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weeks since, will occur to many of our readers as an instance. But when it is considered how very rarely there has been, or how very improbable it is there will be, a war answering in even a moderate degree, to this description, how vain and wicked it is to hazard life in any war that does not answer to this description, - how very seldom therefore in the lapse of ages there can be any such thing as 'fields and waves of glory,'--and at the same time what accursed and unlimited mischief has been done by the universal practice of associating ideas of glory with feats

' of valour, regarded abstractedly from the motive and the cause :--considering all this, we think we can never be wrong in condemping, emphatically, a loose, unqualified way of extol. ling the military character; nor can we be far wrong in condemning the ready assumption, in the case before us, that the daring and aspiring youth' bere meant to be brought in view by the supposition of an opposite character, will be solemnly and punctiliously conscientious in first examining the justice of the cause that calls him to these • fields and waves of glory.? Alas! the probability is that this unfortunate youth,' who is to feel so noble a scorn of the grovelling projects for the security and protraction of life, is entering on war merely as a professional business, which he is to rosecute

i zealously without ever giving his understanding and conscience the trouble of one serious reflection on the justice or injustice of its enterprises, his duty being simply to execute what he is appointed to; and our Author's incautious mode of repeating the common and pernicious language about 'glory,' will certainly tend to confirm his thoughtless confidence in the rectitude of such a plan of life.

We baruly need notice that our Author's reprehension of the solicitude for long life, is accomp nied by an inculcation of the importance of health He would join in the applause of De Witt, who is described as " careful of his health, and negligent of his life.

He exbiiits his Dial in the three characters, oi Monitor, Remembrancer, and Comforter; and proceeds to illustrate the mode and benefit of its application in these characters io e:ch of the successive stages of life. There is much force and beauty in the admonitions addressed to the young, and considerable oint and dexterity in the manner of making the Bioscope warn them against the presumption that they shall live through the whole sequel of years lying beyond the point wbich the index marks as their present year. We trapscribe an elegant and pleasing paragraph, describing with some truth, but we fear with a considerable mixture of poetr, the happy combination of religion with the feelings of child wood.

• And here let me observe, that there is no season of life in which he bright comforts of religion, afforded in the prospect of a life in

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heaven, are so sensibly and purely felt, as in that of a guileless and religious childhood. That this should be so, will not surprise us when we reflect, that Christ himself has pointed out that age as the best representation of the inhabitants of heaven. That it is so in fact, all those can testify, whom God has blessed with the commerce of young minds, grounded in religion, and practised to religious obedience. The spring of youth is more congenial to the temperature of celestial joy, than either the summer, the autumn, or the winter of years. And, if a relish for that joy be imbibed in that age, it will tincture, with the lustre and serenity of spring, all the succeeding seasons of life. A chastened exaltation of the mind, will be the natural and certain consequence of such a temper ; than which nothing can so well nt us for duly combining our services to God and man, while we remain here, under our discipline of trial.'

There is in the admonitory reflections on each of the succeeding periods of life a very cogent seriousness, which acquires a still more impressive solemnity as the work auvances towards a view of its concluding stage. There are soine pointed observations on the reluctance to admit the application of the epithet old to given periods of life--for instance to the age of sixtywhich nature bas most clearly brought within the limits subject to that denomination. There is a striking reprehension of a delusion which we will quote the Author's own words to expose.

. And here I shall take occasion to remark, that there is not a more common or more delusive error, (and which, however soothing it may be to the imagination, is most treacherous to the reason,) than that of looking forward to old age as a station, in which we are to halt, and take oui rest, at the close of the journey of life For, first, we may never attain old age, and then, how mischievous must be the illusion of living always with a view to a period at which we never shall arrive.

“ The laws of probability.” said Mr. Gibbon, at the age of fifty-two, * so true in general, so fallacious in particular, still allow me about fifteen years. I shall soon enter the period which, as the most agreeable of his long life, was selected by the judgement and experience of the sage Fontenelle.” But the sage Fontenelle said so upon the retrospect, and not on the prospect. Mr. Gibbon died within five years.

• But suppose that we shall attain to old age ; still, we shall find it no stationary post, or place of halting. To look to old age as a station, and to console ourselves, as we travel on in life, with the prospec of that imaginary station, is as if a man were journeying from Bath to London, and looked forward for his repose between Kensington and Hyde-Park Corner. The three or four last miles of that journey, will well answer to the last years of the journey of life. The traveller will certainly only look for his epose when he shall be arrived at his home in the Capital.' 'And so in the journey of life. The last years of life neither promise, nor adıninister, any period of retreat in themse ves : for life proceeds as fast, (nay, sensibly faster,) in old age as in any other part of its course; it can then only be in the near prospect of retreat, not in the possession of it.'

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Mr. Gibbon is again introduced, with some very serious comments on the state of feeling in which lie expressed his anticipations of the closing period of life, and the gloomy thoughts which he sometimes sent forward beyond its termination. A forcible contrast is drawn between this disconsolate perverter of the public mind, and Addison, viewed in his last moments. It was proper to select a literary person for the bright side of the contrast, but we may doubt whether Addison was exactly the one required, and may feel some little defect of sympathy with the very animated sentiments with which our Author contemplates the character.

With the most cordial and respectful admiration of the excellent Sir W. Jones, our Author censures bis celebrated Andrometer, as a visionary and deceptious' scheme of life.

Nearly forty pages are occupied with the Epistle of Paulinus, bishop of Nola, written about the year 400, to Celantia, a Roman Christian lady, who had urged him to draw out for her a brief and easily applicable rule of Christian life. Our Author says it has never before been translated into English. It certainly was worth translating ; it is earnest, simple, practical, and devout; but not in the least distinguished by any thing prominently eloquent or powerfully intellectual.

The Elementary View of General Chronology,' is a very useful addition, furnishing much information in a brief and perspicuous form.

We dismiss the book with a very cordial respect for the Author, and a confidence that the book will contribute very essentially to the mosi important improvement of many readers. It is the work of a practised, a very amply instructed, and a devout thinker. It conveys a kind of admonition emphatically necessary, and not often conveyed so well. We have seldom seen seriousness so graceful.

Art. V. Philosophical Transactions of the Royal Society of London,

for the Year 1812. Parts I. and' II. Mathematical Papers. PAPERS devoted to the more abstruse sciences, in the periodi

cal volumes of the London Royal Society, are gradually increasing both in number and in importance; but the Philosophical Transactions are still inferior in this respect to the successive volumes published by the French Institute, and even to those laid before the world by the Academy of Sciences at Petersburgh. We rejoice, however, to notice an obvious improvement; and hope the time is not far distant, when the talents of English mathematicians will be as well known and appreciated on the Continent, through the medium of the London Philosophical Transactions, as they were previously to the unhappy proceedings in the Royal Society which caused the secession of so many valuable members in the year 1784.

The first two papers in the present volume, are by Mr. James Ivory, of the Royal Military College.

1. * On the grounds of the method which Laplace has given in the second chapter of the third book of his Mécanique leste, for computing the Attractions of Spheroids of every description.'

II. · On the Attraction of an extensive class of Spheroids.'

These papers occupy eighty pages, and are extremely va-, luable. All who have attended to the theory of physical astromony, are aware hoth of the importance and of the difficulty of the general problem relative to the attractions of spheroids, when applied to the figures and actions of the planetary bodies. Much was done by Maclaurin, Euler, Dalembert, Lagrange, and Legendre, in succession. But Laplace, in his admirable disquisition on the figure of the planets, has been regarded as having made the nearest approximation to a complete solution : his investigations, indeed, having, with one or two exceptions, been acquiesced in, and adopted, by all his contemporaries. Laplace, in his inquiries, did not seek irectly an expression of the attractive force, but investigated the value of another function, from which the attractive force in any proposed direction may be inferred by means of easy algebraic operations. His method is extremely ingenious and elegant; but Mr. Ivory shows that it is not to be relied upon, because it comprises an inaccurate theorem.

. I cannot grant (says he) that the demonstration which he has given of his proposition is conclusive. It is defective and erroneous, because a part of the analytical expression is omitted without examination, and rejected as evanescent in all cases; whereas it is so only in particular spheroids, and

not in any case on account of any thing which the author proves. Two consequences have resulted from this error; for, in the first place, the method for the attr.ction of spheroids, as it now stands in the Mécanique Céleste, being grounded on the theorem, is unsupported by any demonstrative proof; and secondly, that method is represented as applicable to all spheroids differing but little from spheres, whereas it is true of such only as have their radii expressed by functions of a particular class.'

Mr. Ivory proceeds, though with all that deference which is due to the very extraordinary genius and acquirements of M. Laplace, to retrace the steps of his investigation; whence, and by occasionally diverging for a short period into another tract, he renders evident both the error of the profound French philosopher, and the cause of it. He shews fully, that Laplace's theorem, which, in the law of attraction which obtains in nature, is contained in Equation (2) No. 10, Liv. 3, Mécanique .

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leste, is exclusively confined to that class of spheroids which, while they differ little from spheres, have, moreover, their radii expressed by rational and integral functions of a point in the surface of a sphere. Our ingenious examiner, however, admits, that notwithstanding the defect in the theorem,' the real utility and value of Laplace's selection of the problem of attractions will not be much diminished by its failing in that degree of generality which its author conceived it to possess.'

An Appendix to this paper contains an account of some investigations of M. Lagrange's, directed to the same object; and shows that they fully confirm the reasonings and observations of Mr Ivory

In the second of these papers, Mr. Ivory proceeds to investigate the attractions of a very extensive class of spberoids, of which the general description is, that they have their radii expressed by rational and integral functions of three regular coordinates of a point in the surface of a sphere. This class comprehends the sphere, the ellipsoid, both sorts of elliptical spheroids of revolution, and an indefinite number of other figures, as well such as can he generated by the revolution of curves about their axes, as others which cannot be so described. The problem of attractions is well known to contain two cases. 1 When the density of the attracting body is uniform throughout. II. When it varies according to any given law. The first of these is that in which the difficulties occur, and is that to which Mr. Ivory has directed his attention. His mode of procedure is exceedingly elegant and ingenious; and in the course of it, he has struck out a real and important discovery; for he has demonstrated that the attraction of a homogeneous ellipsoid upon any external point, rehaterer, muy be reduced by an ingenious and simple transformation to that of a second ellipsoid upon a point within it. It is not a little curious to remark that, while this discovery seems comparatively to have been little regarded among English mathe raticians, it has been highly extolled by our Continental neighbours, one of whom, M. Legendre, when speaking of it, says, “ Thus the difficulties of analysis which the problem eshibited disappear at once; and a theory which belonged to the most abstruse parts of mathematics, may now be explained in all its generality in a manner almost entirely elementary.'

Mr. Ivory's method consists in causing the surface of a second ellipsoid to pass through the external point. The principal sections of this second ellipsoid, are situated in the same planes, and referred to the same foci, as the corresponding sections of the given solid. Then upon the surface of the first ellipsoid a point is taken, such that each of its co-ordinates is to the corresponding ordinate of the exterior point, in the same ratio as

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