leed, the judicious variation and felicitous choice of rhythm throughout this poem, make it evident that a distinct untransserable character, and a peculiar power of expression attach to the different forms of versification, apart from the purpose for which they are employed, and constituting their adaption to particular subjects, while they shew that Mr. Hogg is well acquainted with his business as a versifier. There are passages in this part of his work, however, of no ordinary merit ; and we think it probable that with many the whole canto will be the favourite one. It is more didactic than the rest, and contains some fine strokes of satire, and some beautiful sentiments. The idea of the planet Venus, as The land of lovers, known afar, And named the evening and the morning star; Is very happy. The warlike sphere that wades in crimson like the sultry sun,' detains our poet too long, though it is made the subject of some fine descriptive passages. We can make room, however, only for the following very striking lines, which are introduced as illustrative of the idea, that 'there are prisons in the deep below.' 0! it would melt the living heart with woe, For keeping back that knowledge they disdained.' p. 86. We think our readers will concur with us in ascribing no ordinary character to such poetry as this. The conclusion of the third part leaves Mary' within the 'grave alone.' The Poet concludes, · Here I must seize my ancient harp again, And chaunt a simple tale, a most uncourtly strain.' Part the Fourth is, accordingly, in the varied measure of the modern metrical romance, and forms an appropriate sequel to the wondrous tale. The opening of it describes the terror and confusion which prevailed at Carelha, when Mary was first missing. Her maidens knew "The third night of the moon in the wane. As lightly they rode on the beam of the moon.'Her breathless form is at length found prostrate on the sward, • as if in calm and deep devotion.' Her death-like appearance is beautifully described; but * All earthly hope at last outworn, The body to the tomb was borne.' We will not forestal the sequel, but leave our readers to satisfy their curiosity by perusing the volume for themselves ; only just remarking that the effect of her mysterious return, at the hour of the ghost one sabbath night,' the exclamation of her lady mother, who instantly recoguizes the foot of her daughter, but checks herself with · The grave is deep, it may not be!' And their meeting, when the door of the hall is opened, are in the most picturesque style of romantic adventure, and exquisitely touching « That mould is sensible and warm, 0! that embrace was fraught with wonder !' Our limits warn us to conclude this article ; and we have said enough to shew our estimate of Mr. Hogg's poetical genius. We rely upon him to justify our praise by his subsequent productions. If we have in any measure over-rated his abilities, It has not been owing to our having any private acquaintance with the man, or any partiality to the Author, save that partiality which we may be pardoned for feeling, in meeting with a production so delightfully adapted to the wildest rovings of our untamed fancy, and distinguished at the same time by so high a tone of purity and moral feeling. An Ode to Superstition closes the volume. It is in the Spenserian stanza, and is interesting, not only on account of its intrinsic merit, but as developing some of the peculiar traits and sentiments of the Author's mind. We should have been glad to have entered at large into the subject in its relations to poctry, as we deem it one which has not obtained adequate attention, but we must reserve our remarks for another occasion. Mr. Hogg has meritoriously abstained from eking out his volume with notes, but a brief explanation of some local references, and of a few Scottish or provincial words, would have been very acceptable to his Southern readers. Art. VI.-1. New Mathematical Tables, containing the Factors, Squares, Cubes, Square Roots, Cube Roots, Reciprocals, and Hyperbolic Logarithms, of all Numbers, from 1 to 10,000 ; Tables of Powers and Prime Numbers; an extensive Table of Formulæ, or General Synopsis of the most important Particulars relating to the Doctrines of Equations, Series, Fluxions, Fluents, &c. &c. By Peter Barlow, of the Royal Military Academy. 8vo, pp. Ixii. 336. Price 18s. boards. London, G. & S. Robinson. 1814. 2. Mathematical Tables, containing the Logarithms of all Numbers, from 1 to 10,000; the Logarithmic Lines and Tangents to every Degree; a Traverse or Table of Difference of Latitude and Departure; with a Table of Rhumbs. By the Reverend William Alleyne Barker, 2tmo. p. 226. London, Reynolds, Oxford street, 1814. IN proportion to the augmentation of the stock of mathe matical knowledge, arises the expediency of tabulating results. Among the ancients, when the whole of mathematics consisted of plane and solid geometry, the conic sections, and a few elementary applications to mechanics, optics, and astronomy, men might carry all the principles, theorems, and problems in their minds, without any such burden as should drive them to seek adventitious aids; but in consequence of the wonderful extension given to the abstruse sciences during the last two centuries, circumstances have considerably changed. An investigator of sound and well exercised intellect, will remember principles, will be expert in his processes, and can, therefore, always deduce results : but that he may not find it absolutely necessary to waste his time and strength in de. ducing what has been inferred before, it is advisable, not merely that the most valuable particulars should be exhibited in the logical order in which they occur in our best treatises, but that theorems and other useful results should be thrown into synopses and tables, where they may at once be found; and employed in the investigation of the new problems upon which men of theory and men of practice are constantly einployed. To find the square root or the cube root of any proposed integer, requires an operation which every school-boy may perforın; yet it would be exceedingly irksome for the mathematical investigator of some problem in pneumatics or hydraulics, to be arrested in the midst of an inquiry, till he could carry through 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 inore 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 formula for the determination of altitudes by the barometer and thermometer, 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 å 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 reduceckto the form y: — =6, all possible values of y fall within the limits 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 introduction 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 |