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such an operation to seven or eight places of decimals. A similar remark would apply to every species of mathematical research. The importance and value therefore of compendiums like Mr. Barlow's, must be generally felt.
The tables comprehended in this volume are ten in number. Of these, the first contains the factors, squares, cubes, square roots, cube roots, and reciprocals, of all numbers from 1 to 10,000. The second exbibits the first ten powers of all numbers under 100. The third contains the 4th and 5th powers of all numbers from 100 to 1000. The fourth is intended to facilitate the solution of the irreducible case in cubic equations. The fifth is a table of all prime numbers under 100,000. The sixth contains the hyperbolic logarithms for all numbers under 10,000. The seventh is a table of differential co-efficients. The ninth is a comprehensive table of weights and measures, English and Foreign : and the tenth exhibits the specific gravities of more than 300 different bodies. Besides these, which, as our readers will perceive, are formed for utility, there is another table which we deem of much importance, and therefore mention it out of its natural order. This is the Table VIII, which is very comprehensive indeed, and might with a little more extension be denominated a synopsis of matbematical science. It seems to have been suggested by Jones's
Synopsis Palmariorum Matheseos, published in 1706, and Martin's “ Young Student's Memorial Book," published in 1736. Mr. Barlow has brought together within the compass of 00 pages, a most valuable collection of formulæ relating to the extraction of roots, the binomial theorem, roots of quadratic, cubic, biquadratic, indeterminate and other equations, interest and annuities, progressions and summation of series, figurate numbers, logarithmic and trigonometrical series, sines, tangents, secants, &c. of one or more arcs, plane and spherical trigonometry, mensuration of planes and solids, descent of bodies in free space, motion down inclined planes, vibrations and lengths of penduluns, motion of projectiles, centres of gravity, gyration, oscillation and percussion ; fluxional formula relating to forces, exponentials, trigonometrical quantities, rectifications, quadratures, &c. with a comprehensive and highly useful collection of fluents: the table concludes with a synopsis of the elements of our planetary system.
This table is, in truth, so copious and excellent, that we regret to remark that it is not complete. We shall specify a few more particulars which we could have wished to see introduced; and shall cherish the hope that the ingenious Author will experience such encouragement as will induce him to enrich his work with a supplementary sheet. It might contain a theorem or two for equation of payınents, rules for removing
the ambiguities in spherical trigonometry, equation and most obvious properties of the conic sections, theorems relative to the mechanical powers, approximative formulæ for the determination of altitudes by the barometer and therinometer, precepts and theoreins for the use of the table of specific gravities, formulæ for central forces, and for the foci catoptrics and dioptrics, and a table of atmospherical refractions in altitude.
The tables, however, as they now stand, will be found of extreme utility, and they are preceded by an introduction which will greatly facilitate the use of them. In this introduction the Author first points out the means employed in the computation and verification of the tables, acknowledging as he proceeds the several sources from which he derived any assistance; and then explains the application of the tables themselves, exhibiting very perspicuous formulæ and precepts for the direction of the student. In this part of the work he has not confined himself to what is old and well known; but has introduced a few investigations which are both novel in their nature and useful in their tendency. Among these we read with much pleasure his explication of the seeming paradox respecting the irreducible case of cubic equations, and his satisfactory manner of proving that when a cubic falling under that case, is reduced to the form yz - Y=c, all possible values of y fall within the linnits 1 and 1.1549. From this property he deduces his rule for the solution of this class of cubics, and enables the student, by means of a table of six pages, to solve all such equations in little more than half the time that would be required by any other method with which we are acquainted. The iatroduction likewise contains some admirable rules for the solution of equations in general; and some very acute observations by which it is shown decisively that Newton's approximating rule is by no
means so defective as later mathematicians have usually thought it, and that Lagrange's method, on the other hand, notwithstanding its elegance in theory, is nearly useless in practice.
On the whole, we warmly commend Mr. Barlow for the labour and talent displayed in this volume: and we sincerely hope he will find himself mistaken in apprehending that the nature of the subject ' precludes every idea of adequate remuneration,
Mr. Barker's little volume, though of humbler pretensions than Mr. Barlow's, is nevertheless calculated to be useful, especially to military men. It is evidently formed upon the plan of the “ Tables de Logarithmes pour les nombres et pour les “ Sinus," published by Jerome Lalande in 1802 ; and like that compendious manual, is neatly printed and stereotyped. We cannot better describe it than by quoting six lines from the Author's preface.
· The Table of Differences is placed, first, as more convenient when not immediately required for use: the logarithms are to sit places of decimals; the obtuse as well as acute angles are inserted : and, instead of the difference between each minute, in the table of Sines, &c. the value of one second is given.'
From these peculiarities, it is evident, that Mr. Barker intended his tables for the use of those who are not very expert in logarithmic computations. Such persons would have consulted this little volume with still greater convenience, if the tables had been preceded by a few trigonometrical formulæ and precepts, siunilar to those in the portable tables, published by Lacaille, in 1760; or to some of those given in the introduction to the comprehensive and excellent Mathematical Tables of Dr. Hutton.
Art. VII. A Treatise on Spiritual Comfort. By John Colquhoun, D.D.
Minister of the Gospel. Leith, 12mo. pp. 414. Edinburgh, Ogle. AMONG the almost endless variety of treatises on reli
gious subjects, which daily issue from the press, there are comparatively few which treat of experimental religion. In polemical discussions, the disputants on each side, are numerous; and talents of the first order have frequently been employed in controverting or in defending all the articles of the Christian faith. Neither is the number small, nor are the names ignoble, of those who have, in modern times, employed their pens in pleading the cause of practical Christianity, and in pointing out the duties and obligations of a religious life. Not many, however, are sufficiently hardy to encounter the obloquy and the odium which inevitably attach to the writing of works on Christian experience, or are willing to expose themselves on this account, to the terrific charges of enthusiasm, fanaticism, and methodism, which they cannot hope to escape, if they presume to meddle with topics so unfashionable. It would be no uninteresting subject of inquiry, to investigate the causes of this modern refinement, which distinguishes so many professing Christians of the present day, from their pious ancestors; for the prejudice to which we refer, is not confined to the irreligious, of whom nothing better can be expected ; nor to the selfdenominated rationalists, whose frigid and philosophical system utterly excludes all feeling ; but it operates, in some degree, upon the minds of those even who profess a cordial attachment to evangelical religion.
Without entering fully into the investigation of this subject, we may venture to remark, that two causes seem to have concurred to produce this state of feeling among the more cultivated and intelligent class of religious professors. The first is, the injudicious manner in which many, whose zeal has not been duly regulated by knowledge, have written upon the subject; and the second, what may, for want of a better term, be calle the rationalizing system, which prevails among many of the above-mentioned class of characters. A very different order of of beings from the Halls and the Hopkinses, the Owens and the Baxters, of a former age, have undertaken, in modern times, to display the interior of the Christian's character; to describe his conflicts and supports, his trials and deliverances, his sorrows and his joys. By these religion has not unfrequently been grievously caricatured, and occasion has been given to the common adversary to triumph or to blaspheme.
The impressions and operations of genuine piety upon the mind, have sometimes been strangely blended with the visionary flights of a perturbed and heated imagination; and this heterogeneous mixture of good and evil, has been exhibited to the public as constituting Christian experience. The consequence has been, that many persons, disgusted with what is fallacious, and, strictly speaking, fanatical, have rejected that also which is true and scriptural, and have alike discredited the whole. Nor is it less evident, that the habit acquired by highly intellectual characters, of exercising their judgement alone in the pursuit and investigation of every kind of truth, has a natural tendency to produce the result of which we speak, even in persons of reputed piety. They are imperceptibly led to the conclusion, that religion is chiefly, if not exclusively, a matter of the understanding, and that it has little to do with the affections. They pronounce all that humbler Christians say about their feelings, religious cant; and too hastily judge, that the varying, and frequently sudden emotions of mental depression, or of spiritual comfort, of which they speak, are either hypocritical or illusive.
We are happy to find in the treatise of Dr. Colquhoun on "Spiritual Comfort," a work that is exempt from those injudicious statements to which we have adverted, and which is calculated, by its solidity and sobriety, to decrease the force of prejudice, and to silence gainsayers. Its theology is completely that of the old school, in regard both to the systematic arrangement of its contents, and the technical style in which it is written. Seldom have we seen a tract that reminded us more forcibly of the writings of some of our nonconformist divines. This will not, probably, be considered as a circumstance of recommendation by those whose taste is formed on the superficial essay of the
modern school; but we freely confess, it is no ordinary excellence in a work of this description.
In the following extract, the Author of this treatise has, in a plain but perspicuous manner, stated the object of his work, and described the persons for whose benefit it was written, and on this account we have been induced to select it, as enabling the reader to judge of the character and of the execution of the whole.
• The persons for whose use this Treatise is more immediately intended, are they, who have, by the Holy Spirit, been convinced of the guilt, malignity, and demerit, of the sin which dwelleth in them, as well as of the iniquities that are committed by them : who have also been convinced of the utter insufficiency of their own righteous, ness, for their justification in the sight of God, and who have been enabled to embrace Jesus Christ, as their righteousness and strength. All of this description are earnestly desirous of advancing in holiness; but many of them seem to be far from being duly sensible of the high importance of spiritual consolation, to the love and practice of holi. ness. They are soon apprehensive of danger, if they feel iniquities prevailing against them ; but they yield, without alarm, to that dejection of spirit, which is often occasioned, either by inward conflicts or outward trials ; not considering, that disquietude of soul paves the way for despondency, and despondency for utter despair ; all which are, in a high degree, injurious to the spiritual welfare of the soul. Trouble of mind, especially when it proceeds the length of despondency, strengthens the unbelief and enmity of the heart against God; and so disqualifies the Christian for performing acceptably, the duties incumbent upon him. Although God doth not suffer any of his children, ever to fall into the horrible gulf of absolute despair, yet some of them have brought themselves to the very brink of it; so as greatly to dishonour their holy profession, to injure their own souls, and to hurt the souls of many around them, who are always too ready to impute their dejection of spirit to the holy religion which they profess. Thus, they often discourage the hearts of some, who are seeking Jesus; and strengthen the prejudiccs of others, who are enemies to him.
- The sovereign antidote to that sinful and grievous distemper of mind, is the spiritual and holy consolation, which is offered and promised in the gospel. Much of the sacred Volume was written for this end, that the saints might be comforted, and that they, “through patience and comfort of the Scriptures, might have hope. God, in the exceeding riches of his grace, hath given in his word, and confirmed by his oath, many great and precious promises ; in order that all “who have fled for refuge, to lay hold upon the hope set before them.” might not only have consolation, but strong consolation. He hath spoken in his holiness, on purpose that they might rejoice ; that they might be so • filled with all joy and peace in believing,' as to serve him with gladness; and thereby, to recommend faith and holiness to all around them.' p. 1-2.