(Appendix to this Volume the longer additional and illustr...)
Appendix to this Volume the longer additional and illustrative notes which I have written for the new edition of the Elements. Some of those notes would have been inconveniently long as footnotes; others would have been inconveniently placed. For example, although the Note on Screws relates naturally to A rt. 416 and that on the Kinematical Treatment of Curves to A rt. 396, I have placed the Note on Screws before the Note on Curves because Hamilton sremarks on screw motion in the earlier Article required some development in order to make the Note on Curves easily intelligible. Accordingly the order of the notes has been arranged with reference to the notes themselves rather than with reference to the text. The selection and treatment of the subjects of these notes have been subordinated to the illustration of quaternion methods. I have not hesitated to sacrifice brevity for suggestiveness, and above all I have tried to render the notation as explicit as possible. An analysis of the Appendix will be found on pages xlv-xlix. For greater convenience I have provided an Index to the whole work referring to the pages, the volumes being distinguished by the numbers i and ii. I take this opportunity of testifying to the extraordinary accuracy both of matter and of printing in the first edition of the Elements. Every portion of the work bears evidence of Hamilton sunsparing pains. I cannot recall a single sentence ambiguous in its meaning, or a single case in which a difficulty is not honestly faced. I see no sign of diminished vigour or of relaxed care in those portions of the work written in his failing health. My task as editor has convinced me of the extreme caution with wliich any endeavour should be made to improve or modify the calculus of Quaternions. In conclusion, I desire to express my thanks to the College Printer, Mr. George Weldrick, for the great care (Typographical errors above are due to OCR software and don't occur in the book.)
http://www.amazon.com/gp/product/B008LFCI10/?tag=2022091-20
http://www.amazon.com/gp/product/197411161X/?tag=2022091-20
(Excerpt from Elements of Quaternions, Vol. 1 The volume ...)
Excerpt from Elements of Quaternions, Vol. 1 The volume now submitted to the public is founded on the same principles as the which were published on the same subject about ten years ago: but the plan adopted is entirely new, and the present work can in no sense he considered as a second edition of that former one. The Table of Contents, by collecting into one view the headings of the various Chapters and Sections, may suffice to give, to readers already acquainted with the subject, a notion of the course pursued: but it seems proper to offer here a few introductory remarks, especially as regards the method of exposition, which it has been thought convenient on this occasion to adopt. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
http://www.amazon.com/gp/product/1334017239/?tag=2022091-20
Astronomer mathematician physicist
He was born on 4 August 1805 at Dublin, Ireland. His father, Archibald Hamilton, who was a solicitor, and his uncle, James Hamilton (curate of Trim), migrated from Scotland in youth. Hamilton’s father, William Hamilton, professor of astronomy at the University of Glasgow, died in 1790, leaving William to be raised by his mother, Elizabeth Hamilton.
At the age of seven he had already made very considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle, who was an extraordinary linguist, almost as many languages as he had years of age.
He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.
About this period he was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a portion of his time to classics.
Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; for Laplace's great work, rich to profusion in analytical processes alike novel and powerful, demands from the most gifted student careful and often laborious study.
How many more such honours he might have attained it is impossible to say; but he was expected to win both the gold medals at the degree examination, had his career as a student not been cut short by an unprecedented event.
This was his appointment to the Andrews professorship of astronomy in the university of Dublin, vacated by Dr Brinkley in 1827.
After receiving a degree from the University of Edinburgh in 1807, Hamilton went to Balliol College.
Oxford, with a Snell Exhibition.
He quickly acquired the reputation of being the most learned authority in Oxford on Aristotle, and the list of books that he submitted for his final examination in 1810 was unprecedented.
He did not, however, receive a fellowship, primarily because of the unpopularity of Scots at Oxford.
Because he had little interest in his career in the law, he applied in 1821 for the chair of moral philosophy at Edinburgh vacated by the death of Thomas Brown.
The articles, which appeared between 1829 and 1836, were the basis of his international reputation, a reputation that forced the town council to elect him in 1836 to the chair in logic and metaphysics, which he held until his death.
Hamilton’s three most important articles for the Edinburgh Review were those on Cousin (1829), on perception (1830), and on logic (1833).
In the first two, he revealed his unique philosophical position, a combination of the Kantian view that there is a limitation on all knowledge and the Scottish view that man has, in perception, a direct acquaintance with the external world.
The first paper deals with the possibility of human knowledge of the absolute.
Hamilton tried to show that neither of these views is coherent and that there is something incoherent about the very notion of thought about the absolute.
Hamilton had little trouble in showing that Brown neither understood Reid’s position nor could offer arguments that disproved either Reid’s position or the position mistakenly attributed to him by Brown.
They are, however, far less important for the history of thought in general and for the history of science in particular than his work in logic.
Hamilton was one of the first in that series of British logicians—a series that included George Boole, Augustus De Morgan, and John Venn—who radically transformed logic and created the algebra of logic and mathematical logic.
Nevertheless, his place in it must be recognized. The traditional, Aristotelian analysis of reasoning allowed for only four types of simple categorical propositions:(A) All A are B. (E) No A are B. (I) Some A are B. (O) Some A are not B. Hamilton’s first important insight was that logic would be more comprehensive and much simpler if it allowed for additional types of simple categorical propositions.
In particular, Hamilton suggested that one treat the signs of quantity (“all, ” “some, ” “no”) in the traditional propositions as modifiers of the subject term A and that one introduce additional signs of quantity as modifiers of the predicate term B. Hamilton called this innovation the quantification of the predicate.
Other logicians before Hamilton had made the same suggestion, but Hamilton was the first to explore the implications of quantifying the predicate, of admitting eight simple categorical propositions:(1) All A are all B. (2) All A are some B (traditional A).
(3) Some A are all B. (4) Some A are some B (traditional I).
(5) Any A is not any B (traditional E).
(6) Any A is not some B. (7) Some A are not any B (traditional O).
(8) Some A are not some B.
The first important inference that Hamilton drew from this modification had to do with the analysis of simple categorical propositions.
There were, according to the traditional, Aristotelian logic, two ways of analyzing a simple categorical proposition such as “All A are B”: extensively, that is, as asserting that the extension of the term A is contained within the extension of the term B; or comprehensively, that is, as asserting that the comprehension of the term B is contained within the comprehension of the term A.
In either case, the proposition expresses a whole-part relation.
But the new Hamiltonian modification, because it distinguished (1) from (2), (3) from (4), (5) from (6), and (7) from (8), enables one to adopt a different analysis of these propositions.
According to this new analysis, each of these propositions asserts or denies the existence of an identity-relation between the two classes denoted by the quantified terms.
Thus, “All A are all B” asserts that the classes A and B are identical, while “Some A are not some B” asserts that there is a subset of the class A which is not identical with any subset of the class B.
One result, therefore, of the quantification of the predicate is that simple categorical propositions become identity claims about classes.
This is just the analysis of simple categorical propositions that Boole needed and used in formulating the algebra of logic.
Hamilton’s work facilitated a considerably simplified analysis of the validity of reasoning.
The traditional, Aristotelian analysis of mediate reasoning, for example, involved many concepts (such as the figure of a syllogism, major and minor terms) that were based on the distinction between the subject of a proposition and its predicate.
This subject-predicate distinction had some significance when simple categorical propositions were understood as expressing asymmetrical whole-part relations.
But given the new analysis, where these propositions are understood as expressing symmetrical identity relations, there is little point to a distinction between the subject and the predicate of a proposition.
As a result, the complicated traditional rules for the validity of syllogistic reasoning disappear.
One is then left, as Hamilton pointed out in his theory of the unfigured syllogism, with two simple rules for valid syllogisms: If A = B and B = C, then A = C; and If A = B and B ≠ C, then A ≠ C. Similarly, the traditional Aristotelian analysis of immediate reasoning, based upon the complicated distinctions between simple conversion, conversion per accidens, and contraposition, is replaceable by the simple rule that all eight propositions are simply convertible.
Hamilton’s quantification of the predicate is no exception to this rule.
The simplest example of this is the inference to the identity of classes A and C from premises asserting that they are both identical with some class B. Traditional analysis did not even recognize the existence of propositions asserting that two classes are identical; it could not, therefore, explain the validity of such an inference.
As is well known, Kant and most other important eighteenth-century philosophers thought that nothing of importance had been done in formal logic since the time of Aristotle, primarily because of the completeness and perfection of the Aristotelian system.
This is partly due to the fact that both it and the Boolean algebra of logic, which it so greatly influenced, have been superseded by Frege’s far more powerful quantificational analysis—an analysis so different that Hamilton’s theory has no relevance to it.
It is, however, also due to a certain internal weakness in Hamilton’s initial quantification of the predicate, which was pointed out by Hamilton’s great adversary, Augustus De Morgan, during their long and acrimonious quarrels. There really were two quarrels between Hamilton and De Morgan.
The first had to do with Hamilton’s charge that De Morgan had plagiarized some of Hamilton’s basic ideas.
De Morgan then received from Hamilton, in the form of a list of requirements for a prize essay set for Hamilton’s students, a brief account of Hamilton’s quantification of the predicate.
At about this time, De Morgan asked Whewell to return the draft of his paper and then made some changes in it.
In his reply De Morgan claimed that he made the changes before he received the communication from Hamilton.
Although it is not clear as to who was right about the date of the changes, it is clear that De Morgan did not plagiarize Hamilton’s ideas.
Even if De Morgan’s ideas were suggested by Hamilton’s communication, they are so different from Hamilton’s that no one could consider them to be a plagiarism.
The second, far more important, quarrel was about the relative merits of their innovations in logic.
This arose out of the first, since De Morgan was not content with pointing out the differences between the two systems.
Although he offered several criticisms of Hamilton’s list of eight basic categorical propositions, there was only one that was really serious—that Hamilton’s first proposition is not a simple categorical proposition because it is equivalent to the joint assertion of the second and third propositions.
Thus “All A are some B” and “Some A are all B” can both be true if, and only if, “All A are all B” is also true. Hamilton was slow in responding to this argument, primarily because he had suffered an attack of paralysis in 1844 that made it very difficult for him to do any work.
In 1846 De Morgan sent a draft of one of his most important papers on logic to William Whewell, who was supposed to transmit it to the Cambridge Philosophical Society.
When, however, in 1852, he published a collection of his articles from the Edinburgh Review, he included in the book an appendix in which he argued that De Morgan’s criticisms were based on a misunderstanding of the eight propositional forms.
De Morgan thought that “some” meant “some, possibly all. ”
If it did, then he would certainly be right in his claim that the first proposition is equivalent to the joint assertion of the second and third propositions.
But Hamilton said that he had meant in his forms “some, and not all. ”
Consequently, the conjunction of “All A are some B” and “Some A are all B” is inconsistent, and neither of these propositions can be true if “All A are all B” is true. The controversy rested at this point until some years after Hamilton’s death.
Then, De Morgan renewed it in a series of letters in Athenaeum (1861–1862) and in his last article on the syllogism, in Transactions of the Cambridge Philosophical Society (1863).
De Morgan began his new attack by casting doubt on the claim that Hamilton had meant “some, but not all. ”
He did this by showing that much of what Hamilton had to say about the validity of particular inferences made sense only if we suppose that he meant “some, perhaps all. ”
There is little doubt that De Morgan was right about this point.
After all, there are five, and only five, relations of the type discussed in categorical propositions that can hold between two classes: (1) the two classes are coextensive, (2) the first class is a proper subset of the second class, (3) the second class is a proper subset of the first class, (4) the two classes have some members in common but each has members that are not members of the other, and (5) the two classes have no members in common.
Thus Hamilton’s propositions 6–8 seem to be superfluous. De Morgan’s final critique clearly showed that Hamilton had not exercised sufficient care in laying the foundations for his new analysis of the validity of reasoning.
This was quickly recognized by most logicians; Charles Sanders Peirce, the great American logician, described De Morgan’s 1863 paper as unanswerable.
While there is no doubt that De Morgan’s critique helped lessen the eventual influence of Hamilton’s work, it should not prevent the recognition of both the intrinsic merit of Hamilton’s work and its role in the development of mathematical logic in Great Britain during the nineteenth century.
He is known mainly for his work on quarternions, but he also wrote on optics, on dynamics, and on the theory of the solutions of an algebraic equation of degree five. Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.
He was twice awarded the Cunningham Medal of the Royal Irish Academy.
(Appendix to this Volume the longer additional and illustr...)
(Excerpt from Elements of Quaternions, Vol. 1 The volume ...)
Hamilton was a Whig, and the Tory town council therefore chose his opponent, John Wilson.
He was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much time.
He was a member of the United States National Academy of Sciences and member of the Saint Petersburg Academy of Sciences.
He had married Helen Bayly and had several children.
