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HAM discussion is, that we can in no proper sense have a science of the infinite and its relations, as we have of the finite. In several points of view this conclusion is, without doubt, one of the most important advances hitherto made in the history of speculative philosophy. But having taken this step, we can hardly rest here; a variety of vital and pressing questions at once arises for treatment and solution. The absolute or infinite is cast beyond the sphere of thought and science; it is still, however, allowed by Hamilton to remain in some sense in consciousness, for it is grasped by faith, and faith is a conscious act. The question, accordingly, at once meets us—In what sense and how far can there be an object within consciousness, which is not properly within thought or knowledge? In other words, how far is our faith in the infinite intelligent and intelligible? This point demands further and more detailed treatment than it has met with either at the hands of Sir W. Hamilton himself, or any one who has sought to carry out his principles.

In his treatment of the science of logic, the dialectical genius of Hamilton shines forth with peculiar lustre. He has adopted and carried out with great vigour and precision the positive view of the formal character of the science; thus distinguishing, on the one hand, pure logic from psychology or the theory of the mental processes, and on the other, from that modified branch of logic known as the inductive. In his vigorous discrimination of these points, Hamilton has done great service to the cause of scientific accuracy and precision. He has, however, gone much further than this; for he has introduced into the science principles which produce a complete renovation in its fundamental positions, and simplify, while they add largely to, its development. The fixity of logical doctrine had become almost proverbial. It was reserved for the author of the new analytic to break up the commonplace routine of the study, and set at nought, in the interest of a fuller and more symmetrical development of the science, the traditions of two thousand years. The two great points from which the new developments of logical science arise in the view of Hamilton, are—1. The application to notions, propositions, and reasonings of the doctrine of comprehension, or of attribute as opposed to that of extension or class; and, 2, The theory of a quantified predicate, or the designation of the quantity of the predicate as well as that of the subject in propositions, and its applications—in the enlargement of the number of the propositional forms; the simplification of the doctrine of the conversion of propositions; the renovation of the doctrine of opposition, and immediate inference; and the amplification and simplification of the theory of syllogism. The new logical doctrines, and the modifications of the old, which result from the introduction into the science of these two principles, are only now beginning to receive from logicians that attention and critical examination which their importance demands. While every student of logic will admit the advantages which have resulted to this branch of philosophy from the accurately scientific manner in which it has been handled by Hamilton, it should, at the same time, be remarked, that the literal exhibition of logic proper, in all the symmetry of a pure science, and in the completeness of its abstraction from material considerations, has been attended, to a considerable extent, with a superficial treatment of the deeply interesting and vital questions connected with scientific induction. Mixed or modified logic, in the hands of Hamilton, appears more as a mere precarious supplement to pure logic, than, as what it properly is in its integrity, the theory of the nature and process of science. Hamilton affords us no full discussion either of the psychology, the logic, or the metaphysics of induction.

The lectures on metaphysics and logic, lately published from the author's manuscripts, occupy a different position from the writings given to the world by Mr. W. Hamilton himself. These lectures were not designed for publication. They were made up, in fact, of the material out of which he was in the habit of elaborating more finished discussions. They were written from day to day to meet the exigencies of class instruction, and were thus designed for a comparatively temporary purpose. This will account for the easy, unelaborate, and even occasionally careless style in which they are written, and also in part for the numerous and lengthy quotations which they contain; though this latter circumstance must also be referred to the author's comparative disinclination for composition. To a person of Sir W. Hamilton's habit of mind and standard of taste, philosophical composition was a much less congenial task than minute and comprehensive research into the history of opinions and the accompanying energy of reflection. Hence, in communicating to a class of students elementary instruction in philosophy, and, above all, in the technicalities of logic, he adopted the language of those writers who appeared to him to state the received opinions clearly and well, and afford a medium of lucid communication. The long quotations in the logical lectures from Krug and Esser—writers of no especial mark or originality—are cases in point. With all these drawbacks, the lectures form no unworthy monument of the author's philosophical genius, and are probably fitted to render his opinions more generally accessible to ordinary readers than his other writings. Viewed in reference to the purposes for which they were composed, they must be pronounced to have been eminently successful. They revived the taste for philosophical studies in Scotland—nay, in Britain; they opened up new, accurate, and elevated views of the scope of those studies to the minds of the youth of the university; they served to give its proper place to speculative philosophy as an independent science, and yet as casting illumination over all the fields of human knowledge; they proved by their practical influence that philosophy is not a mere training of the understanding to fit it for a place in the course of respectable and mediocre routine, but a discipline and occupation of the highest faculties of the man, as an intelligent and moral being, which no other branch of human study can supply.

Besides the writings referred to in the course of this notice, Sir W. Hamilton was the author of the following pamphlets:—1. "Letter to the Right Honourable the Lord Provost of Edinburgh, on the Election of a Professor of Mathematics; comprising observations on the value of mathematical science as an object of liberal study," 1838. 2. "A Correspondence relative to certain proceedings of the Town Council of Edinburgh affecting the philosophical professors of the University, and in particular the professor of logic and metaphysics," 1839. 3. "Be not Schismatics, be not Martyrs by Mistake; a Demonstration that the Principle of Non-Intrusion, so far from being fundamental in the Church of Scotland, is subversive of the fundamental principles of that and every other Presbyterian Church Establishment," 1843. 4. "A Letter to Augustus de Morgan, Esq., on his claim to an independent rediscovery of a new principle in the Theory of Syllogism," 1847.—J. V.  HAMILTON,, was born of Scotch parentage in Lincoln's inn, London, 28th January, 1728-29, and educated at Winchester school under the care of Dr. Burton. He was admitted a gentleman commoner at Oriel college, Oxford, 1st March, 1744-45, Dr. Bentham being his tutor. He afterwards became a member of Lincoln's inn, where he made deep research into the springs and principles of the English constitution, studied municipal law, and investigated the charters of our great trading companies. After his father's death, 15th January, 1754, he was elected a member of the British parliament for Petersfield in Hampshire. In 1755, in the course of a debate which arose on an address to the crown, in which the minister. Lord Hillsborough, took occasion to introduce an indirect approbation of the treaties concluded with the emperor of Russia and the landgrave of Hesse Cassel, he made his celebrated speech which earned for him the epithet of Single-speech. When the treaties came regularly before the house in February, 1756, he again took part in the debate, and on the 28th April following was appointed one of the lords of trade. In 1760 his acquaintance with Dr. Johnson commenced, with whom he lived in intimacy till Johnson's death. In 1761 he accepted the office of principal secretary to George, earl of Halifax, lord-lieutenant of Ireland; and the fame that he had acquired in England followed him into the Irish house of commons, where he made five speeches on various occasions, which fully satisfied the expectations of his auditors. In 1763 he was made chancellor of the exchequer in Ireland, which office he held till 1784. About this time the authorship of Junius's letters was, with very little probability, attributed to him. In 1792 his constitution was shaken by a paralytic stroke, and he died at his house in Upper Brooke Street, 16th July, 1796. Malone published his works and life, 8vo, London, 1808.—N. H.  HAMILTON,, archæologist and diplomatist, was educated at Harrow, and compelled to leave school prematurely by an accident which made him lame for life. Ill health prevented him from graduating at either university. In 1797 as an attache and private secretary, he accompanied to 