APPARENT ANOMALY IN ALGEBRA.

THE apparently anomalous result of a number's being equal to half of itself, or that two is equal to four, obtained by treating a simple equation according to the rules of Algebra, viz: that adding to, or subtracting from, both sides of an equation, the same quantity does not destroy its equality; or that multiplying or dividing both numbers of an equation by the same factor or divisor, does not destroy its equality, is found to be no exception to the general laws of equations.

The following is the apparent exception. Let the equation be

Multiply both numbers by x,

x= a, (1.) x2 a x, (2.) Subtract a2 from both numbers - a2 — a x — a2, (3.) Decompose into binomial factors, (x+a)(x—a)=a(x — a) (4.) Divide by (x-a) and we shall obtain x+a=a, (5,) or that a number is equal to twice itself or one half itself. Every equation of the second degree is the product of two binomial factors, real or imaginary. In equation (1) of the first degree, the simple binomial factor is x -a=0 which is multiplied in equation (2) by the binomial x 0 = 0, thus (x — a) (x — 0) : the same as equation (2.)

= x2 a x

The subtraction in equation (3) is an artifice to facilitate the decomposition into factors; and by this same subtraction, each number becomes separately equal to zero; thus, x2 — a2 — 0, and a x-a2 = 0.

So that (x+a) (x—a) — a, which is satisfied by xgiven by the hypothesis; but the value of the other factor (x+a) does not appear from this, therefore we will resolve equation (3) by completing the square

The first value is that of x in the factor (x-a) and the second is the value of x in the factor (x+a). Also the binomial factors in the right hand member of equation (4) are (a+0) and (x-a.)

As a and a both enter the equation alike, by resolving equation (3) with respect to a, we shall obtain a similar result relative to a in terms of x. R. F. I.

NOTE. In the article on PLATO AND HIS ETRICS, page 27, line 6 from top, for "other" read "any other"; page 28, last line, for "horrors must be spared" read "honors must be shared"; page 30, line 2d from top insert "his" before "history"; page 36, line 5th from top, for "nation" read "notion."

THE

AMERICAN REVIEW.

No. II.

MARCH, MDCCCXLIII.

The Private Journal of AARON BUPR, during his Residence of Four Years in Europe, with selections from his Correspondence; in two vols. 8vo. New York: Harper and Brothers, 1838.

Memoirs of AARON BURR, with Miscellaneous Selections from his Correspondence; by MATTHEW L. Davis; in two vols. 8vo. New York: Harper and Brothers, 1837.

Reports of the Trials of Colonel AARON BURR, Late Vice-President of the United States for Treason, and for a Misdemeanor, etc., in the Circuit Court of the United States, held at the City of Richmond in the District of Virginia, in the Summer Term of the year 1807, etc., taken in short hand by DAVID ROBERTSON, Counsellor at Law. Philadelphia: Hopkins and Earle,

1808.

An Examination of the Various Charges exhibited against AA ON BURR, Esq., Vice President of the United States, and a Development of the Character and Views of his Political Opponents. By Aristides. Printed for the Author. 1804. A Narrative of the Suppression by COL. BURR, of the History of the Administration of John Adams, Late President of the United States, written by JOHN WOOD, etc.: New York, Denniston and Cheetham. 1802. A View of the Political Conduct of AARON BURR, Esq., Vice President of the United States. By the Author of the "Narrative." New York: Denniston and Cheetham. 1802.

THE manner of the world is, to make all remarkable men consistent prodigies. The approved are almost Deified; the disapproved are Demonized; the shades in the character of one are lost in overpowering light; the occasional rays of brightness in the other are merged in predominant darkness. Hence the tendency to an im

pression that great men are the creatures of Destiny; instead of being models to imitate, examples to encourage our ascent of the steep path of virtue, or beacons to warn us against the easy descent of vice; instead of teaching the necessary and just connection between that which is good and that which is honorable, and between that which is evil and that which is of bad report, we are rather taught, in the words of Sallust, that "Fortune bears sovereign influence over every thing; it is she that brightens or obscures all things, more from caprice and humor than a regard to truth and justice." When men are thus made the sport of fate, and are rendered Gods or Monsters, how many truth-telling lessons are lost to the world? Cotemporary history is the fruitful cause of this error. When passion, prejudice, and selfish motives are at work, the tide of public opinion ever flows strongly in one direction, and it is only at some distant period that the returning wave is seen. By an indiscriminate condemnation of the good and bad qualities of a man, evil is always done; for the candid acknowledgment of a virtue increases the weight of just censure of a vice. Posterity, therefore, in pronouncing judgment upon the character of the subject of these remarks, if justified by truth in so doing, will not hesitate, righteously to temper that judgment with mercy.

Aaron Burr was born on the sixth day of February, 1756, at Newark, in the State of New Jersey. His maternal grandfather, the Rev. Jonathan Edwards, was a man of ardent piety, distinguished talents, and eminent as a theologian and metaphysician. Sarah Pierrepont, the wife of Mr. Edwards, was alike remarkable for uncommon beauty of person, and her enthusiastically religious disposition; and their daughter Esther, the mother of Aaron Burr, though deeply imbued with those religious impressions which had become almost characteristic of the family, yet possessing much sprightliness of wit, vivacity and cheerfulness of manner, her religion cast no shade of melancholy over her life, but adorned her character with new charms and graces. She was married on the 29th of June, 1752, to the Rev. Aaron Burr, then President of the College of New Jersey. This union, endeared by congenial pursuits, habits, and temperament, was severed by the early death of Mr. Burr. Within a

short period occurred the death of Mr. Edwards, which was speedily followed by that of Mrs. Burr, who died in the twenty-seventh year of her age, sixteen days after the decease of her father, and six months after that of her husband. She was carried off by a violent disorder "which the physician said he would call by no name but that of a messenger sent suddenly to call her out of the world." Six months more, and Mrs. Edwards also Her remains were interred at Princeton, where "the father and mother, son and daughter, were laid together in the grave, within the space of a little more than a year." By these extraordinary events, Aaron Burr and his sister Sarah, at a tender age, were deprived of parental protection. Nothing is more interesting in the history of great men than the tracing of those quiet, gradual, and almost imperceptible parental influences which, in the plastic period of youth, impart an indelible impression to the character, and give a tone and impulse to the embryo hopes, wishes, feelings, and aims-all-pervading, never-ending. As it is the lot of woman to be the mother of human life, so it is hers to nurture and mould the mind-to become the second mother of the soul to direct its young and trembling efforts-to stamp it with marks not to be obliterated by the waves of the ocean of life-to repress the evil and encourage the good -to aid in giving that form to the mind, which, when the miniature expands to mature dimensions, remains its peculiar spiritual form for ever. The Rev. Aaron Burr is described as "possessed of a slender and delicate constitution; joining to uncommon talents for the despatch of business, a constancy of mind that constantly secured to him success." He was greatly respected in every station and relation of life. He was a man of acknowledged talents, of sound, practical good sense, of unimpeachable integrity, and of ardent piety. Polished in his manners, he had uncommon powers in conversation, and possessed the happy art of inspiring all around him with cheerfulness. As a reasoner, he was clear and solid, and as a preacher, animated, judicious, fervent, and successful. He had warm affections, was greatly endeared to his family and friends, and was open, fair, and honorable in all his intercourse with mankind." In remarking the close and striking resemblance of intellectual

character between the parent and child, the features of diversity, it may also be observed, are precisely those which the authority of a father, and the beneficient influence of an amiable, pious, and intelligent mother, would have been instrumental in correcting. In the easy pecuniary circumstances in which young Burr was left on the decease of his parents, may be traced also the cause of much of that independence and carelessness as to the opinions of society, which poverty would have moderated and checked, if not destroyed. In the struggle for his daily bread, a man encounters afflictions, rebuffs and trials, which, while they chasten and temper, also refine and elevate his character-teach him the necessity of trust and reliance upon a higher power, debar him the means of sensual indulgence-impose upon him the practice of austere living, and direct all the energies of his mind, rather to the conquest of the difficulties which meet him, than to a self-abandonment to his passions.

Without parental admonition-without the teachings of adversity, this child was thrown upon the world. At the age of four years, he was placed under the care of Dr. Shippen, of Philadelphia. His first exploit consisted in absconding from his home in consequence of a quarrel with his kind guardian. When about ten years old, having become smitten with a fondness for the life of a sailor, he repeated the same experiment, and engaged as a cabin-boy on board of a vessel about to sail from New York. Whilst engaged in the duties of his new occupation, perceiving the approach of his uncle, who had discovered his place of retreat, he clambered to the masthead, whence he refused to come down, "until all the preliminaries of a treaty of peace were agreed upon.'

He entered Princeton College, the institution over which his father and grandfather had successively presided, at the early age of thirteen, and from this time he may fairly be considered as his own master. During the early portion of his collegiate career, he appears to have been a close and indefatigable student, a habit subsequently relaxed for more frivolous and dissipating pursuits. His irregularities were carried to characteristic extremes; at times sleeping but six hours out of the twenty-four, and devoting the remainder to retirement and study, and at others abandoned to pleasure, amuse