something very important is being witnessed. Of all the foundational writings concerning mathematical logic, this one is the most accessible to the nonexpert and it has the most to offer the nonexpert. The secondary literature on Boole is lively and growing, as can be seen from an excellent recent anthology (A BOOLE ANTHOLOGY by J.Gasser 2000) and a complete bibliography that is now available (Nambiar 2003).Boole's manuscripts on logic and philosophy, once nearly inaccessible, are now in print (Grattan-Guinness and Bornet 1997). This is a good time to start to study Boole.
It is true that Boole had written on logic before, but his earlier work did not attract much attention until after his reputation as a logician was established. Today he is known almost exclusively for his logic. In 1848 he published a short paper "The Calculus of Logic" (Boole 1848) and in 1847, at his own expense, he published a pamphlet The Mathematical Analysis of Logic (Boole1847). By the expression `mathematical analysis of logic' Boole did not mean to suggest that he was analyzing logic mathematically or that he was using mathematics to analyze logic. Rather his meaning was that he had found logic to be a new form of mathematics, not a form of philosophy as had been previously thought. More specifically, his point was that he had found logic to be a form of the branch of mathematics known as mathematical analysis, which includes algebra and calculus. (For a short description of this branch of mathematics, see the article "Mathematical Analysis" in the 1999 Cambridge Dictionary of Philosophy (Audi 1999, 540-41).
Although this book begins mathematical logic, it does not begin logical theory. The construction of logical theory begins, of course, with Aristotle whose logical writings were known and admired by Boole.In fact, Boole explicitly accepted Aristotle's logic as "a collection of scientific truths" (1854, 241) and he regarded himself as following in Aristotle's footsteps. He thought that he was supplying a unifying foundation for Aristotle's logic and that he was at the same time expanding the range of propositions and deductions that were formally treatable in logic. Boole thought that Aristotle's logic was "not a science but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest" (Boole 1854, 241). As has been pointed out by Grattan-Guinness (2003; Grattan-Guinness and Bornet 1997), in 1854 Boole was less impressed with Aristotle's achievement than he was in 1847. In "The mathematical analysis of logic" (Boole 1847) Aristotle's logic plays the leading role, but in The Laws of Thought (Boole 1854) it occupies only one chapter of the fifteen on logic. Even though Boole's view of Aristotle's achievement waned as Boole's own achievement evolved, Boole never found fault with anything that Aristotle did in logic, with Aristotle's positive doctrine. Boole's criticisms were all directed at what Aristotle did not do, with what Aristotle omitted doing.Aristotle was already fully aware that later logicians would criticize his omissions, but unfortunately he did not reveal what he thought those omissions might be (Aristotle, Sophistical Refutations, Ch. 34).
The new 2003 edition by Prometheus Books(ISBN 1-59102-089-1, Paper ...)contains an accessible 25-page introduction by a modern logician. ... Read more

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( 20 ) OF EXPRESSION AND INTERPRETATION. A Proposition iB a sentence which either affirms or denies, as, All men are mortal, No creature is independent. A Proposition has necessarily two terms, as men, mortal; the former of which, or the one spoken of, h culled the subject; the latter, or tha! which is affirmed or denied of the subject, the predicate. These are connected together by the copula w, or m not, or by some other modification of the substantive verb. The substantive verb is the only verb recognised in Logic; all others are resolvable by means of the verb to be and a participle or adjective, e. g. " The Romans conquered"; the word conquered U both copula and predicate, being equivalent to "were (copula) victorious" (predicate). A Proposition must either be affirmative or negative, and must be also either universal or particular. Thus we reckon in ail, four kinds of pure categorical Propositions. 1st. Universal-affirmative, usually represented by A, Ex. All Xs

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What sould I do?
I'm 16 years old and I'm a boy and I'm from Switzerland, that's it ... Read more

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Originally published in 1800. CALCULUS OF FINITE DIFFERENCES by GEORGE BOOLE. PREFACE: IN the following exposition of the Calculus of Finite Dif ferences, particular attention has been paid to the connexion of its methods with those of the Differential Calculus a connexion which in some instances involves far more than a merely formal analogy. Indeed the work is in some measure designed as a sequel to my Treatise on Differential Equations. And it has been composed on the same plan. Mr Stirling, of Trinity College, Cambridge, has rendered me much valuable assistance in the revision of the proof sheets. In offering him my best thanks for his kind aid, I am led to express a hope that the work will be found to bo free from important errors. GEORGE BOOLE. QUEEN'S COLLKOE, CORK, April 18, 1800. PREFACE TO THE SECOND EDITION: WHEN I commenced to prepare for the press a Second Edition of the late Dr Boole's Treatise on Finite Differ ences, my intention was to leave the work unchanged save by the insertion of sundry additions in the shape of para graphs marked off from the rest of the text. But I soon found that adherence to such a principle would greatly lessen the value of the book as a Text-book, since it would be impossible to avoid confused arrangement and even much repetition. I have therefore allowed myself considerable freedom as regards the form and arrangement of those parts where the additions are considerable, but I have strictly adhered to the principle of inserting all that was contained in the First Edition. As such Treatises as the present are in close connexion with the course of Mathematical Study at the University of Cambridge, there is considerable difficulty in deciding the question how far they should aim at being exhaustive. I have held it best not to insert investigations that involve complicated analysis unless they possess great suggestiveness or are the bases of important developments of the subject. Under the present system the premium on wide superficial reading is so great that such investigations, if inserted, would seldom be read. But though this is at present the case, there is every reason to hope that it will not continue to be so; and in view of a time when students will aim at an exhaustive study of a few subjects in preference to a super ficial acquaintance with the whole range of Mathematical research, I have added brief notes referring to most of the papers on the subjects of this Treatise that have appeared in the Mathematical Serials, and to other original sources. In virtue of such references, and the brief indication of the subject of the paper that accompanies each, it is hoped that this work may serve as a handbook to students who wish to read the subject more thoroughly than they could do by confining themselves to an Educational Text-book. The latter part of the book has been left untouched. Much of it I hold to be unsuited to a work like the present, partly for reasons similar to those given above, and partly because it treats in a brief and necessarily imperfect manner subjects that had better be left to separate treatises. It is impossible within the limits of the present work to treat adequately the Calculus of Operations and the Calculus of Functions, and I should have preferred leaving them wholly to such treatises as those of Lagrange, Babbage, Carmichael, De Morgan, & c. I have therefore abstained from making any additions to these portions of the book, and have made it my chief aim to render more evident the remarkable analogy between the Calculus of Finite Differences and the Differential Calculus. ... Read more

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George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day. The utility of "Boolean algebra" in computing has greatly increased curiosity in the nature and extent of his achievements. His work is most accessible in his two books on logic, "A mathematical analysis of logic" (1947) and "An investigation of the laws of thought" (1954). But at various times he wrote manuscript essays, especially after the publication of the second book; several were intended for a non-technical work, "The Philosophy of logic", which he was not able to complete. This volume contains an edited selection which not only relates them to Boole's publications and the historical context of his time, but also describes their strange history of family, followers and scholars have treid to confect an edition. The book will appeal to logicians, mathematicians and philosophers, and those interested in the histories of the corresponding subjects; and also students of the early Victorian Britain in which they were written. ... Read more

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a frustrating read
While Boole is considered one of the founders of computing, few people have actually read his papers. This book offers a chance to remedy that omission. It gives a sampling of papers that show Boole's thinking, and how this changed over his lifetime. The papers also somewhat indirectly reflect how logic was perceived by other mathematicians in Victorian Britain.

The book can be a trifle frustrating. Boole and others of his time had no hardware or software in which to apply their logic. Which is a tribute to how much he was able to accomplish, through pure deductive reasoning. But one cannot help wondering how much further he would have been able to pursue his research, if he had access to even simple functioning computers. Very analogous to Babbage and Lovelace. In contrast to Claude Shannon, who was able to see his research instantiated in hardware and software during his lifetime. ... Read more

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Modern mathematical logic would not exist without theanalytical tools first developed by George Boole in TheMathematical Analysis of Logic and The Laws of Thought. Theinfluence of the Boolean school on the development of logic, alwaysrecognised but long underestimated, has recently become a majorresearch topic. This collection is the first anthology of works onBoole. It contains two works published in 1865, the year of Boole'sdeath, but never reprinted, as well as several classic studies ofrecent decades and ten original contributions appearing here for thefirst time. From the programme of the English Algebraic School toBoole's use of operator methods, from the problem of interpretabilityto that of psychologism, a full range of issues is covered. TheBoole Anthology is indispensable to Boole studies and willremain so for years to come. ... Read more

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CELEBRATING BOOLE AFTER 150 YEARS
This review begins with three excerpts from a review by the Finnish historian of logic Risto Vilkko (pronounced REES-toh FEEL-koh) in THE REVIEW OF MODERN LOGIC. His whole review is well worth reading.
(1) James Gasser's A Boole Anthology is a collection of seven classical and ten recent studies on George Boole's work, its background, and its intellectual frame of reference. Two of the classical articles were published during the 1860s, with the rest appearing during the latter half of the 20th century. The recent ones are based on lectures given at Lausanne on the occasion of the conference entitled "Boole 1997: One Hundred and Fifty Years of Mathematical Analysis of Logic". Gasser's anthology is a valuable contribution to the renaissance experienced by Boole studies since the publication of Desmond MacHale's biography of Boole in 1985. In addition to MacHale's book, three recent new editions of Boole's ground-breaking The Mathematical Analysis of Logic (1996; 1998; 2001), an important selection of his manuscripts on logic and its philosophy [Grattan-Guinness & Bornet 1997], and the present volume are telling examples of the current lively interest in the life and thought of George Boole. Indeed, in the editor's words, "the present anthology constitutes an attempt to capture some of the 'buzz about Boole'" (p. vii).
(2) All in all, Gasser's Boole Anthology draws us a picture of George Boole, not as the founder of modern logic, but as an important mediator between traditions, and an initiator of an on-going and ever increasing development of the field of logical theory. On the one hand he was a historically sensitive scientist whose work emerges seamlessly from the Scholastic-Aristotelian tradition. On the other his novel ideas in the field of the algebra of logic paved the way for the break-through of a modern, mathematically-oriented logic at the turn of the 20th century. He was not an isolated miracle but an exceptionally gifted mathematician who was greatly influenced by several of his contemporaries. All the contributors of the volume seem to agree with Dummett that "there can be no doubt that Boole deserves great credit for what he achieved, in the sense that in those historical circumstances what he did must have been very difficult to do" (p. 79).
(3) James Gasser's A Boole Anthology is an important collection of first-rate Boole-studies.
Although, I realize how risky it is to disagree with someone as wise and learned as Prof. Vilkko, I think that the picture he gets from Gasser's BOOLE ANTHOLOGY is incorrect.I think that Boole should be honored as the founder of mathematical logic. As people come to a deeper and more nuanced view of the nature of logic, they will find that Boole had contributed much more to its transformation into a mathematical science than they had thought. So many of Boole's ideas have become so deeply entrenched in modern thinking about logic that it takes concentration to identify them and to realize that we owe our knowledge of them to Boole.
Gasser's book will help people to decide whether Prof. Vilkko is right. Another book that will help is the 2003 Prometheus edition of Boole's 1854 LAWS OF THOUGHT, ISBN 1 59102 089 1, which includes an introduction by John Corcoran, who by coincidence is one of the contributors to the Gasser anthology. Let me end this review by quoting the last paragraph of Corcoran's introduction: "It has been said that Galileo's greatest achievement was to persuade the world's scientists that physical reality is mathematical, or at least that science should be pursued mathematically.In his words, "The Book of Nature is written in mathematical characters."In a strikingly similar spirit, Boole stated: "it is certain that [logic's] ultimate forms and processes are mathematical" (Boole 1854, 12).Perhaps Boole's greatest achievement was to persuade the world's logicians that logical reality is mathematical, or at least that logic should be pursued mathematically." - Frango Nabrasa, Manatee, FL. ... Read more

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This brief text assists students in understanding Boole's philosophy and thinking so they can more fully engage in useful, intelligent class dialogue and improve their understanding of course content. Part of the Wadsworth Notes Series, (which will eventually consist of approximately 100 titles, each focusing on a single "thinker" from ancient times to the present), ON BOOLE is written by a philosopher deeply versed in the philosophy of this key thinker. Like other books in the series, this concise book offers sufficient insight into the thinking of a notable philosopher, better enabling students to engage in reading and to discuss the material in class and on paper. ... Read more

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Chapters: People From Lincoln, Lincolnshire, George Boole, Mark Byford, Jim Broadbent, Lee Swaby, Ashley Bayes, John Ward, Jamie Clapham, Simeon Hodson, Charles Seely, Toby Jepson, Lee Chapman, Lee Frecklington, Dave Gilbert, the Casuals, Marlon Beresford, Derek Fatchett, Steve Race, Paul Mayo, Julian Hails, Joanna Eden, Rob Locke, Harry Houghton, Paul Palmer, Neville Marriner, Allison Pearson, Steve Froggatt, Steve Wilkinson, Boz Burrell, James Rodwell, Sam Scorer, Sheila Gish, Stuart Reddington, Simon Marsden, Ben Dixon, Hermann Arthur Jahn, Kelly Adams, Graham Clapham, John Disney, Shane Clarke, Sam Mullarkey, John Quayle, Marion Rose Halpenny, Lucy May Barker, Charlie Millington, Frederick Groves, Clive Goodyear, Daniel Cox, John Hartley, Dennis Townhill, Berechiah de Nicole, Isobel Lang, Leonard James Keyworth, Charles Clarke, Daniel Brevint, Nick Butler, James Upton, Nathan Adams, Dennis Peacock, Radulf of Lincoln, Andy Hutchinson, Richard Armstrong, Ray Dring, Tom Wilkinson. Source: Wikipedia. Pages: 193. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: George Boole (pronounced ) (2 November 1815 8 December 1864) was an English mathematician and philosopher. As the inventor of Boolean logicthe basis of modern digital computer logicBoole is regarded in hindsight as a founder of the field of computer science. Boole said, ... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning ... George Boole's father, John Boole (17791848), was a tradesman of limited means, but of "studious character and active mind". Being especially interested in mathematical science and logic, the father gave his son his first lessons; but the extraordinary mathem...More: http://booksllc.net/?id=38298 ... Read more

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Written by a great English mathematician, this classic text begins with the differences of elementary functions and explores interpolation, mechanical quadrature, finite integration, and the summation of series. Several useful tests for the convergence and divergence of series are developed, as is a method for finding the limits of error in series expansions. The latter half of the book discusses difference-equations, including linear, mixed, and partial difference-equations, and concludes with applications to problems in geometry and optics. The text pays particular attention to the connection of the calculus of finite differences with the differential calculus, and more than 200 problems appear in the text (some with solutions). Unabridged republication of the classic 1872 edition.
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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This scarce antiquarian book is included in our special Legacy Reprint Series. In the interest of creating a more extensive selection of rare historical book reprints, we have chosen to reproduce this title even though it may possibly have occasional imperfections such as missing and blurred pages, missing text, poor pictures, markings, dark backgrounds and other reproduction issues beyond our control. Because this work is culturally important, we have made it available as a part of our commitment to protecting, preserving and promoting the world's literature. ... Read more

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishings Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the worlds literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Correspondence Without Communication.
The title of this review is taken from a 1986 essay in the journal HISTORY AND PHILOSOPHY OF LOGIC about this fascinating book.Boole and De Morgan were two brilliant, objective, and dedicated mathematician-philosophers struggling at the same time to modernize logic. They had a strange relationship. In 1847 after working independently, each published what was to become a landmark book on formal logic, astoundingly on the very same day. De Morgan had already discovered the crucial concept "universe of discourse" before Boole did any work on logic. But it was Boole who coined its name after learning it from De Morgan. The concept was absent from Boole's 1847 work , but it fit perfectly into Boole's thinking and it came to play a prominent role in his more famous 1854 book. Without this interaction, the history of logic may have been very different. Aside from this one point neither understood much of what the other was thinking.They had an extensive correspondence (published here for the first time), which began in 1842 although "universe of discourse" is not mentioned until 1861(page 89 in this book). But despite heroic and honest efforts they hardly communicated. Both had radical and radically different approaches to revamping logic. Perhaps each had such a tenuous grip on his own new ideas that neither could understand the other without risking losing everything. For those who know a little logic and have read the 1847 and 1854 works by the two men this book could be an eye-opening experience. My feeling is that in the right hands this book could prove to be a crucial data point for theories of the growth of scientific thought. ... Read more

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( 4 ) CHAPTER II. direct theorems of finite differences. 1. The operation denoted by A is capable of repetition. For the difference of a function of x, being itself a function of x, is subject to operations of the same kind. In accordance with the algebraic notation of indices, the difference of the difference of a function of x, usually called the second difference, is expressed by attaching the index 2 to the symbol A. Thus AAws= AX- In like manner AAX = AX, and generally AA"~X = AX (1), the last member being termed the nth difference of the function ux. If we suppose ux = »3, the successive values of ux with their successive differences of the first, second, and third orders will be represented in the following scheme: Values of X 1 2 3 4 5 6 . 1 8 27 64 125 216 At*. 7 19 37 61 91 ... AX 12 18 24 30... AX 6 6 6... It may be observed that each set of differences may either be formed from the preceding set by successive subtractions in accordance with the

Table of Contents

CONTENTS; DIFFERENCE- AND SUM-CALCULUS CHAPTER I; pagk; NATURE OF THE CALCULUS OF FINITE DIFFERENCES 1; CHAPTER II; DIRECT THEOREMS OF FINITE DIFFERENCES 4; Differences of Elementary Functions, 6 Expansion in factorials, 11; Generating Functions, 14 Laws and relations of E, A and j^, 16; Secondary form of Maclaurin's Theorem, 22 HerBchePs Theo-rem, 24 Miscellaneous Expansions, 25 Exercises, 28; CHAPTER III; ON INTERPOLATION, AND MECHANICAL QUADRATURE 33; Nature of the Problem, 33 Given values equidistant, 34 Not equidistant-Lagrange's Method, 38 Gauss' Method, 42 Cauchy's Method, 43 Application to Statistics, 43 Areas of Curves, 46 Weddle's rule, 48 Gauss' Theorem on the best position of the given ordinates, 51 Laplace's method of Quadratures, 53 References on Interpolation, &c 55 Connexion between Gauss' Theorem and Laplace's Coefficients, 57 Exercises, 57 B F D b; X; CONTENTS; CHAPTER IV page; FINITE INTEGRATION, AND THE SUMMATION O ... Read more