exacted from every English peasant and yeo- | and there is no mode of encouraging them so man. What manly exercise now forms part of the discipline of youth? The governors of many of those states which call themselves free would be afraid to place arms in the hands of the population at large, or to encourage them to learn the use of implements of defence; otherwise the rifle would now be, in the hands of an English peasant, what the bow was in former times. The hour will arrive when rulers, who have been accustomed to place their whole reliance upon standing armies, and to distrust the loyalty of their own population, will have reason to regret the decay of that self-relying spirit which they have laboured to extinguish. I do not advocate the revival of pugilistic combats, though much may be said in favour of that barbarous amusement; but I cannot read Virgil's account of the games practised by the followers of Eneas, without feeling how immeasurably superior was the spirit which is breathed in the following lines— Hi proprium decus et partum indignantur honorem In a well-governed community not only should the population be encouraged to practise all sorts of gymnastic exercises; but also they should be trained to military evolutions, and to the use of arms. For such purposes days ought to be set apart, and prizes ought to be distributed by the municipal authorities. The acquisition of money has become the sole object of pursuit in modern days. Mammon now rules the civilized world with imperious sway. It should be the aim of the statesman to impart nobler emotions, more generous aspirations, than those which the love of gain can inspire. There are some who affect to disapprove emulation in every form-whether in a boatrace or in an academy. Yet even such squeamish moralists may assist in providing recreation for the people. They cannot object to throw open to the multitude zoological collections, botanic gardens, museums of painting and sculpture, or to encourage attendance upon lectures directed to the advancement of literary and scientific knowledge. It ought to be the pride, as it is the duty, of an enlightened government to encourage all such pursuits, legitimate as that which calls into action the co-operation of the people themselves. Hence the municipal representatives of the people should not only be empowered, but stimulated, to provide in each locality such arrangements as shall contribute in the highest attainable degree to the health, recreation, and intellectual improvement of the population. There is no village, however small, in which something might not be done to promote the enjoyment of the inhabitants. These things are in some countries left undone, merely because no organization has been formed for carrying such objects into effect. "What is everybody's business is nobody's," says the proverb. It appears like intrusion on the part of an individual to do that for the public which the public neglects to do for itself; and if a benevolent or public-spirited individual hazards such an intrusion, some sinister motive will generally be imputed to him. Take the simplest instance that can be brought forward in illustration of this observation. It generally happens that, in the vicinity of every village, there are spots of favourite resort, which attract by their beauty of scenery, or by some other charm. It naturally occurs to every one that seats should be provided in such places for the accommodation of the public, yet seats are not provided. There is no public body authorized to make such arrangements, and each individual says to himself, "It is not my business. Why should I be called upon to expend my private funds for the accommodation of the public?" Or, if he be willing to incur the expense, he is deterred by the consideration that some unworthy motive will be attributed to him, in case he undertake to provide the desired accommodation. Were political institutions organized with a view to promote the happiness of the people, much would be done that is now left undone; much would be left undone that is now done. To exact taxes which shall be squandered upon the parasites of government, and to coerce those who offend against laws enacted for the maintenance of an artificial state of society, which is often repugnant to the requirements of nature, is too generally the principal, if not the sole object to which the whole energy of civil administration is directed. If taxes were levied with a view to promote the well-being and enjoyment of all classes of the community, they would be paid without reluctance, and universal contentment would render superfluous many of the ex pensive appliances now employed for the re- | sculpture, and natural history, &c. In the straint and coercion of a discontented population. There is, perhaps, no country in Europe in which so little has been done to promote the amusement of the people as in the United Kingdom. Upon the Continent there are few towns of any considerable size in which arrangements have not been made, either by the central government or by the municipal authorities, to give to the inhabitants the pleasures afforded by public promenades and gardens, military music, theatres, museums of painting, United Kingdom, on the contrary, even the public squares are for the most part reserved exclusively for the enjoyment of the privileged few, instead of being thrown open to the whole population; and access to the repositories of art, nay, even to the glorious old cathedrals which were erected during the time which we presumptuously designate as "the dark ages,” can seldom be procured except by payment of a fee on admission. Yet we boast of modern refinement, civilization, progress, and philanthropy. SIR WILLIAM ROWAN HAMILTON. BORN 1805 DIED 1865. ing Trinity College in 1822 he carried everything before him, and attracted the attention of Dr. Brinkley, himself an able mathematician, who took him under his personal care. But Hamilton soon began those original investigations which surpassed the productions of his masters, and gained the admiration of the whole scientific world. Even so early as 1824 Dr. Brinkley reported: "This young man, I do not say will be, but is the first mathematician of his age." It was in this year he contributed a paper on "Optics" to the Royal Irish Academy. The ability of this paper being so widely recognized, he was induced to work up this and other studies into a complete form, and these he elaborated into his Theory of Systems of Rays, a production which placed him in the front rank of scientific thinkers. [Sir William Rowan Hamilton, astronomer | he had reached his sixteenth year. On enterroyal for Ireland, was born in Dublin in August, 1805. His father, a well-known and able solicitor, traced his descent from a branch of the Scottish Hamiltons, who, like many other Scottish families, settled in the north of Ireland in the reign of James the First. The mother was related to Hutton the mathematician, from whom in some measure Hamilton's talent in this direction may have been inherited. Intended for the East India Company's service, he was at a very early age placed under the care of his uncle, the Rev. James Hamilton, curate of Trim, to be educated by him. The curate evidently gave his pupil the full benefit of his own learning, for when only four years of age young Hamilton had already made some progress in Hebrew. His studies were further extended to a knowledge of Greek and Latin, with the rudiments of French, Italian, Spanish, and German, Syriac, Persian, Arabic, Sanscrit, Hindustani, and Malay, so that at the age of fourteen he was master of several of these languages and tolerably familiar with the others. In 1819 he wrote a letter to the Persian ambassador, who declared in astonishment that he could not believe there was a man in England equal to writing such a letter in Persian. But it was not in linguistic attainments that the chief fame of Hamilton was to rest. At an unusually early age he displayed a taste and aptitude for the study of mathematics. In this abstruse science, although self-taught, he made such progress as to have covered the ground of the usual university course before In 1827, while still an undergraduate, and working for a fellowship, he was elected to the chair vacated by the retirement of his friend | Dr. Brinkley, and he thus became professor of astronomy and astronomer royal for Ireland. He was then only twenty-two years of age. He took up his residence at the observatory at Divesink, near Dublin, where his sisters, ladies of unusual abilities, resided with him until his marriage in 1833 with Miss Bayly, daughter of the rector of Nenagh. On the occasion of the first meeting of the British Association held in Dublin in 1835, Mr. Hamilton, who as president delivered the usual address, was knighted by Lord Mulgrave, the lord-lieutenant. In 1837 he was elected president of the Royal Irish Academy, and from it received a pension of £200 a year, | ment not superfluous; or rather, it is indiswhich was afterwards continued to his widow. pensable that as a clear definition, or at least He was a member of most of the scientific exposition, of the precise force of each of these societies of Europe, and for his researches re- old marks, used in new senses, should be given, ceived the much-prized distinction of being as it is in my power to give. Perhaps, indeed, elected an honorary member of the Academy I may not find it possible to-day to speak with of St. Petersburg. what may seem the requisite degree of fulness of such exposition of more than the two first of these four signs; although I hope to touch upon the two last of them also. The numerous contributions he made to scientific societies and publications are proofs of his extraordinary genius and abilities, but those on which his fame chiefly rests are his Lectures on Quaternions delivered in 1843. Hamilton's labours and theories have not only stood the test of recent investigation, but have been extended and worked upon by men eminent in science at home and abroad. He was a mathematician, a metaphysician, a natural philosopher, a poet, and an orator. In 1865 his health began to decline, and shortly after his last appearance in public at the opening of the Dublin Exhibition, an attack of gout, to which he was subject, carried him off on the 2d September of the same year, aged sixty. He was fond of poetry and of poets, and was pleased to count amongst his friends Coleridge, Southey, Wordsworth, Mrs. Hemans, and others. He was not devoid himself of some poetic talent, but perhaps it is well that too much ability as a versifier did not step in to distract his attention from abstract science. A list of his works would occupy too much of our space: suffice it to say that there was no mathematical discussion of importance during his time to which he did not make a contribution.] ON SYMBOLS. (FROM "LECTURES ON QUATERNIONS."]) The object which I shall propose to myself in the lecture of this day is the statement of the significations, at least the primary significations, which I attach in the calculus of quaternions to the four following familiar marks of combination of symbols, + which marks or signs, are universally known to correspond in arithmetic and in ordinary algebra to the four operations known by the names of addition, subtraction, multiplication, and division. The new signification of these four signs have a sufficient analogy to the old ones to make me think it convenient to retain the signs themselves; and yet a sufficient distinction exists to render a preliminary com 1 By permission of Messrs. Hodges, Foster, & Co. First, then, I wish to be allowed to say, in general terms (though conscious that they will need to be afterwards particularized), that I regard the two connected but contrasted marks or signs + and -, as being respectively and primarily characteristics of the SYNTHESIS and ANALYSIS of a STATE of a progression, according as this state is considered as being derived from, or compared with, some other state of that progression; and with the same kind of generality of expression I may observe here that I regard in like manner the other pair of connected and contrasted marks already mentioned, namely, × and ÷ (when taken in what I look upon as their respectively primary significations), as being signs or characteristics of the corresponding SYNTHESIS and ANALYSIS of a step in any such progression of states, according as that step is considered as derived from, or compared with, some other step in the same progression. But I am aware that this very general and preliminary statement cannot fail to appear vague, and that it is likely to seem also obscure, until it is rendered precise and clear by examples and illustrations, which the plan of these lectures requires that I should select from geometry, while it allows me to clothe them in astronomical garb; and I shall begin by endeavouring thus to illustrate and exemplify the view here taken of the sign—, which we may continue to read, as usual, MINUS, although the operation of which it is now conceived to direct the performance is not to be confounded with arithmetical, nor even, in all respects, with common algebraical subtraction. I have said that I regard primarily this sign, or minus, as the mark or characteristic of an analysis of one state of a progression when considered as compared with another state of that progression. To illustrate this very general view, which has been here propounded at first under a metaphysical rather than a mathematical form, by proceeding to apply it under the limitations which the science of geometry suggests, let SPACE be now regarded as the field of the progression which is to be studied, and POINTS as the states of that pro gression. You will then see that in conformity with the general view already enunciated, and as its geometrical particularization, I am led to regard the word "minus," or the mark, in geometry, as the sign or characteristic of the analysis of one geometrical position (in space), as compared with another (such) position. The comparison of one mathematical point with another, with a view to the determination of what may be called their ordinal relations, or their relative position in space, is in fact the investigation of the GEOMETRICAL DIFFERENCE of the two points compared, in that sole respect, namely position, in which two mathematical points can differ from each other. And even for this reason alone, although I think that other reasons will offer themselves to your own minds when you shall be more familiar with this whole aspect of the matter, you might already grant it to be not unnatural to regard, as it has been stated that I do regard, this study or investigation of the relative position of two points in space as being that primary geometrical operation which is analogous to algebraic subtraction, and which I propose accordingly to denote by the usual mark (-) of the wellknown operation last mentioned. . . To illustrate first by an astronomical example the conception already mentioned of the analysis of one geometrical position considered with reference to another, I shall here write down, as symbols for the two positions in space which are to be compared among themselves, the astronomical signs and t, which represent or denote respectively the sun and earth, and are here supposed to signify, not the masses nor the longitudes of those two bodies, nor any other quantities or magnitudes connected with them, but simply their situations, or the positions of their centres regarded as mathematical POINTS in space. To make more manifest to the eye that these astronomical signs are here employed to denote points or positions alone, I shall write under each a dot, and under the dot a Roman capital letter, namely A for the earth and B for the sun, as follows: O B A and shall suppose that the particular operation of what we have already called analysis, using that word in a very general and rather in a metaphysical than in a mathematical sense, which is now to be performed, consists in the proposed investigation of the position of the sun, B, with respect to the earth, A; the latter being regarded as comparatively simple and known, but the former as complex, or at least unknown and undetermined; and a relation being sought which shall connect the one with the other. This conceived analytical operation is practically and astronomically performed to some extent whenever an observer, as for example my assistant (or myself) at the Observatory of this University, with that great circular instrument of which you have a model here, directs a telescope to the sun; it is completed for that particular time of observation when, after all due micrometrical measurements and readings, after all reductions and calculations, founded in part on astronomical theory and on facts previously determined, the same observer concludes and records the geocentric right ascension and declination, and (through the semidiameter) the radius vector (or distance) of the sun. In general, we are to conceive the required analysis of the position of the ANALYZAND point B with respect to the ANALYZER point A to be an operation such that, if it were completely performed, it would instruct us not only IN WHAT DIRECTION the point B is situated with respect to the point A, but also AT WHAT DISTANCE from the latter the former point is placed. Regarded as a guide, or rule for going (if we could go) from one point to the other, . . . the result of this ordinal analysis might be supposed to tell us in the first place HOW WE SHOULD SET OUT, which conceived geometrical ACT of setting out in a suitable direction corresponds astronomically to the pointing or directing of the telescope in the observation just referred to. And the same synthetic rule, or the same result of a complete analysis, must then be supposed also to tell us, in the second place, HOW FAR WE OUGHT To Go in order to arrive at the sought point B, after thus setting out from the given point A in the proper direction of progress (this direction being, of course, here conceived to be preserved unaltered), which latter part of the supposed guidance or information corresponds to the astronomical inquiry, how far off is the sun, or other celestial object, at which we are now looking with a telescope properly set. ... Now the whole sought result of this (conceived) complete analysis of the position B with respect to the position A, whether it be regarded analytically as an ordinal relation, or synthetically as a rule of transition, is what I propose to denote or signify by the symbol B-A, formed by inserting the sign minus | tively the initial and final states. We may then, as often as we think proper, paraphrase (in this theory) the geometrical symbol B-A by reading it aloud as follows, though it would be tedious always to do so, “B analyzed with respect to A, as regards difference of geometrical between the two separate symbols of the two points compared, the symbol of the analyzand point B being written to the left of the mark -, and the symbol of the analyzer point A being written to the right of the same mark; all which I design to illustrate by the follow-position." But for common use it may be ing fuller diagram, sufficient (as already noticed) to retain the shorter and more familiar mode of reading, "B minus A," remembering, however, that (in the present theory) the DIFFERENCE thus originally or primarily indicated is one of POSITION and not of magnitude; which, indeed, the context (so to speak) will always be sufficient to suggest or to remind us of whenever the symbols A and B are recognized as being what they are here supposed to be, namely, signs of mathematical points. Small Farmers; The Agricultural Labourer Viewed; Hints on Emigration to Canada, &c. He also translated a volume of Monod's sermons from the French. He contributed regularly to Blackwood's Agricultural Magazine, Chambers's Journal, and other periodicals. Notes and Gleanings of the County Wexford was his last work. The Royal Dublin Society awarded him a gold medal in recognition of his services, and he had a pension from the Literary Fund. He died on October 24, 1875, in his eighty-seventh year. This brief sketch is taken from the memoir in Webb's Compendium of Irish Biography.] [The Rev. William Hickey is perhaps better | of Everyday Life; Irish Cottagers; Plea for known as "Martin Doyle," the nom de plume under which he published the greater number of his works. He devoted his pen principally to works with the praiseworthy view of helping to raise the material condition of his poorer fellow-countrymen. A clergyman himself, and the son of a clergyman, he also deserves praise for having avoided in his works subjects which might tend to excite religious animosities; and in the discharge of his parochial duties he was equally careful in avoiding matters of controversy. The exact date of his birth is uncertain, probably it took place about the year 1787. He was the son of the Rev. Ambrose Hickey, rector of Murragh, county Cork. He graduated at St. John's College, Cambridge, and he also took the degree of M.A. in Trinity College, Dublin. Ordained in 1811, he ministered in succession in Dunlecky, county Carlow; Bannow, Kilcormick, Wexford, and Mulrankin, to the last of which he was appointed in 1834; there, too, he remained for the rest of his life. He was prominent in founding societies for the encouragement of agriculture, and as early as 1817 began to employ his pen in promoting the same cause. His first work was on the State of the Poor in Ireland. There followed a large number of works on kindred subjects: Hints to Small Farmers; Common Things HOME REFLECTIONS ON HOME (FROM "THE COMMON THINGS OF EVERYDAY LIFE.") Every man's proper dwelling is the domestic hearth, wherever this is or whatever its character, whether in town or country. The occupations of life necessarily cause the members of a family to be separated many hours of the day. After the care and toils of business are ended, how gladdening to turn to the peace and quiet of a happy home, to have |