80, the sentence to express by equations the ratio of the given quantities first employed to those which we would introduce," is 04- cure, and if it were not for the plainness of the subject, would be unintelligible. It would not be difficult to increase the list of errata which is given at the end. In p.95, at the top, the second and third lines should read thus, “I take their half sum and designate it by X, since their difference DE=0 is given," &c. Or it may be altered thus,-"I take their sum and designate it by 2x, since their difference DE=C is given," &c. 'This error is in the original as well as in the translation. In the translation, p. 77, instead of “by the method,” &c. we should read “according to the method,” &c. (selon la methode, &c.) In p. 76, (original 294) near the foot of the page, the translation perverts the sense, and is also inconsistent with itself. The original is not plain, but should probably be rendered thus: “We treat here, however, only of general rules; we can often construct in a manner much more simple than by always setting out from the same principles," &c. In p. 130, near the foot, the translation “making with each other an angle equal to that of the conjugate diameters," is not the sense of the original.* We have already intimated that the treatise before us, is very far behind the actual state of the branch of mathematics of which it professes to treat. During the fifty years which have succeeded its publication, numerous and very important improvements have been introduced. To prevent its falling too far in the rear, M. Peyrard introduced into the edition of 1805, no less than seventy pages of additions, some of which should certainly have been retained by the translator. We are particularly surprised, that he should have permitted himself to publish a treatise for the use of our colleges, which does not even contain the equation of a right line. We are disposed, however, to acknowledge a certain degree of obligation to the translator. If the treatise which he has published is imperfect, still he has contributed to make known a branch of mathematics which had hitherto received almost no attention in this country. We presume this treatise will soon be superseded by one far more complete. Our remarks in this paper are particularly designed for Mr. Farrar's volume; but we must not close without a short notice of those of Bourdon and Biot. The first of these is very extensive and of very great value, and contains the system prepared by the author for the use of the royal colleges of France. The first chapter is employed in explaining those methods of * Vide p. 375 application which vary with the nature of the problems, but which by their simplicity, often have the advantage over general methods. After having laid down the principles relative to the construction of algebraic expressions, several questions are resolved, the discussion of which is suitable to initiate young men into the manner of interpreting the singular results of algebra applied to geometry. After this, coine plane and spherical trigonometry, the discussion of which concludes the first section of the work. The third chapter introduces analytical geometry, properly so called, that is, the method which consists in resolving the questions of geometry by the aid of equations of a point, of lines, and of surfaces. It embraces all the principles relative to the point, the right line and the circle situated upon a plane. Although the circle is only a particular case of one of the curves of the second degree, the author has presented a distinct view of it, in order to accustom learners, by an analytical investigation of properties which are already known to them, to read in equations and the results of their combinations, whatever these equations and these results are capable of representing. The problem of tangents is resolved by a general method, which the author says is no where else to be found. The problem of the transformation of the coordinates in two dimensions serves as an introduction to the next chapter, which is chiefly employed in general views upon curves of the second degree. With a view to avoid considerations which are too abstract, he gives purely geometrical definitions of the ellipse, the hyperbola and the parabola, which are constructed according to these definitions, and the equations of which are afterwards investigated. After tracing the analogies of these curves, he demonstrates by the transformation of the coordinates, that these are the only curves which any equation of the second degree with two variables is able to represent. The identity of the curves of the second degree with the sections of a cone and a plane are then established. The fifth chapter, which is the most important, comprehends the principal properties of the conic sections. The analogy between the equations of the ellipse and the hyperbola, leads to an abridgment of labour and to the avoiding of tedious repetitions, by discussing the properties of these curves in connexion. The relations between the ordinates and abscissas of these curves, their quadrature, the properties of supplementary cords, the relation of these cords with conjugate diameters, tangents and their properties with respect to radii vectores, the properties of the ellipse and of the hyperbola referred to a system of conjugate diameters, of the hyperbola referred to its asymptotes, &c.; such are the principal views belonging to the two first curves. With respect to the parabola, whose analogy to the ellipse and the hyperbola is more distant, the author treats of it separately. The sixth chapter, which is in some measure a supplement to the preceding, embraces the classification of the curves of the second degree by the separation of the variables, the construction and discussion of particular equations; the investigation of the centre, the axes, the diameters, the asymptotes, &c. of the curve corresponding to a given equation; the determination of a conic section according to certain conditions; in fine, the construction of determinate equations of the third and fourth degrees with a single unknown quantity. The seventh and eighth chapters comprise analytical geometry in three dimensions. The seventh treats of the point, the right line and the plane considered in any manner in space; while the eighth contains a succinct view of surfaces of the second degree preceded by the problem of the transformation of the coordinates in three dimensions, and some general views upon certain curve surfaces, such as the sphere, cylindrical and conical surfaces, conoidal surfaces, and surfaces of revolution. The “Essai” of M. Biot occupies nearly the same ground with Bourdon's 'Treatise, the contents of which we have been sketching, and we, therefore, presume it will not be necessary to enter into a particular description of it. It is chiefly designed for those who are preparing for the French Polytechnic school, and was composed by the author for his scholars while he taught in the central school of the department of Oise. The first edition appeared in 1802, but the succeeding editions have received many and great improvements. His method of treating the subject is much more abstract than that of Bourdon, and his work, to be read with ease, requires considerable acquaintance with mathematics. The evidence is frequently of that species, which is not easily described, but to which one must become accustomed, before he can peruse with understanding, the works of Lacroix, Monge, Lagrange and La Place. If, however, the instrument in the hands of Biot, is more difficult to be wielded, than as it is presented by several authors, it is on the other hand, in the same proportion more powerful. He has introduced some historical notices and general views in various places, of which we have freely availed ourselves when they suited our purpose. It is our opinion, that the perusal of Bourdon first, and of Biot afterwards, will be the best course for those who wish to become thoroughly initiated in the elenients of analytical geometry. Since writing the above, the last edition (1826) of Professor Farrar's volume has fallen into our hands. We immediately examined it, in the expectation that the errors which we have noticed, would be corrected in it. But it appears, that although it had been used at Cambridge from 1820 to 1826, only one of the errors which we have pointed out, has been discovered. Art. III.-A Commentary on the Epistle to the Hebrews. In two volumes. By Moses STUART, Associate Professor of Sacred Literature in the Theological Seminary at Andover. Andover, 1827. It is not without stantly to call them. To advance its It is not without reason, that even in our enlightened days, as we are pleased constantly to call them, great importance should be attached to the claim of antiquity. To advance its long and continued existence as the sanction for a custom, is only another method of stating that the experience of ages has tested its utility and proved its wisdom. We are not ashamed to confess our veneration for old opinions, whilst at the same time we think with St. Cyprian, that “custom ought not to hinder that truth should prevail, for custom without truth is but agedness of error." In this country, the glory of our career will depend upon the skill with which we may unite the wisdom of the past with the increasing knowledge of our own times. To adapt old and well-tried principles and forms to the new wants and changing fashions of society, should be the object and end of all innovation. In the search after this necessary and desirable adaptation, we should not, without judgment, follow the ancients, for “not because they went before us in time, therefore in wisdom, which being given alike to all ages, cannot be prepossessed by them,” nor should we, still worse, be led astray by our own vanity, and abate as nuisances all customs which militate against our own untried opinions. In the formation and regulation of our schools and colleges, we have ample scope for a fair trial of our skill at improvement. We have commenced our literary course untrammelled by long venerated usages, disconnected from all political or religious bias, and assisted by the experience of a civilized and enlightened nation, engaged in the same pursuits, feeling the same wants, nay, speaking the very language which we inherit. We believe that the European sytems of education are not suited in all their details to our state of society; we believe that, in the course of ages, many abuses have become incorporated therein, which the enlightened men of that continent would rejoice to remedy. We see at this time in England, the liberal party engaged in the laborious and expensive design of rearing new establishments in London, to supply the deficiencies and get rid of the abuses of the old universities. With these advantages and this past and passing experience, there must be some defect in ourselves, some weak point in our national character, if we cannot so organize our literary institutions, as to enable them to meet the wants of the community and the improvements of the age in the nature and measure of instruction, as well as in the modes of training youth for the business of life. The history of education, including the progress of literary institutions, would furnish materials not merely for an interesting essay, but for a most important book. We shall not attempt it here it will be sufficient to point out one or two remarkable changes in American seminaries, suggested by the valuable work before us. The first colleges erected in this country, were designed exclusively for the education of Ministers of the Gospel. Our later institutions have been established upon a more enlarged plan, but we have not got what, in European phraseology, can be termed an University. Legal, medical and theological lectures are attached to several of our colleges, but the most distinguished institutions for the three learned professions, are all separate and exclusive. We have now sixteen medical colleges, many law schools, and at least twelve theological seminaries. The concentration of professional knowledge, and the increase of competent practical instructors in these institutions, is felt and acknowledged ; all are doing good to their country and rising into reputation. The work at the head of our article, confines our remarks at this time to the theological establishments recently founded and growing up in the United States. Most of these seminaries recommend themselves by their excellent arrangements for the promotion of liberal learning among clerical men. Without any disparagement to the clergy as a sacred body, we may be permitted to say, there has been VOL. III.-N0. 6. 40 |