That religious inquiries may be pursued to a point, on which human reason will meet with difficulties that it cannot resolve, is most true. But what is the moral caution which this fact should impress upon our minds? Is it, that we should receive these inexplicable difficul ties with implicit faith? No such thing. The true moral admonition is this that we ought not to reject truths, which it has pleased God to reveal, and which we can understand because, when extending our researches beyond what is revealed, we are stopped by difficulties which our reason cannot penetrate or explain.' The Barrister was at first accused of great severity in terming the methodists a sect of Anti-moralists: but, on their own confession, this charge is not without foundation. They admit that such doctrines as Original Sin and the Atonement may be very pernicious when taught separately or by themselves: but they endeavour to keep them in credit by the singular assertion that, though in themselves so very pernicious, they are capable of being more than neutralized, and even of becoming very salutary, when taken in connection with others. By what process of theological chemistry this conversion of poisons into good and wholesome nutriment is effected, we cannot ascertain. Doctrines confessedly bad in themselves are edge-tools, with which it is dangerous to play; popular preachers should keep clear of them; and evangelical preachers cannot employ them, because doctrines bad in themselves never proceeded from the lips of Christ and his apostles. The Barrister is completely justified when he says; To affirm of the Christian system, that its doctrines are individually productive of evil; that they are beneficial only in the mass; and that it is their connexion that constitutes their excellence, is the language of gross and shameful imposition, from whatever great political religious party in this kingdom such language may proceed, it is an insult on the common sense and moral judgment of every man in whose mind any trace of these attributes remains. - Look to the genuine revelation of CHRISTIANITY. Take the whole system to pieces to its last spring, - examine it in all its parts, asunder, as distant as you please; the more you thus examine it, the more clearly will you discern its beauty and its usefulness.’— - hold them The genuine doctrines of the Gospel will be found, upon a close and rational examination, to contain the motives which should excite and stimulate our obedience to its precepts; it is therefore utterly impossible, but that each doctrine taken separately must contribute to the good resulting from their united operation and effect.' - -- Profound views of sin become the boast of our modern evan-. gelical preachers: but in our judgment their profound views are very shallow and indiscriminating. These men seem to have a horrible confused notion of sin in the abstract, and of its damning nature: but they do not deem it worth their while to attack sin in detail. Vice and sin appear not with them to be convertible terms. In short, according to the New Creed, a man may abstain from all the grosser vices, and yet be the chiefest or vilest of sinners. - -Such are the terms which our Saints empley when they speak of themselves: but, if the hearer were to yield assent to this account of superlative vileness, the reply would probably be similar to that of the methodistic Mrs. Ranby in Celebs, and the cant of the sect would stand confessed. Nothing, indeed, is more deserving of reprehension than this silly style of affected humility; which tends to destroy all differences in characters, to place the vilest and the most respectable on a perfect level, and to raise in each a similar spirit of self-accusations and despair of Divine Forgiveness. Let us hear the Barrister speak against their preposterous method of "alarming the conscience?" Can the conscience of a good man put forth a sting as sharp as that which fills the worst of criminals with agony and alarm? Is he to be brought to the same self-accusations which spread a horror over the solitude of the robber and the assassin? Is the honest trader, and the exemplary father of a family, to utter the same confessions of guilt with the veteran swindler, and the licentious profligate? The HOLIEST OF MEN to despair of forgiveness in themselves, equally with the WORST OF SINNERS!!! Where, in the Gospel, is this required? In which of the Evangelists of Christianity is this doctrine to be found? These men turn religion upside down. They give us a system, in which every thing is inverted ;-a system abounding in every thing that can puzzle our reason, and perplex our faith. What man of plain understanding can bring himself to the conviction, that the best and the worst of them should load themselves with the same accusations, and be filled, -as consistently they ought, with the same remorse ? The thing is not possible. It runs counter to every dictate of common feeling, and of common sense. The whole proclamation of divine truth is against it. "Blessed," says our Saviour, "are the pure in heart for they shall see God.". But the Evangelists of our day deny this point blank. According to them, the most pure in heart must as much despair of forgiveness in themselves as the most impure!!! Thus, they pluck away from the true Christian all the hope that sustains him, and make the promises of the Gospel of none effect.' - By the New Evangelists, great stress is laid on Faith, as something abundantly more meritorious than the noblest virtue : but we may here ask them, is not Faith an act of the mind, and how can this act of the human mind be more meritorious than that which prompts to benevolence or any other virtue? The reply, perhaps, will be, Faith is produced by the spirit: but we will ask again, are not Good Works denominated by the Apostle fruits of the spirit?" Their whole string of assertions about faith faith and works manifests strange misconceptions, and ought to be seriously reviewed. When we approach the conclusion of this fourth part of the Hints, we perceive the reason of their being addressed to the Legislature as well as to the Public at large. Aware, as every sensible man must be, that the fomes mali resides in the Thirtynine Articles, which are here very justly pronounced to be a stumbling-block to the friends of the Establishment and a stalking-horse to its enemies,' the author does not hesitate to recom mend a revision of them : Let the Articles which were framed in an unenlightened age, and at a period of bigotry and bloodshed; let them be fairly, and without prejudice, examined; and if any one is found to contain any expression which seems to undervalue the importance of good works, or, which is the same thing, to undervalue that practical obedience to the laws of God, without which religion is an empty name, — let such expression be withdrawn. If there is any Article that experience has proved to be more productive of religious dissension than of reverence to God or allegiance to the state let such cause of offence and disunion be removed. If any article should be found to have separated conscientious and worthy men from the Established Church, by demanding an implicit and specific faith on points not fundamental-let such article be so revised as to restore the right of private judgment, and the freedom of religious inquiry.' This measure has hitherto been proposed without effect; and probably, in the present instance, the appeal to the Legislature will be equally in vain. Our rulers are extremely partial to old forms; not so much because they are what they ought to be, as because they are old, and because innovations are supposed to be dangerous. Our opinion on this subject has been often given. We are confident that the Established Church could lose nothing, but would to a certainty acquire additional strength and respectability, by the alterations proposed. Methodists will exult while they have the Articles to prop up their unscriptural creed; and their preachers will have some pretext for calling themselves true Churchmen, even while they are undermining the church. The friends of the Establishment would act wisely to consider the complexion of the times but if nothing can be done with old forms, we hope that the Barrister, and those who think with him, will not relax in their endeavours to expose the errors of the Methodistic system, and to bring the rational part of the community to espouse a scriptural, intelligible, and universally practical creed. ART ART. XI. Problems in some of the higher Branches of Algebra. 4to. 5s. sewed. Johnson and Co. B ETWEEN the progressive improvements of the analysis of finite quantities and of the analysis of infinitesimal quantities, the difference in rapidity is almost immense. Towards the close of the seventeenth century, Newton invented his fluxionary or differential calculus; and since that time, the calculus has made such wonderful advances, that it has satisfactorily solved almost all the numerous, complicated, and minute phænomena of the heavens. At the beginning of the sixteenth century, the cultivation of algebra under Lucas de Burgo was renewed. In the middle of that century, Cardan and Lewis Ferrari solved, the one, cubic, and the other, biquadratic equations: but since that time, more than two hundred and sixty years have elapsed, and no improvement of any moment in the doctrine of the solution of equations has been made. We stop at the term which arrested the progress of Lewis Ferrari; and other obstacles than the length of the calculations prevent us from solving generally equations which pass the fourth degree. Besides the method used by Ferrari, other methods, indeed, have been invented. Descartes suggested a mode of solution, apparently different from that of Ferrari, and now better known and more used; and on Harriott's principle of the generation of equations, Euler and La Grange have of late years given solutions: - but no one of these methods can properly be said to supersede the others: they are attended with nearly the same trouble in practical application, and they all require the solution of an equation of three dimensions. Euler, solving on the same principle, and by the same process, equations of the second, third, and fourth degree, was not without hopes, as we may collect from his memoir in the Petersburgh Commentaries, that the method might be extended to equations of higher degrees: but multiplied failures and enormous calculations have damped and almost extinguished the ardour of attempting the general solution of equations. It is not plain that algebra, as an art of computation, would be improved if such resolution could be effected; and besides, physical astronomy and its attendant sciences now divert our attention from the sterile plains of algebra, and invite industry and ingenuity to researches of greater pleasure and profit. Ferrari, independently of any consideration of the structure of the biquadratic equation +q-rx+s, solved such equa tion. Descartes, who was contemporary with Harriott, supposed the equation to be formed by the multiplication of two REV. DEC. 1810. Dd quadratic 9 ↑ quadratic factors, ✯'+ƒx+£2 + —— 27, and x2 — ƒx 2 f 2 q r 2 + + ; and he deduced an equation involving 2 2f f of this form, ƒ° — Pƒ + Qƒ2 -R, which is called a bicubic equation, and is solved by the solution of a cubic equation. Having hence obtained the values of f, which, if a, ß, ,, be roots of the original equation, are a + B, a+v, &c. he easily determined the roots a, ß, &c. or solved the biquadratic. + In the cases of quadratic, cubic, and biquadratic equations, it is easily proved that the number of the roots of an equation is equal to the dimensions of the equation: that is, in a cubic equation, x3+qx-ro, we have three algebraical quantities; any one of which being substituted for x in the above equation, the equation becomes identical. Suppose these roots to be a, B, 7, and that we assume a function of these roots, as Aa+ B3 Cy, and permuting the quantities a, ß, y, form AB+ Ba Cr if we can determine the values of these functions, we shall be able also to determine the roots by a simple process of elimination. In like manner, if the four roots of a biquadratic equation be a, f, y, d, and, assuming a function of the roots, as Aa + B3+ Cy + Dồ, we be able to find three or four values of such function, we shall also obtain the roots themselves. On considerations like these, mathematicians in modern times have attempted the solution of equations, and from such attempts many curious and useful formulæ have origi nated. In the methods of Descartes and Ferrari, the reducing or subsidiary equation is in fact an equation of three dimensions; and, unless the reducing equation be of lower dimensions than the original equation, nothing is gained by the deduction of the reducing equation. If we assume the function A + BB+Cy+D, then such function admits of 24 values; or the reducing equation would be of 24 dimensions, and consequently more difficult of solution than the biquadratic. The preceding function, therefore, must be simplified; and if we make BA and DC, it will become A (a+E) + C(2 + d) which function admits of only six different values. If, now, we can make these values of the form a,-a; b, — b; &c. the equation of six dimensions will become a bicubic equation, Let, then, the original biquadratic be x+ + qx2 —rx +5, and let |