only that their spots might be more noticed; so much can nature do for our intellectual part, and so little for our moral." Every one, who has read the histories and biographical collections of the ancients, must know that they are fitted to excite those melancholy remarks. Modern European history and biography are, undeniably, open to the same commentary. We have run through" the sketches, in the numerous volumes of Memoirs, published on the continent of Europe, of late years, and the biographical dictionaries now in a course of publication at Paris; and have risen from them, with the sad reflection, that, while nature and education could do so much for the intellectual part, bad constitutions of government and society, vicious example and safe opportunity, and political convulsions, so completely pervert or extinguish the moral.

It is a striking feature, how many of our celebrated personages were the architects of their own fortunes-rose from obscurity, indigence, or illiteracy. Thus, in the French revolution, a large portion of the successive leaders, both in the civil and the military departments, were of humble origin. But how different their course and their fate, and their relation to the destinies and character of their country, from those of the American patriots! Most of them were swept away by the tempests of faction, which they profligately contributed to raise, or vainly struggled to quell, after having worshipped at the most unholy and fatal shrines. The survivors have furnished ample materials for a Dictionnaire des Girouettes, a dictionary of Weathercocks; forming a most deplorable, disgusting picture of selfishness and impudence; of repeated and shameless apostacy, not between domestic parties, or men, with the same principles, but from political and religious institutions and forms, doctrines, modes and persons, diametrically opposite to each other.

With regard to the prevalence of particular merits, some diversity may be observed in the great divisions of our country. In the East, there appear to have been more learning, diligence, authorship, religious zeal, systematic frugality, and early republican principle and resolution-more founders or benefactors of schools and colleges; more ingenious mechanicians. The profession of the Law seems to have flourished, in all parts, with nearly a proportional share of most able members :-that of Medicine has been richest in the middle states. South of the Susquehanna, we find the most briliant orators; more active and chivalrous military leaders; more liberal and accomplished gentlemen, high-minded, devoted patriots, who exposed their persons and large estates, and endured all losses and priva

tions, with a zeal, firmness, and even alacrity, that bring to mind the best of the cavaliers in the civil wars of England. Of Merchants, we might name many, scattered over the seaboard, who have exerted a noble and enlightened liberality, and who bore an important share in the work of the Revolution, or in that of subsequent legislation. In the West, we discover a large fund of heroic character, romantic and eventful enterprise, and gallant exploit:-the personal annals of Ohio, Kentucky, and Tennessee, are replete with objects and results, which excite curiosity and admiration. In the history of their Indian wars, the reader has, indeed, a repetition of much of what is so minutely told of those of New-England, in her many volumes on that subject; the incidents and characters bear a resemblance; yet the scenes of action, and the actors, are so far different, as to beget a peculiar interest and important variety. It may be hoped that the western states will do justice to themselves, by early embodying their authentic traditions.

We rejoice, that biographical details are acquiring more vogue and importance in the United States; and trust, that this circumstance will animate the citizens, who possess valuable memoirs or documents, to give them to the world. A nation is judged, as much, perhaps, by its single characters and lives, its exalted and striking men, as by the sum of its power, wealth, learning, or civilization. Perhaps, too, the chief interest and consequence of History lie in the virtues or vices, the designs and acts, and the fortunes and adventures, of individuals; rather than in the aggregate of public occurrences, or the mere movements and effects of battles, sieges, state policy and vicissitudes, general felicities or disasters of what kind soever. We Americans should make our Biography as full as possible,--at least the honourable part of it,for the credit of our political and social institutions, and as a broader standard of comparison with Europe; an instrumentality in which, as we have already intimated, we have great reason to be satisfied with it, and shall have greater, when it shall be enlarged. The study involves many and precious lessons; mementoes, elevating or depressing, lively or serious. It reminds us of life and energy, glory and power; but, as constantly, of misfortune and death.

Optima quæque dies miseris mortalibus ævi

Prima fugit: subeunt morbi, tristisque senectus
Et labor, et duræ rapit inclementia mortis.

The records of the venerable dead, in books, are like the monuments in Westminster Abbey, which, if they warmed and excited the authors who visited them, have occasioned, and justly, much solemn moralizing,-very sober conclusions on

the end of all human prowess and strife. We need not state the impression, which is felt in thinking of the multitude of names, in Bayle's Dictionary, for example, which were once so celebrated, and deemed so important in the annals of policy, religion, war, literature, philosophy, and which are now utterly forgotten, or generally unknown. Well might Marcus Antoninus exclaim,-even in his time-"How many men of high renown, with whose praises the world once rang, are now consigned to oblivion; and how many bards and panegyrists, who promised immortality to their names, have, themselves, long since disappeared in the gulf of time!".

ART. II.-Elements of Analytic Trigonometry, Plane and Spherical. By F. R. HASSLER, F. A. P. S. New-York: published by the Author. James Bloomfield, printer. 1826. 8vo. pp. 192.

TRIGONOMETRY, in its several branches, and in the number of its practical applications, forms, perhaps, the most important department of mathematical science. Whether we view it as applied to the simpler business of the land surveyor, and the elementary problems of practical navigation, or extended to the mensuration of the spheroidal surface of the earth, and of the angular positions of those great and distant bodies, that are spread throughout the regions of unbounded space, we feel, in every case, sensible of its value. But it is in the efficient and indispensable aid that it affords to the calculus of modern mathematicians, that we find the highest and most marked instances of its usefulness.

We derive the basis of Trigonometry, like that of almost every branch of our knowledge, from the ancient Greeks; it has, indeed, received from time to time, improvements of the most important character; but we must consider ourselves, even at the present moment, under the deepest obligation to that remarkable people, as well in this instance, as in nearly all the sciences we at present cultivate. Whether it took its rise among them, or whether their Trigonometry was derived from some more early nation, it is impossible, at this day, to determine. The Greeks have been accused of arrogating to themselves, the discoveries and the science of others, and of carefully hiding from posterity the sources whence they obtained them. The Egyptians are frequently considered as the precursors and instructers

of the Greeks. Laboured arguments and eloquent treatises have been drawn up, to maintain the early civilization and advance in knowledge, of this mysterious people. More close investigation of their monuments and remains, seems, however, to lead us daily to the conclusion, that much of this superiority and intelligence is entirely imaginary, and that those of their works, on which the record of their remote progress in science has been supposed to be inscribed, are the productions of a later period-the imperfect reflections, instead of the faint dawnings of the light that blazed in Greece.

The eloquent Bailly, abandoning the idea that Egypt was the cradle of European science, has sought it in a nation residing in central Asia, whose name, whose monuments, and whose language, are long forgotten, and of which no trace remains but the imperfect fragments of their astronomic science, preserved, but not understood, by the Chaldeans of the age of Alexander, the Chinese, and the Hindoos of our own times. A late most able and competent judge, Delambre, has stripped this fabled nation of its honours, and shown most conclusively, that the writings of the later Greeks, are, in truth, the basis of all the boasted learning of Hindostan and China, which countries probably received them through the intervention of the Arabs, who preserved, cultivated, and even improved some of the branches in which the Greeks made so marked and rapid a progress.

of

Wherever we may seek for the origin of the Mathematics, it is to be confessed, that we are to look to the writings of the Grecian geometers for its earliest and most authentic recordsfor its most elegant and accurate methods. If the progress the moderns has rendered many of their writings obsolete; pointed out shorter and more advantageous roads to the common end; and finally extended the applications of the science to objects that might, at first sight, appear beyond the reach of human investigation; we may, notwithstanding, consult them with advantage, as the finest models of close and accurate argument, and as the best exercise that has ever been discovered for the improvement of the reasoning faculty.

The time of the origin of trigonometry among the Greeks, is extremely well defined, and marked by circumstances and facts so pointed, as to leave no doubt as to its exact era. Archimedes was born in the year 287 before Christ. He is said to have travelled in his youth into Egypt, to acquire the knowledge taught by the priests of that country. However this may be, he was certainly not only acquainted with all the science of his own country, but made to it many additions. His Are

narius is a work written for the purpose of showing the extension of which the system of numerical symbols employed by him, was capable. In it, he undertakes to prove, that the number of grains of sand contained in the globe of the earth, is not, as was at the time maintained, infinite, but capable of expression in the notation then in use; and that, if it were even applied to the number of grains contained in the greater sphere, whose radius is the distance of the earth and sun, they would still admit of a numerical expression. In the course of his investigation, it becomes necessary for him to determine the diameter of the sun. The observation, he states he made for the purpose, is, perhaps, less rude than might have been anticipated from the state of the mechanic arts; but when he attempts to obtain the result, he is compelled to have recourse to a mere graphical process, and shows conclusively, that he is unacquainted with any mode of determining, by calculation, the measure of the angle at the vertex of an isosceles triangle, when the base, and the two equal sides are given. As this is one of the easiest cases of practical plane trigonometry, we may of course infer, that any mode of applying calculation to the determination of the unknown parts of triangles, from others that are given, was as yet undiscovered, with the sole exception of the cases that may be solved by the aid of the famous proposition of Pythagoras.

This proposition has reached us in the collection of the geometrical knowledge of his times, made by the celebrated Euclid this knowledge he methodised, arranged, and reduced to one common method. In his elements of geometry, we find several other propositions essential as the foundation of trigonometry, but no trace of any method of calculation, other than the case above mentioned. His first book contains, besides the Pythagorean proposition, several that are important in showing the nature of triangles, gives the determination of the sum of their angles, and the value of their surface in terms of their base, and perpendicular altitude. In his second book, we find a remarkable proposition, in respect to the difference between the sum of the squares of any two sides of any triangle, and the square of the third side; a proposition that is, however, incapable of being reduced to practice, without the aid of a method for calculating the unknown parts of triangles from others that are given. This proposition is now concisely stated as follows: b2 + c2 2 bc. cos. A. find the useful theorem, that the line

a2 =

In his third book, we