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Art. VIII. The New Series of the Mathematical Repository.- No. 9. By Thomas Leybpurn, of the Royal Military College. 8vo. pp. 96. 2 folding plates. Priqe 4s. sewed. Shenyood and Co. 1809. TT is not, we are aware, the custom of -reviewers to notice periodical publications; but there is something so interesting in the plan, and so respectable in the execution of the Work before us, that we do not hesitate to make a deviation from our usual practice in its favour. The Repository, of which the present number terminates the second volume of the new series, is devoted exclusively to mathematics. . It is divided into three parts. The first contains solutions of questions (in every department of mathematical inquiry) proposed regularly in each successive halfyearly number of the work; the second consists of original essays; and the third comprehends mathematical memoirs; extracted from works of eminence, and translated in general from other languages.

The only works now at all resembling the present, established in this country are, we believe, the Ladies' Diary, -and the Gentleman's Diary— both Almanacs published annually by the Stationers' Company ; of these little productions, the former has maintained its utility and its reputation for more than a century, the. latter for 70 years ; each having been conducted from its commencement by a succession of mathematicians of considerable eminence. It was to these publications, together with some others long since defunct, (as the Palladium, the British Oracle, the Mathematician, Halliday's Miscellanea Curiosa Mathemalica, Turner's Exercises, Hutton's Mathematical Miscellany, &c.) and ihe woik now on our table, that we alluded in a former article*, as having been manifestly beneficial in promoting and diffusing the pure mathematical spirit.

In the Ladies' and the Gentleman's Diary, the portion devoted to mathematical investigation is so small, that their utility in this respect is of necessity much limited. On this account we are glad to observe the regular appearance of the Mathematical Repository, which, while its successive numbers are of such a size as to allow of extent and variety of discussion, is yet of a price that brings it sufficiently, within the purchase of every scientific man.

The original essays in the 2nd volume of this work amount to twenty one, some of which are highly curious and valuable. Among these we may specify Mr. Gough's application of properties of parallelograms to the moments of

* Vol. V. p. 1098.

forces; Mr. Cunliffe's theorems for finding the sums of in • finite series by means of circular arcs and hyperbolic logarithms; the application of a new dynamical principle by Mr. Bayley; and Mr. Barlow's demonstration'of a theorem, in the JDiophantine analysis. In this theorem it is affirmed, that "every integral number whatever, is either a square, or the sum of two, three*, or Jour squares"—a part.only, it may be observed, of a more general theorem originally given by. Mr. Fermat. Mr. Barlow, however, has the honour of being the first who has furnished a complete and satisfactory, demonstration of the theorem which has engaged his atten« tion.

The * memoir' in the present volume extracted from a foreign work is an extremely interesting one on Elliptic Transcendentals, by M. Legendre. Our only complaint respecting it is, that the portions in which it is published are too minute.

We cannot here enter into a particular examination of the first volume of the new series of the Repository, Dor of the three volumes in 12mo which constitute the first series. It is sufficient to say, that a large proportion of their contents may be read with advantage by young mathemati* cians, and with pleasure by the man of confirmed science.

We are happy, therefore, to commend this laudable attempt of the Editor to furnish amusement and instruction to his brother ' philomaths:' and we are conscious that when we warmly recommend the Mathematical Repository to those who delight in such speculations as it contains, we are at once discharging a duty to Mr. Leybourn and conferring a benefit upon the public.

.Art. IX. Remarks on Prisons. By Stevenson Macgill, D. D. Minister of the Tron Church of Glasgow. 8vo. pp. 80. price Qs. Longman & Co. Hamilton, Ogle, 1810.

A VERY general attention will ere long, We hope, be excited, to the •

established arrangements and practical administration of the national prisons. Dr. Macgill, as an able co-operator in the important work, of awakening and instructing the public mind on this subject, is intitled to our warmest thanks. He explains, in a concise but perspicuous manner, the true principles of incarceration ; and adverts successively to the objects which should Le kept in view when that measure is adopted. These are,

1, the most effectual and least severe methods of securing the person $

2, the various precautions to be observed both in the construction and management of prisons, so as to promote the health of those who are sonlincd; 3. the protection of the moral character and interests of the

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prisoners from further iojury; 4. regulations for securing a vigilant superintcndance of the actual management. On all these points, Dr. Macgill'Jgood sense and benevolence have furnished many very important hints; and we hope a time will come, when much of the advice contained in his publication will be enabled, through the powerful interposition of the legislature, to assume the authoritative aspect of law, and enforce iw claims to obedience. On the first topic, that of Health, Dr. M. observes,

■ « Imprisonment, of itself, from its natural effect on the spirits, will, in many cases, injure the constitution. This circumstance should render the community more anxious to provide such means of health, as the nature of a place of confinement will permit; and still more to guard against practices and customs which must be directly pernicious It is Do part of the punishment designed for prisoners of any description, that they should be given up to rheumatisms, scurvy, dropsies, fevers, and consumptions; that their strength should be wasted with sickness, their constitution undermined, and their lives embittered with excruciating pains, with debility of body and of mind, and their necessary consequences, poverty and wretchedness. Yet, in such an unfortunate manner have many ot our prisons been constructed and managed, that often the unhappy prisoners have been punished, not-only with confinement according to their sentence, but with sufferings, and even death, of the most direful kind; or, when death has been protracted, with a long train of diseases and afflictions, during die few years which they dragged out of their miserable existence. If justice and humanity revolt at such treatment, even of the guilty; with what feelings should we think that it may possibly be the fate of the innocent and unfortunate!' pp 7, 8.

The remarks on diet, separation of prisoners, modes of employing their time, and encouraging industry, the exclusion of intoxicating liquors, though not professing to be very new, are judicious and well arranged: and we are pleased to observe the stress which is laid on providing effectual means of religious instruction. We approve of the recommendation, that a better order of persons be employed as governors of prisons, that their office be rendered more respectable, and their sala. ries liberal: that all fees be abolished, (excepting, perhaps, such as are paid for extra conveniences,) and that the wages of inferior officers be ascertained. The plan recommended for securing an effectual superintendance, may be open to some objection, inasmuch as there may be no possibility of finding a due number of qualified persons to undertake it gratuitously: in many situations, however, it will undoubtedly be practicable. Every humane reader will; in our opinion, feel it his duty to promote the circulation of this pamphlet.

Art X. Earneit Contention for the True Faith, a Sermon preached at Scarborough, at the Primary Visitation of the most Reverend Edward, Lord Archbishop of York, July 28, 1810, by Francis Wrcnghara, M. A. F. R. S. of Trinity College Cambridge, published at the Re. quest of the Clergy, with the Approbation of his Grace. 4to. pp. 50. price 3s. 6d. Mawman.

N exhorting the clergy to * strive for the faith,' it was natural for Mr. Wrangham to specify the object of contest, and the mode of car

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rying it on. Accordingly he proposes to ascertain—what is the character of the faith referred to, (more properly what is the faith) and— what are the modes of contention recommended, in the third verse of Jude. With regard to the object of contest, Mr. W. with great propriety considers it as consisting of the Christian doctrines,—especially those relating to the person and office of Jesus Christ, the influence of the Holy Spirit, and good works. As to the latter part of the discourse,—the mode of contending for the Christian doctrines, he insists on the necessity of residence,—of cultivating a catholic charity, which, while it avows with a manly spirit its own principles, makes use of kindness and condescension rather than haughtiness and severity, to communicate instruction and gain converts—and of deeply imbuing the mind with Christian sentiments, and revering the doctrines as well as the precepts of the gospel. He states also, as indispensable, solid learning, and a good life, points out the advantages of catechising, and of instituting and superintending parochial schools; and recommends the unremitted exercise of private devotion, and the diligent performance of all the duties of Christian sympathy and benevolence. To say that advice like this is consonant to the gospel, would be idle; nor is it necssary to hint at the benefits that would result from its Universal adoption, or even to recommend a discourse containing lessdni so worthy of general attention.

We are sorry, however, that Mr. W. has evinced by his own example, that it is far less difficult to deliver than to practice the precepts of charity. While we wish to maintain a strict neutrality between the adherents of Calvin and Arminius, yet we cannot but be of opinion, that he has inveighed against the followers of the reformer with undue severity; that he has, though unintentionally we are sure, misrepresented their sentimenf; and that it is rather illogical to consider their virtues not so much the fruit of their principles, as the production of a soil which their principles cannot deteriorate. The Stoics are justly regarded as the most virtuous and rational of all the ancient philosophers. There.was a sublimity in their speculations, and an elevation in their morality, that have been surpassed only by the doctrines and precepts of Christianity. None, however, of the modern sects of Christians approach so near the Stoics as the Calvinists. It has always appeared to us extrabrdihary, that there should have been a multitude of Christian philosophers, who have observed no moderation either in their praise of the Stoics, or in their vituperation of the Calvinists ; who have pronounced the most ostentatious panegyric on the spirit and virtues of Zeno and Seneca, but have seemed anxious to exclude Calvin and Hooker from the rank of rational and virtuous men, while, at the same time, the severe and martly virtues are cultivated by the Calvinists with greater diligencce and' more success, 4than by their favourite philosophers. We think this consideration ought in some measure to cool the heat and irritation of those worthy persons, who will not suffer themselves to reflect on the dogmas of the reformer of Geneva with common patience. It should induce them £o treatthe favourers of his system wiih some degree of respect, and acknowledge their virtues with candour; and, if they are desirous of making proselytes, to employ cogent reasonings and mild persuasion instead of obvious misrepresentation and domineering abuse.

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In point of style, this sermon is a good deal overlaboured. Mr. Wrangham appears to have adhered rather too stiffly to the maxim of Quintilian—Dicturus intucatur afiud quern dicendum sit; and to have imagined majesty and pomp so indispensable, in a discourse before an archbishop and his clergy, that he has made no scruple of sacrificing not merely amenity and grace, but perspicuity itself. In one place he talks of the •contingent infliction of penalties to be enforced by the vengeance of an irritated parishioner'; in another, of a * corollary in direct hostility to the dangerous deductions of the fiduciary sluggard;' and in a third, of the ' obligation of catechising, not perfunctorily exacting a mechanical recital.' Pulpit discourses, of whatever kind, should certainly be quite free from every thing mean and vulgar, and in all circumstances 'retain an air of dignity. But the dignity should arise from the subject, not from the dress. It should be the simple majesty of truth, always the more commanding the less adorned.

To the sermon are appended a copious variety of notes, which may amuse and even instruct those, who will find, in the sermon itself, a recommendation of their own spirit and practice.

Art. XL The Pastoral Care. A Didactic Poem, in Three Parts. Addressed to the Junior Clergy. With three engravings, fcp. 8vo. pp. 170. price 12s. bds. Hatchard. 1808.

IN this performance, we have plenty of good advice, though of the tritest kind. The standard, however, of theological sentiment and clerical deportment is not so exactly conformable, perhaps, to that of the scriptures, as to deserve unqualified approbation*- To afford some notion of its poetical merit, we shall quote a favourable extract, regarding the visitation of condemned prisoners.

* Heaven hath for death-bed penitents a place,
But higher theirs who run the righteous race;
Who this denies, a God unjust obeys,
And cheers survivors in their hell-ward ways.
Nor too much darken, nor too much illume
The conscience shadowed in remorseful gloom:
Four words thy text—" Despair not, nor presume."

« For many a daring culprit owes his loss
To pondering on his brother of the cross;
Who ne'-er perhaps till then had heard of heaven;
Perhaps with few foul deeds to be forgiven;
Perhaps whose breast the witness of the breast
Saw shining from its slough, and loved, and blest.

'Small hopes can hence be founded in a sigh
Half-heaved by criminals about to die:
Woe, to which heaven its mercy will impart,
Must be the resurrection of the heart:
Be that, which could it fill an ampler span,
Would make another and a new born man:
-~r!e his, who in the few short hours he owns,
Deplores, confesses, trenibUs, weeps, atones, j

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