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stituting — for n we have

Log. n-*m x ("=}. + iHs'l—) 3 + *' )

a good converging series when n is any positive "u°lber greater than k. The modulus in both series being denoted by m. Again, when treating of the modulus, at p. *, "might taav« been added, that in any system of logarithms, the modulus u always equal to the Log. of (1 + 1+1+ —3 +1£3% + sJXs^f

These and other curious consequences might have been readily deduced, had the Editor adopted the ingenious method of investigating logarithmic series discovered by Lagrange, vi£>

by making the number ^=a =(l+o—I)" =((l+a—l) )*>

expanding the latter expression into a series by the binomial theorem, and then simplifying the result by supposing ««=<?. Indeed, we are'much surprised that Uiis method, and Lagrange's analogous process for finding the log. when the number is given, have not yet found their way into any of our English books.

Farther, we should have advised the Editor, when treating the subject of Annuities on Lives, &c. instead of referring solely to Demoivre who commonly reasons on a wrong hypothesis, to direct his readers to the valuable performances of Simpson, Price, and Morgan.

But the principal alteration we would beg to recommend in a new edition of this Introduction, is a farther deviation from the geometrical method than the Editor has yet ventured • to adopt; we advise this, not on account of any dissatisfaction with geometrical demonstrations, but for the sake of gaining space; since more topics may be investigated with equal perspicuity in smaller compass by the analytical method, Thus, after it is demonstrated geometrically that,


• ,„+i\ sin. a cos. b —sin. b. cos. a.
tin. [a oj =

+ .

■ , + .. cos. a cos. b — sin. a silt. b.

cot. (a ' 6) =. ■■

..... v __-v r

every thing else relative to the arithmetic of sines and tangents, the expressions for multiple arcs, &c. becomes a matter of pure analysis. In like manner, after deducing the property

of plane triangles, that —- = ■—» by means 6f a. diagram

sin. Bo. (where a and I are the sides respectively opposite the angles. A and B), the whole of plane trigonomptry may •be'jnvestlgated analytically- And again, in spherics, having shewn geometrically'that,, cos. a == co£: h cc^s,. c.-Hcos. A sin. b sin. cj \ -,.-.' ,\co^.^j= cos. a. cos. c'-(-_cps. B sin. A sin. cv i • . /:. cos.r = «os. a cos..£ + cos; C sin. -ft pin. A J.

^i$Tc«Baal»ing doctrine of spherical trigqnpiqetry follouvs easily, *y a-were transformation find substitutioft,of equations. - l?hei alterations here advijsed .need jifa.. h« avoided op the supposition that they will make the introduction too abstruse: yjftuve contrary; we think the change 'will father simplify its general appearance. None of the processes which we recoro,meudl;6he adopted, is so difficult, either in reality or in /appearance, as the method of deducing the logarithmic serieses now introduced ; with these changes, the introduction would have [a gre&er air of uniformity, and room would be gained .for the admission of much more useful matter into, tlie same space.' We have given these hints of improvements, not with. a view of cavilling, or expatiating oh defects.; but from a real "desire to tender more beneficial to tlie public, a work which, 'e\0U in its' present state, we consider as richly deserving encouragement.' - ■" .if. .

"Art. XVIII. The Nature and Importance of a Good Education, A Sermon' preached January 14, 1808, at the Rev. Mr. 'GafFee's Meeting, New . Broad Street, before the Promoters of the Protestant Dissenters-' Grammar-School, lately opened at Mill-Hill, Hendon*-Middlesexj To-which are annexed, the Regulations of the Society, .and^aXjst of .the'Subscribers, &c. By David Bogue, A. M. 8vo. pp. 40. : Price 1^ Co»der, 1808. !.',•

, A Combination of the noblest powers with the best intentions in human character, is the greatest benefit to the community, the highest condition of our nature, and the just object of pursuit in, all systems of educa'tion. To promote in some degree this illustrious and beneficent object, among society at large, is the paramount purpose of our own labours.; every institution, therefore, which is avowedly intended .and evidendy adapted to advance it, has especial claims to our cordial approbation. We do not hesitate in applying this character to the recent academical institution among Protestant Dissenters. And if any thing can increase our satisfaction at the commencement of an undertaking so congenial with our own in the rectitude of its .primary design, it is the consideration, that among the particular class of Qur countrymen, who are precluded by their conscientious opinions from resorting to the grand national Seminaries, an establishment for superior classical and mathematical tuition, has long been peculiarly desirable. A few sentences extracted from an Address whichhas been circulated by the Founders, and prefixed to this Sermon, will enable the reader to appreciate the principles they entertain.' '■''."> "Though the^School takes its denomination from the class of'Christians is which it originated, and ifl'which literary advantages are confessedly most wanting, it is not intended either to exclude all but the children of Protestant Dissenters, or to attempt making proselytes of such children as shall be sent thither from another community. The former would be illiberal, the latter disingenuous; and both would constitute a gross dereliction of that just, candid, and manly system, which Protestant Dissenter* should, Hbove all people, hold themselves bound to maintain, when they say, "Let every one be fully persuaded in his own mind." ._ « The Committee have engaged a very commodious house at Mill-Hill, • in the parish of Hendon, a situation peculiarly pleasant and. salubrious, where they hope many of the rising generation will imbibe the elements of sound literature and the principles of evangelical religion, and thus become a credit to the Institution, the joy of their parents, and blessings in every relation of social life."


'Let us hope that Patrons wh] not be wanted in support of an Institution so adapted to the exigency of the churches, and to which the public' attention would not have been solicited, but from a full conviction that the design is calculated to promote, far beyond the precincts of a party, the spirit ot that religion, which is at once the ornament and the hope of man." pp. v.—vii.

Mr. Boguehas ably discharged the two functions of advocate and counsellor. His discourse is replete with just views and manly sentiments ; it includes a Forcible train of argument in favour of the Institution," and a judicious exposition of principles to be observed in conducting it. The text is Prov. X. I. A wise son maheth a glad father. Having first developed the constituent parts of a good education, good principles, and literary acquisitions—he next specifies its fruits, capacity for business—habits of vigorous application—qualification for public employments—means of innocent relaxation—a higher susceptibility of future improvement—the increase of

knowledge in society—the"power of influencing the public opinion

ability to «erve the church of Christ—the promotion of religion and the divine glory. The style is energetic, rather than elegant; but well adapted to a discourse which is not declamatory, but argumentative. We must be contented with inserting the following specimens. The utility of early grammatical instruction is neatly explained.

« A boy at school, sitting down to learn a lesson in Virgil, with his grammar and his dictionary by his side, presents an object by no means beneath the notice of a philosopher. Invention, judgment, memory, are all called into exercise: to make out his task, they must often be exerted with patient attention and perseverance; and it is only after repeated trials, that he succeeds. Among other effects of such a manner of spending time, the faculties of the mind are strengthened, and the capacity for tvery usefuj employment in life considerably enlarged.' p. 10.

'That th» general sentiments of a country be good, is a matter of the

'highest importance. But this will always depend on two things on tb»

number of christians, and on the influence which by their talents they have acquired over the public mind. It is by a comparative few, that the tone is given. Hitherto, unhappily, in most countries, public opinions have not been on the side of truth, and piety, and goodness: but false ideas on rc»t ligion, on morals, on public institutions, and on the happiness of commun

Vol. IV, Mm

ties, have been prevalent in almost every land; and the great adversary of mankind has successfully contrived to get the moral taste of the nations Into his hands Hence evangelical doctrine is methodlsm, and a life regulated by the gospel of Christ is rigid puritanism. To how long a list might instances of this kind be extended! Of what consequence is it, then', that christians shou'd make themselves masters of public opinion, and give it a decisive tone in favour of every thing that is good.' pp. 18, IS.

* The system of education which it is designed to adopt, will form the pupib for usefulness in the various lines of business; and it will be a valuable introduction to more elevated literary pursuits, for such as intend to devote themselves to the learned professions, and a life of study. 'When you think or speak on the subject, always keep in view the two parts of which it consists, and the importance or the union of both in the scholar. 3T he principles to be taught are infinitely valuable: a life conducted under their influence leads to immortal glory. But if literature be not connected with them, the person will be much less useful than he would have otherwise been, and altogether unfit for some departments, which, when properly filled, conduce in a very high degree to the happiness of man. On the other hand, if learning be alone, if it be not united with good principles, it is a sword in the hand of a madman, who is as likely to attack his friends as his foes: it is a stately ship without a helm or a pilot. The pride of the scholar's heart will mislead him on the journey of life, and he is in imminent danger of employing his talents to the dishonour of God, and the injury of man: when oath are united, every thing that is wise and good may be expected from him.' p. 28.

Appended to the Sermon, are the General Plan, and a List of the earlier Patrons, of this infant Seminary.

Art. XIX Human Lotus best supported by the Gospel. A Semion preached in the Cathedral Church of St. Peter, York $ before the Hon. Sir Soulden Lawrence, Knt. one of the Justices of the Court of King's Bench, March 6, 1808. By the Rev. Francis Wrangham, M. A. F. R*S. of Trinity College, Cambridge. Published at the Request o"J\ the High Sheriff, and the Gentlemen of the Grand Jury. 4to. pp. 31. Price 2s. 6d. York* Wilson; Mawman, &c. 1808.

Tn this discourse, (from Deut. iv. 8.) Mr. Wrangham begins by extol* Jing the British Themis, or legal constitution of England. He then notices the reflections that have been cast on the insufficiency of its minute details, and the inefficacy of its sanguinary sanctions, to enforce, among our countrymen, the due observance of social duties. Hence he takes occasion to remark the great propriety with which legislators have "summoned the fear of tJie Lord in aid of their own denunciations." We were pleased to find Mr. Wrangham cautiously distinguishing the incidental and secular benefits of Christianity, from its chief and eternal object; especially as some of his clerical brethren appear more willing to recommend the religion of Christ, as an instrument for civilizing the barT "barous, and regulating the civilized, than to proclaim it as the dispensation of mercy to the guilty. . The minister of Christ, he observes,

« Must recollect, that the promotion of civil concord, though one of the most valuable of earthly blessings, is with him but a subordinate function r that his primary charge is to foray men, in Christ's stead, to be reconciled to Gad; and that therefore his foremost exertion in the service of his Great Principal must be, to make his hearers not good subjects, but good Christians—Christians in the peculiar acceptation, in which the name was first earned by the disciples at Antio. h; and which still, under a synonymous epithet, honourably stigmatises the evangelical followers of their Lord. In achieving the latter, indeed, he will incidentally have achieved both.

"This will demand a constant and conscientious exhibition of the peeu« liar truths, and precepts of the Gospel^ l'o g jard the sanctity of an oath, and to protect society from the effects of some of the grosser suggestions of the Tempter, it might possibly suffice funder the lack of nobler mitives) to announce a Jehovah ever competent, by his infinite knowledge and justice and power, to detect and condemn and punish the most secret trangressions. But these misrepresentations alone, however they may imperfectly serve the cause of secular and social duty, will never qualify mankind for their better, that is, their heavenly country. With the spirit of fear, which they are adapted to infuse, and which chiefly acts as a curb to prevent the perpetration of evil, must be combined the principle of love—all-bearing, all-believing, all-hoping, all-endunng love—kindled by a frequent display of the glorious work of Redemption in its whole tissue of causes, cost, and consequences, to animate to the doing of that which is.good.

"Let us not shun then, in miserable compromise with the sophistries of the sceptic, or the jibes of the scorner, to d dare to the best of our ability the whole counsel of God What he has commanded us to impart entire, let us not, under the affectation of improving, presume to curtail. There can be no partial reception of the doctrines of Christianity. It must be every thing, or it is nothing."

Mr. W. concludes with exhorting the higher orders of Society, to promote the salutary influence of religious principles among their inferiors, not merely by an exemplary observance of external forms, but by evincing in their whole demeanour a cordial zealous piety.

Ilighly as we are gratifi d with t:e just and pious sentiments of this discourse, we could wish they had been delivered in a more uniformly perspicuous style. The occasion perhaps did not justify that famiiiar and affectionate and zealous mode of address which we deem especially desirable, and doubt not that Mr. Wrangham adopts, in parochial sermons j but the requisite dignity might have been better attained by a somewhat philosophical discussion of the thesis on the same scriptural principles, than by a pomp of diction or an excess of rhetoric.

Mr. W. has subjoined an Appendix, referring to two distinct subjects. First, an account of the establishment of a smallparochial library of moral and religious publications, in the vestry room of his church. The plan and selection of works appear liberal and judicious; the clerk or schoolmaster is librarian, and the books are lent gratuitously. We are to glad learn that so laudable an undertaking has been received with great avidity; and strongly recommend it to general imitation. The other subject of the appendix, is a controversy in which Mr. W. has been involved with the venerable "Society for promoting Christian Knowledge" on the amount

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