preferring the method he has pursued, is, that it seems so im mediately connected with the Binomial Theorem, and the doctrine of Logarithms. This, however, is counterbalanced by the, disadvantages already mentioned, and by the additional one of leading to demonstrations that are synthetical, and little calcu lated to exercise the powers of invention.

We do not think that this inversion of method was necessary for the purpose of demonstrating either the Binomial Theorem, or the series for Logarithms and Exponentials, all which have already been deduced from the usual notion of powers, by more than one author, with great accuracy and simplicity of reasoning. We would particularly refer our readers to a work, not so much known in this country as it ought to be, viz. Principiorum Calculi differentialis et Integralis Expositio Elementaris, by SIMON L'HUILIER * professor of mathematics at Geneva. Several excellent demonstrations of the Binomial theorem have also been given by EULER. One of the most elegant of these is to be found in LA CROIX, Complement des Elemens d'Algebre, sect. 65, where it is followed by another from the Philosophical Transactions for 1796, hardly less commendable.

On the whole, therefore, though we cannot but praise the ingenuity and the skill that appear in the demonstrations of the ninth book, and the attention to strict and logical argumentation, of which the author seldom loses sight, we must regret a want of simplicity, a great deviation from the natural path of discovery, and the substitution of synthetical proof for analytical investigation.

The tenth book treats of the roots of equations, and gives a very distinct and accurate account of them, free from several of the difficulties that occur in this part of Algebra. If such a quantity as a3+a x2+bx+c is reduced into three simple binomial factors, so as to be equal to (x—a)(x—ß)(x—y), α, ß, are said to be roots of the given quadrinomial. Here then is no difficulty about negative roots; because, when any of the simple factors becomes of the form +ß, then the root 3 is accounted negative.

In this way the difficulty about imaginary roots is also removed. The roots of a Polynomial, if they can be found by any general method, must be of a certain form, so as to depend on the coefficients a, b, c, d, &c. or to be deduced from them according to a certain system of operations. Now, this form and VOL. XX. NO. 40. Ee

* Introduction, p. 5. &c.

these operations become impossible when the coefficients are related, in a particular manner to one another, and in that case the root of the Polynomial has no real value. A paradox still remains behind. Now, does this impossible root admit of arithmetical operations being applied to it, as if it actually denoted quantity and how comes it, that when so treated, it leads to true and useful conclusions concerning quantitics that really exist? The solution of this difficulty is not considered by our author, and indeed does not belong to the elements of the

science.

"

In the 10th book are contained several methods, of which the demonstrations are reserved to a subsequent part of the work. Thus, the 6th proposition of the 10th, is Cardan's rule for the solution of cubic equations: but the demonstration is not given till the 21st book, p. 288. The reason of this proceeding does not appear. The method of approximating to the roots of equations is also contained in the 10th book.

We shall pass over the intermediate books, as not containing any thing of which the method is very different from that which is usually followed; till we come to the 15th, which treats of the method of Fluxions. Here the definition of a fluxion is very difficult to be understood, and such as we think, to a learner, must be quite incomprehensible. We do not indeed rememher to have seen a definition, accounted elementary, which is involved in so much obscurity, or which requires so much previous instruction to render it intelligible. It contains in it, indeed, the idea of a Fluxion; but to discover what it contains, requires one to be already familiar with the calculus. How much better it would have been, to call the fluxion of any fuuction the first term of the increment of that function, which, after all, is the idea meant to be conveyed. The fault of introducing synthetical demonstrations occurs here again; we remark, however, a demonstration that lias merit, viz. that the infinite series Ax + Bx2 + Cœ3, &c. &c. is infinitely small when a is infinitely small. The proof is satisfactory, and we believe new.

Trigonometry is not introduced till the 16th book; and it is then only analytical trigonometry, and not that which treats of the arithmetical solution of the cases of plane and spherical triangles. This is referred to the end of the book; and is treated in a manner much too general and concise to be of much practical utility.

1

In the 17th book the method of drawing tangents to curves is considered; and seme properties of the conic sections are also deduced, and of the radius of curvature. The method of finding fluents is delivered in the 18th, and a number of curious theo

rems are laid down;—and here also is introduced something of the arithmetic of impossible quantities. The obscurity of the original idea of a fluxion contributes to render all this part much darker than it ought to be.

The 20th book contains the doctrine of finite differences, which is given, we think, very distinctly, and in a manner that may be truly useful, always making an allowance for the extreme conciseness of the work. The best treatise on this subject which we know of, that can be considered as quite elementary, is one by the ABBé Bossut, contained in the Encyclopedie Methodique, to which the treatise here given has a considerable resemblance. The 21st book demonstrates several propositions before referred to, such as the investigation of Cardan's rule, the investigation of the Binomial theorem, and certain propositions concerning fluents. It concludes with theorems relating to the maxima and minima of variable quantities, some of which are of the most difficult kind,--that, viz. where the quantity that is to be a maximum or a minimum, is not expressed as a finite algebraic function. This includes the very difficult problems called Isoperimetrical; so that this small volume comprehends the elements of the pure mathematical sciences, from the Axioms of Geometry up to some of the highest parts of the Integral Calculus.

Though we have stated several objections to the work, we admit that it has great merit on the whole, and may be very useful to two classes of readers; 1mo, To the students who attend the prelections of a master, to whom it serves as a text-book; and, 2do, To those who are already instructed in the mathematics, but wish to have at hand a portable compendium for reminding them of those formulas and demonstrations which may have escaped their recollection. The work to which that before us may be most readily compared, is the elementary treatise of the Aвré DE LA CAILLE, equally comprehensive, and hardly less concise. The French author does not so much affect originality of method as the Portuguese; and on that account perhaps his work is the more useful. In clearness it very much excels the other; and is, we believe, the best compendium of mathematical science, in the same compass, which has yet been given to the worl... To be second to LA CAILLE's treatise, amounts to high praise; and we have great pleasure in bestowing this encomium on the production of a country from which the sciences have not hitherto received much of their improvement.

*

ART. X. Rejected Addresses; or the New Theatrum Poetarum. 12mo. pp. 126. London, 1812.

A FTER all the learning, wrangling, and solemn exhortation of our preceding pages, we think we may venture to treat our readers with a little morsel of town-made gayety, without any great derogation from our established character for seriousness and contempt of trifles. We are aware, indeed, that there is no way by which we could so certainly ingratiate ourselves with our provincial readers, as by dealing largely in such articles; and we can assure them, that if we have not hitherto indulged them very often in this manner, it is only because we have not often met with any thing nearly so good as the little volume before us. We have seen nothing comparable to it indeed since the publication of the poetry of the Antijacobin; and though it wants the high seasoning of politics and personality, which no doubt contributed much to the currency of that celebrated collection, we are not sure that it does not exhibit, on the whole, a still more exquisite talent of imitation, with powers of poetical composition that are scarcely inferior.

We must not forget however to inform our country readers, that these Rejected Addresses' are merely a series of imita tions of the style and manner of the most celebrated living writers-who are here supposed to have tried their hands at an address to be spoken at the opening of the New Theatre in Drury-Lane-in the hope, we presume, of obtaining the twenty pound prize which the munificent managers are said to have held out to the successful candidate. The names of the imaginary competitors, whose works are now offered to the public, are only indicated by their initials; and there are one or two which we really do not know how to fill up. By far the greater part, however, are such as cannot possibly be mistaken; and no reader of Scott, Crabbe, Southey, Wordsworth, Lewis, Moore, or Spencer, could require the aid, even of their initials, to recognize them in their portraits. Coleridge, Coleman, and Lord Byron, are not quite such striking likenesses. Of Dr Busby's and Mr Fitzgerald's, we do not hold ourselves qualified to judge -not professing to be deeply read in the works of these originals. There is a prose address however from Mr Cobbett, which is admirable one from the Editor of the Morning Post, which was scarcely worth making--and one from the ghost of Samuel Johnson, which is more unequal than most of the others. The total number is twenty-one.

There is no talent so universally entertaining as that of mimickry—even when it is confined to the hyely imitation of the

air and manner-the voice, gait, and external deportment of ordinary individuals. Nor is this to be ascribed entirely to our wicked love of ridicule; for, though we must not assign a very high intellectual rank to an art which is said to have attained to its greatest perfection among the savages of New Holland, some admiration is undoubtedly due to the capacity of nice observation which it implies; and some gratification may be innocently derived from the sudden perception which it excites of unexpected peculiarities. It rises in interest, however, and in dignity, when it succeed in expressing, not merely the visible and external characteristics of its objects, but those also of their taste, their genius and temper. A vulgar mimic repeats a man's cantphrases and known stories, with an exact imitation of his voice, look and gestures; but he is an artist of a far higher description, who can make stories or reasonings in his manner, and represent the features and movements of his mind, as well as the accidents of his body. The same distinction applies to the mimickry, if it may be so called, of an author's style" and manner of writing. To copy his peculiar phrases or turns of expression-to borrow the grammatical structure of his sentences, or the metrical balance of his lines or to crowd and string together all the pedantic or affected words which he has become remarkable for using-applying or misapplying all these without the least regard to the character of his genius, or the spirit of his compositions, is to imitate an author only as a monkey might imitate a man-or, at best, to support a masquerade character on the strength of the dress only; and at all events, requires as little talent, and deserves as little praise, as the mimetic exhibitions in the neighbourhood of Port-Sydney. It is another matter, however, to be able to borrow the diction and manner of a celebrated writer to express sentiments like his own-to write as he would have written on the subject proposed to his imitator-to think his thoughts in short, as well as to use his words-and to make the revival of his style appear a natural consequence of the strong conception of his peculiar ideas. To do this in all the perfection of which it is capable, requires talents, perhaps, not inferior to those of the original on whom they are employed-together with a faculty of observation, and a dexterity of application, which that original might not always possess; and should not only afford nearly as great pleasure to the reader, as a piece of composition, but may teach him some lessons, or open up to him some views, which could not have been otherwise disclosed.

The exact imitation of a good thing, it must be admitted, promises fair to be a pretty good thing in itself; but if the resem