they have wanted the necessary talent, but that they have wasted it upon unimportant objects. The industry and genius expended upon the ingenious puzzles of the Ladies' Diary, would have sufficed to build up the system of the Mecanique Celeste; but hardly one valuable addition was made to that great edifice, of which Newton was the founder, and Laplace the finisher, by an English hand. When they at last awoke to a sense of their deficiencies, the labour of recovering the ground lost by near a century of neglect, was enormous; and in this particular subject, we may cite the work of Woodhouse, in the first edition, to show how far even he was behind the science of the continent. In his subsequent republications, he has, in some measure, supplied the deficiencies; but the additions are not so engrafted, as to make them parts of one harmonious whole.

From the manner in which we have stated that the analytic method in trigonometry arose, in its application to the higher analysis, and from a consideration of its slow and gradual introduction into more elementary works, it will be obvious that the methods, although perhaps the most proper for discovery, are neither homogeneous, nor suited to elementary instruction. It has at last been completed in all its parts, and the means of applying it to every case that can possibly occur in practice, are well understood. The time has therefore arrived, when it is possible to recast the whole subject, and reduce it to one consistent and uniform system. This is the object of the work before us, and the author has been in no small degree successful in its accomplishment. In every other elementary treatise with which we are acquainted, the trigonometric functions have been considered as lines absolutely existing, in some given relation to an arc of a circle of a definite radius. In all the elementary investigations, whether geometric or analytic, this radius bears a most important part; yet, no sooner do we make a direct step into the farther analysis, than it disappears, and the functions are represented in relations to each other, that lines can never assume. Thus, for instance, if we were to seek the value of the sine of an arc, in terms of the cosine and tangent, it would appear to be equal to their product, which, geometrically considered, would be impossible, as it is tantamount to declaring a line to be equal to a rectangle. Euler evidently conceived the angular functions to be the expressions of the ratios between lines, and not as lines themselves, and in this way they are now employed in all calculations. To render the theory consistent with the practice, it is necessary that their properties should be investigated upon a principle derived from

this view of their nature. This has been at last done by Mr. Hassler.

Among the three sides of a right angled triangle, combined by pairs, there exist six possible relations or ratios; each of these may be represented by a vulgar fraction, (proper or improper), of which the sides under consideration, constitute the numerator and denominator; this may, of course, be expressed by the decimal notation, and this expression will be a function of one of the acute angles. If the relation be that of the hypothenuse to the side adjacent to an angle, it is the cosine; if of the hypothenuse to the opposite side, it is the sine; if of the side adjacent to an angle, to that opposite, it is the cotangent; and if of the opposite to the adjacent side, the tangent. The two remaining relations give the secant and cosecant, but they are now no longer necessary in calculation. If to this we add the Pythagorean proposition, that the square of the hypothenuse is equal to the sum of the squares of the two sides, we have the basis upon which Mr. Hassler has built a complete system of the elements of Plane Trigonometry, and of what is frequently called the Arithmetic of Sines; which system requires no farther preliminary knowledge in the learner, than the first forty-seven propositions of Euclid, the four rules of arithmetic, the fundamental principle of proportion, and the solution of simple equations. We question if there be any work in modern mathematics, so simple in its basis, so clear and easy in its steps, and so full and complete in its deductions.

His spherical trigonometry requires, in addition, a knowledge of a few of the propositions of solid geometry, contained in the eleventh book of Euclid, and he has chosen as his fundamental theorem, the equations between the parts of the four right angled plane triangles, surrounding a right angled spherical triangle, instead of the more usual method of deducing the formula for an oblique angled triangle, from the plane triangle touching it at one of its angles. In this mode of proceeding, the analytic steps, that lead to the subsequent formulæ, are more simple than in the other.

The object of the work being purely elementary, the author has not conceived that the many practical purposes to which it may be applied, come fairly within his scope. He has, however, arranged all his practical formulæ in tables, and given instances of their calculation, in such a way as not only to show the manner of using each, but to give clear and precise ideas of the manner of arranging calculations, in such a way as to abridge labour, and render their examination more easy; these examples may be studied, and the chapter on the general

principles of calculation consulted, with advantage, even by those most proficient in the science.

Mr. Hassler then may be considered as having performed the same task in this elementary department, that Laplace has in the more important and elevated branch of celestial mechanics, that of combining and arranging the discoveries of his predecessors, and reducing them to one common method. His work is, indeed, far more humble in its design, but little less useful in its object; for if Laplace have presented the higher mathematics in a more clear and condensed light, Mr. Hassler has, in like manner, illuminated one of the most important preliminary steps to that sublime object; a step not only important, as absolutely necessary to the student of more elevated science, but from its direct application to various useful and practical purposes.

From what has been said in relation to the manner in which analytic trigonometry was built up, and from the numerous names of eminent men, who have been concerned in bringing it to its present state, we must confess that we entered upon the examination of this treatise without anticipating any novelty, except in the treatment, and the methods of investigation. We have, however, discovered one formula entirely new to us, and which, so far as we know, had escaped the research of any former analyst-it is that of a series for the tangent of a multiple arc, applicable to the calculation of tables of that trigonometric function. The usual series for this purpose is expressed in a fractional form, and is, of course, more complex in calculation than those for the sine and cosine. Our author has succeeded completely in rendering this series as simple in its shape as the others, and has thus rendered a most important service to those who may be hereafter engaged in the construction of tables.

Upon the whole, then, we cannot but congratulate ourselves, that it should have fallen to our lot, at the commencement of our career as reviewers, to call the attention of our readers to such a work; a work that will afford to foreign nations a high idea of the state of knowledge in our country; and which, as the production of an adopted citizen, who, although educated in his native land, first applied himself to mathematical science, as a profession, in our country, and drew it up originally for the use of an institution* supported at the public expense, is unquestionably national.

The Military Academy at West Point, where Mr. Hassler acted as Professor of Mathematics, before he proceeded to Europe to procure instruments for the survey of the coast.

praise, overstrained similitudes, and bombastic representations of characters and events.

In regard to those eulogies which we have undertaken to notice, it may be remarked, that there is a very great disparity in their degrees of excellence. The most finished pieces among them, are those of Messrs. Webster, Sergeant, Peleg Sprague, Duer, and Wilkins. These would not discredit any age or nation. Those of Joseph E. Sprague, of Massachusetts, and of William Johnson, of South Carolina, are very respectable productions. The piece of Mr. Wirt, discovers in many parts great fertility of fancy and considerable native genius, but is too florid and declamatory, and is altogether in a style more adapted to poetry than to prose. Mr. Everett's oration also, is not without its beauties, and discovers an imagination enriched with classic learning; but we cannot refrain from the observation, that it is, throughout, bespangled with too much ornament, laboured and artificial; and discovers but too plainly, that the author is more intent upon setting off his own rhetorical powers to advantage, than recommending the good and great qualities of the deceased patriots. Concerning the remainder of these orations, it may be said, that while some of them are by no means destitute of merit, for the most part they are so tawdry and inflated, and so stuffed with conceits, puerilities and overstrained hyperboles, that the fruit of whatever good sense may be contained in them, is completely overshadowed and obscured, by the superabundant efflorescence in figures. Their authors appear to be tugging, sweating, and spurring onwards their lagging invention, to reach the sublime; but unhappily, failing in the attempt, sink into bathos.

To proceed, and make good our assertions, by a recurrence to the several pieces in their turn. That circumstance which so greatly increased the sensibility of the American people, upon their late privation, was rather a disadvantage to the orators, who were called upon to utter the sentiments, and awake the sympathies of their fellow-citizens on that solemn occasion; viz. that they were constrained to portray the characters, and commemorate the lives and deaths of two distinguished patriots at the same time, and of patriots too, who performed similar parts in the same drama. This breaks in upon that unity of subject, which attracts attention, and enlists the feelings in behalf of the personages concerned; it subjects the speaker to the necessity of a second recurrence to the same events, which, under other circumstances, would be tedious, irksome and inelegant. Under this disadvantage of the subject, however, several of the speakers have acquitted themselves with considerable address,

and detailed the events in both lives with sufficient neatness, clearness, and beauty.

To begin with the several exordiums.-We are presented in these with specimens of almost every imaginable kind, from the most finished introduction, to a model of the reverse. Mr. Webster's introduction is natural, manly, and striking; Mr. Duer's neat, perspicuous, and interesting; that of Mr. Wilkins impressive; and that of Mr. Sergeant exceedingly good. If Mr. Sergeant had gone through the toil of composing a regular and complete eulogium, and had succeeded as well throughout the whole, as he has done in his introduction; we should regard it as one of the finest of panegyrics. We shall present it to our readers, in order that they may judge of it for themselves.

"Time, in its course, has produced a striking epoch in the history of our favoured country; and as if to mark with peculiar emphasis this interesting stage of our national existence, it comes to us accompanied with incidents calculated to make a powerful and lasting impression. The dawn of the fiftieth Anniversary of Independence beamed upon two venerable and illustrious citizens, to whom, under Providence, a nation acknowledged itself greatly indebted for the event which the day was set apart to commemorate. The one was the author, the other the ablest advocate, of that solemn assertion of right, that heroic defiance of unjust power, which, in the midst of difficulty and danger, proclaimed the determination to assume a separate and equal station among the powers of the earth, and declared to the world the causes which impelled to this decision. Both had stood by their country, with unabated ardour and unwavering fortitude, through every vicissitude of her fortune, until the glorious day' of her final triumph crowned their labours and their sacrifices with complete success. With equal solicitude, and with equal warmth of patriotic affection, they devoted their great faculties, which had been employed in vindicating the rights of their country, to construct for her, upon deep and strong foundations, the solid edifice of social order and of civil and religious freedom. They had both held the highest public employment, and were distinguished by the highest honours the nation could confer. Arrived at an age when nature seems to demand repose, each had retired to the spot from which the public exigencies had first called him-his public labours ended, his work accomplished, his beloved country prosperous and happy-there to indulge in the blessed retrospect of a well-spent life, and await that period which comes to all. But not to await it in idleness or indifference. The same spirit of active benevolence, which made the meridian of their lives resplendent with glory, continued to shed its lustre upon their evening path. Still intent upon doing good, still devoted to the great cause of