« PreviousContinue »
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 formula?, 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 formulae 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, thai 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 conduction 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 •uch 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.
To conclude; we cannot too strongly recommend the introduction of this treatise, as a text book, into the colleges and universities of the United States. We have expressed our opinion of the geometric method, and believe it must be abandoned; a step to which has already been made in the translation of Lacroix for Harvard University. But the work before us is far simpler in its basis than that of Lacroix; more elementary and direct in its attainment of the parts applicable to ordinary calculations; and far more extensive in its views and objects.
Art. Ill *ft Selection of Eulogies, pronounced in the several States, in Honour of those illustrious Patriots and Statesmen, John Adams and Thomas Jefferson. Hartford: published by D. F. Robinson &. Co., and Norton & Russell. 1826. 8vo. pp. 436.
We have before us a volume bearing this title, published at Hartford, in Connecticut, together with a single eulogy in a pamphlet form, delivered atCharlestown, in Massachusetts, by Edward Everett esq.; comprising altogether twenty panegyrical orations.—It is a circumstance of national congratulation, and encouraging to our patriotic hopes, upon the decease of our distinguished benefactors, to find the learning and eloquence of our country so readily enlisted in the task of celebrating their talents and services. One of the natural and most amiable tendencies of the human heart, is, to bestow honours up6n the meritorious dead—and, perhaps, the gratitude of nations is seldom more usefully employed, than when erecting monuments over the ashes of their departed worthies—If Christian states cannot, like the Pagans, elevate their heroes and wages to stations among the gods, and, after the ceremony of their deification, anticipate beneficial influences from their future sway within the system of nature, they can, at least, by public demonstrations of gratitude and affection, sow the seeds which shall produce future sages, statesmen and patriots, who shall become emulous of the talents and virtues of their predecessors, and be animated to like achievements in their country's service. This was one of the expedients by which Greece and Rome anciently produced that bright succession of heroes, statesmen, philosophers, poets and orators, who have conferred upon their names such deserved celebrity.
The orators, therefore, who have devoted their learning and eloquence to pronouncing funeral orations, over our two lamented presidents, have deserved well of their country. The task which they undertook, is a delicate and interesting one, but very difficult of execution. Perhaps, there is scarcely any literary performance, which, in order to a finished execution, requires a more skilful operation, or a more masterly hand, than that of a funeral oration. It is an easy task, indeed, to the orator, to indulge in hyperbolical encomiums, and to render himself and his audience giddy by glittering images, bombastic phrases, and overwrought delineations of ideal greatness—but aptly to decide upon the distinctive merits of those whose panegyr ric we undertake to pronounce, seize with accuracy their traits of character, assign them their due proportion of praise, and extol their talents and virtues in a style elevated, indeed, and sufficiently ornamented to meet the public expectation; but not swoln or florid ; chastened and simple, but not tame ; requires a mind not only of the highest order of native endowments, but also furnished with the treasures of literature. As Cicero justly remarks, in his treatise upon oratory, there is a style suited to each kind of speaking and writing—Nam, says he, et causa capitis alium quendam verborum sonum requirunt, alium rerum privatarum atque parvarum; et aliud dicendi genus deliberationes, aliud laudationes, aliud judicia, aliud sermones, aliud consolatio, aliud objurgatio, aliud disputatio, aliud historia desiderat .
In philosophical disquisitions, a just taste expects nothing more than neatness, simplicity and perspicuity; in history, these qualities, added to a talent at description; in oratory, a more free indulgence of ornament, and in panegyrics, the highest degree of legitimate decoration. As panegyrical orations are delivered with the professed purpose of pleasing, and have not within their scope either to instruct or persuade, a liberal criticism must allow them a range of embellishment, limited only by the principles of sound sense and correct taste. This rule seems to have its foundation in the acknowledged barrenness of the subject, as well as in that state of admiration and enthusiasm to which the minds of all, on such occasions, are excited. It is expressly allowed by Quintilian upon those grounds :—* Quoque, says he, quid natura magis asperum, hoc pluribus condiendum est voluptatibus: et nimis suspecta argumentatio, dissimulatione ; et multum ad fidem adjuvat audientis.voluptas. As each kind of writing has its characteristic excellencies, so each also has excesses to which it immediately tends. The natural tendency of eulogies is to excessive exaggeration in