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general theorem for raising a quantity consisting of two terms to any power m.

The same genera! theorem will also serve for the evolution of binomials, because to extract any root of a given quantity is the same thing as to raise that quantity to a power whose exponent is a fraction that has its denominator equal to the number that expresses what kind of root is to be extracted. Thus, to extract the square root of a+b, is to raise ab to a power whose exponent is. Now, a+b being found as

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above; supposing m=4, you will find a+b+; therefore x+a=x+nax3-1 = a + 1 × a − b b + 1 x − 1 × α- { b2 + +n.

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This proof applies only to those cases in which n is a whole positive number; but the rule extends to those cases in which n is negative or fractional.

Ex.1. u+x2=a3+8 a1x+28 eʻr2+ 56 a3 x3 +70 a* x2 + 56 a3 x3 + 28 a2x2 + 8 ax2+x3.

Ex. 2. 1+x=1+nx+n.

n

Let x+b.x+c.x+d. &c. = x2-1 +
Pr2-1+Q x2- 1 +&c. and x+x+b. +"."."—2x2+&c.

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x2+Px"1+Qx2+&c. are the same +ax11+aP-2+&c. series; therefore, AP+a,B=Q+a P, &c. that is by introducing one factor, x+à, into the product, the coefficient of the second term is increased by a, and by introducing x+b into the product, that coefficient is increased by b, &c. therefore the whole value of A is a+b+c+d+&c. Again, by the introduction of one factor, x+a, the coefficient of the third term, Q, is increased by a P, i. e. by a multiplied by the preceding value of A, or by axb+c+d+&c. and the same may be said with respect to the introduction of every other factor; therefore upon the whole,

B=a.b+c+d+&c.
+b.c+d+&c.
+c.a+&c.

In the same manner,

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and so on; that is, A is the sum of the quan- ber, the coefficients of the terms taken

backward, from the end of the series, are respectively equal to the coefficients of the corresponding terms taken forward from the beginning.

Thus, in the first example, where a +x is raised to the 8th power, the coefficients are 1, 8, 28, 56, 70, 56, 28, 8, 1.

In general, the coefficient of the n+1
N.N- 1.n 2.....3.2.1

1.2.3......n-2.n—1.n

th

=1.

ten: for, adds he, though kings, princes, and great personages be few; yet there are many other excellent men, who deserve better than vague reports and barren elogies.

Biography, or the art of describing and writing lives, is a branch or species of history, in many respects as useful and important as that of history itself; inasmuch as it represents great men more distinctly, unencumbered with associates: and descending into the detail of their actions and characters, their virtues and failings, we obtain a more particular, and, of course, a more 1th term, interesting acquaintance with individuals than general history allows. A writer of lives may, and ought, to descend to minute circumstances and familiar incidents. He

term is

The coefficient of the nth term is

n.n-1.n
1.2.3....

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n-2.n-1
n.n-1.n- 2. .3. n.n-1

1.2.3..... N·

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1.2

&c.

The sum of the coefficients 1 +n+n. is expected to give the private, as well as

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the public life of those whose actions he records; and it is from private life, from familiar, domestic, and apparently trivial occurrences, that we often derive the most accurate knowledge of the real character.

The subjects of biography are not only the lives of public or private persons, who I have been eminent and beneficial to the world, but those also of persons notorious for their vice and profligacy, which may serve, when justly characterised, as warnings to others, by exhibiting the fatal consequences which, sooner or later, generally follow licentious practices. As for those who have exposed their lives, or devoted their time and talents for the service of their fellow-creatures, it is but a debt of gratitude to perpetuate their memories, by making posterity acquainted with their me rits and usefulness. In the lives of public persons, their public characters are principally, but not solely, to be regarded; the world is interested in the minutest actions of great men, and their examples both as public and private characters, may be made subservient to the well-being and prospe rity of society.

It has been a matter of dispute among the learned, whether any one ought to write his own history. There are instances both ancient and modern that may be adduced as precedents for the practice: and the reason assigned for it is, that no man can be so much the master of the subject as the person himself: but, on the other hand, it is a very difficult task for any one to write an impartial history of his own actions. Plutarch mentions two cases in which it is allowable for a man to commend M m

himself, and to be the publisher of his own merits; which are, when the doing of it may be of considerable advantage either to himself or to others. Notwithstanding this high authority, the former case is unquestionably liable to great objections, because a man is to be the judge in his own cause, and therefore very liable to exceed the limits of truth when his own interests are concerned, and when he wishes to render himself conspicuous for virtue or talents. The ancients, however, had a peculiar method of diverting the reader's attention from themselves, when they had occasion to record their own actions, and of thus rendering what they said less invidious, which was, by speaking of themselves in the third person. Among the moderns a practice has been introduced, which cannot be too strongly reprobated, though sanctioned by men of great talent, integrity, and real worth, namely, of making the memoirs of themselves the vehicle of abuse of their contemporaries; every one of whom would, no doubt, be able to give a very different, and perhaps plausible reason, for the several actions which the biographer has undertaken to scrutinize and condemn.

Dr. Priestley has constructed and published a "Biographical Chart," of which our plate is given as a specimen. This chart represents the interval of time between the year 1200 before the Christian æra, and 1800 after Christ, divided by an equal scale into centuries. It contains about 2000 names of persons, the most distinguished in the annals of fame, the length of whose lives is represented by lines drawn in proportion to their real duration, and terminated in such a manner as to correspond to the dates of their births and deaths. These names are distinguished in to several classes by parallel lines running the whole length of the chart, the contents of each division being expressed at the end of it. The chronology is noted in the margin, on the upper side, by the year before and after Christ, and on the lower by the same æra, and also by the succession of such kings as were most distinguished in the whole period. See Plate BIOGRAPHY.

For a more full account we refer to Dr. Priestley's description which accompanies the chart; from which we shall make a short extract, that cannot fail to entertain the reader.

"Laborions and tedious as the compilation of this work has been (vastly more so than my first conceptions represented it to me),

a variety of views were continually opening upon me during the execution of it, which made me less attentive to the labour. As these views agreeably amuse the mind, and may, in some measure, be enjoyed by a person who only peruses the chart, without the labour of compilation, I shall mention a few of them in this place.

"It is a peculiar kind of pleasure we ręceive, from such a view as this chart exhibits, of a great man, such as Sir Isaac Newton, seated, as it were, in the circle of his friends and illustrious contemporaries. We see at once with whom he was capable of holding conversation, and in a manner (from the distinct view of their respective ages) upon what terms they might converse. And though it be melancholy, it is not unpleasing, to observe the order in which we here see illustrious persons go off the stage, and to imagine to ourselves the reflections they might make upon the successive departure of their acquaintance or rivals.

"We likewise see, in some measure, by the names which precede any person, what advantages he enjoyed from the labours and discoveries of others; and, by those which follow him, of what use his labours were to his successors.

"By the several void spaces between such groups of great men, we have a clear idea of the great revolutions of all kinds of science, from the very origin of it; so that the thin and void places in the chart are, in fact, no less instructive than the most crowded, in giving us an idea of the great interruptions of science, and the intervals at which it hath flourished. The state of all the divisions appropriated to men of learning, is, for many centuries before the revival of letters in this western part of the world, exactly expressed by this following line of Virgil :

· Apparent rari nantes in gurgite vasto, But we see no void spaces in the division of statesmen, heroes, and politicians. The world hath never wanted competitors for empire and power, and least of all in those periods in which the sciences and the arts have been the most neglected.

"But the noblest prospect of this nature is suggested by a view of the crowds of names in the divisions appropriated to the arts and sciences in the two last centuries. Here all the classes of renown, and, I may add, of merit, are full; and a hundred times as many might have been admitted, of equal attainments in knowledge with their

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