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general theorem for raising a quantity con tities a, b, c, &c. B is the sum of the prer sisting of two terms to any power m. dacts of every two; C is the sum of the proThe same general theorem will also serve
ducts of every three, &c. &c. for the evolution of binomials, because to
Leta=b=o=d=&c. then A, or a +b extract any root of a given quantity is the +o++&c. =na; =ab tacB tbc same thing as to raise that quantity to a + &c.=ax the number of combinations power whose exponent is a fraction that has of a, b, c, d, &c. taken two and two, =n.
n-1 its denominator equal to the number that
a'; in the same manner it appears expresses what kind of root is to be extracted. Thus, to extract the square root of that C=n. "71."5%'
, &c. And a+b, is to raise a tb to a power whose exponent is z. Now, atom being found as sta.rt0.xtc. &c. to n factors = above ; supposing m=s, you will find a +61 stal"; therefore Ital=x + nazsa: =at +ixa-16+}X-4 Xa-16+ tn." "°-+*.**."
b *'*--*-4a-10+, &c. = a1 + mom ?+&c.
This proof applies only to those cases in +
which n is a whole positive number; but 8 až 100
the rule extends to those cases in which n is To investigate this theorem, suppose n negative or fractional. quantities, I tu, +b, x + c, &c. multiplied together; it is manifest that the first Er. 1. a = a +84'I+28+ term of the product will be t'", and that 56 a' x + 70 &*** + 56 u'm + 28 a 2 + gros - , -, &c. the other powers of .r will 8 ax' tao. all be found in the remaining terms, with different combinations of a, b, c, d, &c. Er... 17 zd"=1+na+n."5122
Let x+bortoortd.&c. = mart Pr--+Q+---+&c. and I +2.3to. +.."."***+&c. rtor+d.&c.=r+Am-+B.22 + &c. then ra + Ar-' + B.09–2,
*s-2 +&c. Ex. S. 2? +27"="+nads-+*. and r taxx*-' + P 7*- + Q rast n-1
42*4x4 +.&c. +Px**+Qxr-. +&c. are the same t'ax-'+aPx9–2 +&c. S series ; If either term of the binomial be negative, therefore, A =Pta, B=Q+aP, &c. its odd powers will be negative, and consethat is by introducing one factor, &tà, into quently the signs of the terms, in which the product, the coefficient of the second those odd powers are found, will be term is increased by a, and by introducing changed. x +b into the product, that coefficient is increased by b, &c. therefore the whole Ex. * a -21% = Q — 8 a' x + 28 ao 33 value of A is a +b+c+d+ &c. Again, - 56 m 2 + 70c* ** — 56 a'r + 28 e I by the introduction of one factor, I + a, the -8ax' to. coefficient of the third term, Q, is increased by a P, i.e. by a multiplied by the preceding Ex. 5. a? — X = q*• – nesmrt value of A, or by a xototd F&c. and
028*** &c. the same may be said with respect to the introduction of every other factor; therefore upon the whole,
If the index of the power to which a binoB=a.5+c+d F&c.
mial is to be raised, be a whole positive pumtb.ctd F&c.
ber, the series will terminate, because the +c.d+ &c.
coefficient *.*71.474.&c. will beIn the same manner, C=a.6.c. +d+ &c.
come nothing when it is continued torti
factors. In all other cases the number of +a.c.d+ &c.
terms will be indefinite. +b.c.d+ &c.
When the index is a whole positive pumand so on; that is, A is the sum of the quan. ber, the coefficients of the terms taken
n.n-1.n - 2.
backward, from the end of the series, are ten: for, adds he, though kings, princes, respectively equal to the coefficients of the and great personages be few; yet there are corresponding terms taken forward from many other excellent men, who deserve the beginning.
better than vague reports and barren Thus, in the first example, where a tris elogies. raised to the gth power, the coefficients are Biography, or the art of describing and 1, 8, 28, 56,70, 56, 28, 8, 1.
writing lives, is a branch or species of hisIn general, the coefficient of the n+1°
tory, in many respects as useful and impor
tant as that of history itself; inasmuch as
3.2.1 term is
it represents great men more distinctly, un1.2.3. .12.n-1.n
encumbered with associates : and descendThe coefficient of the nth term is
ing into the detail of their actions and cha
racters, their virtues and failings, we obtain -2....3.2.
a more particular, and, of course, a more =n; of the n — n -2.n-1
1th term, interesting acquaintance with individuals n.n-1.n—9....3
than general history allows. A writer of &c.
lives may, and ought, to descend to minute 1.2.3..
circumstances and familiar incidents. He The sum of the coefficients 1 +n +n is expected to give the private, as well as
the public life of those whose actions he *+&c. is 2“.
records; and it is from private life, from For if x =a=1, then Ita=1+1"
familiar, domestic, and apparently trivial
occurrences, that we often derive the most =9"=1+r+n."=++&c. accurate knowledge of the real characSince x ta' = "+nax-tn.
The subjects of biography are not only +&c.
the lives of public or private persons, who
have been eminent and beneficial to the And x-al" = x" — nax-'tn.
world, but those also of persons notorious
for their vice and profligacy, which may u? 23m2 &c.
serve, when justly characterised, as warnBy addition, x +al” +3-a" = 2.7" ings to others, by exhibiting the fatal con
sequences which, sooner or later, generally +2.n. uxr" -- +- &c.
follow licentious practices. As for those who have exposed their lives, or devoted
their time and talents for the service of = x + n. 2
their fellow-creatures, it is but a debt of u? x*–2 +&c.
gratitude to perpetuate their memories, by By subtracting one series from the other,
making posterity acquainted with their me.
rits and usefulness. In the lives of public x+"
pally, but not solely, to be regarded; the n-2 a'r"
world is interested in the minutest actions
of great men, and their examples both as The trinomial a +b+c may be raised to
public and private characters, may be made any power by considering two terms as one
subservient to the well-being and prospefactor, and proceeding as before.
rity of society. Thus, a + 6 +0"=a+n.otc.ca - It has been a matter of dispute among
the learned, whether any one ought to tn. 6+812.0" –2 + &c. and the
write his own history. There are instances 2 powers of b + c may be determined by the both ancient and modern that may be adbinomial theorem.
duced as precedents for the practice: and
the reason assigned for it is, that no man BIOGRAPHY, a very entertaining and can be so much the master of the subject instructive species of listory, containing as the person himself: but, on the other the life of some remarkable person or per- hand, it is a very difficult task for any one sons.
to write an impartial history of luis own Lord Bacon regrets, that the lives of actions. Plutarch mentions two cases in eminent men are not more frequently writ. which it is allowable for a man to commend
Orta" + x
=naz-s+n."persons, their public characters are princi
hinself, and to be the publisher of his own a variety of views were continually opening inerits ; which are, when the doing of it upon me during the execution of it, which may be of considerable advantage either made me less attentive to the labour. As .to himself or to others. Notwithstanding these views agreeably amuse the mind, and this high authority, the former case is un- may, in some measure, be enjoyed by a questionably liable to great objections, be- person who only peruses the chart, without canse a man is to be the judge in his own the labour of compilation, I shall mention cause, and therefore very liable to exceed a few of them in this place. the limits of truth when his own interests “ It is a peculiar kind of pleasure we reare concerned, and when he wishes to ren ceive, from such a view as this chart exhider himself conspicuous for virtue or talents. bits, of a great man, such as Sir Isaac NewThe ancients, however, had a peculiar me- ton, seated, as it were, in the circle of his thod of diverting the reader's attention friends and illustrious contemporaries. We from themselves, when they had occasion sec at once with whom he was capable of to record their own actions, and of thus holding conversation, and in a mammer (from rendering what they said less invidious, the distinct view of their respective ages) which was, by speaking of themselves in upon what terms they might converse. And the third person. Among the moderns a though it be melancholy, it is not unpleaspractice has been introduced, which cannoting, to observe the order in which we here be too strongly reprobated, though sanc see illustrious persons go off the stage, and tioned by men of great talent, integrity, to imagine to ourselves the reflections they and real worth, namely, of making the me- might make upon the successive departure moirs of themselves the vehicle of abuse of of their acquaintance or rivals. their contemporaries; every one of whom “We likewise see, in some reasure, by the would, no doubt, be able to give
names which precede any person, what addifferent, and perhaps plausible reason, for vantages he enjoyed from the labours and the several actions which the biographer has discoveries of others; and, by those wbich undertaken to scrutinize and condemn. follow him, of what use his labours were to
Dr. Priestley has constructed and pub- his saccessors. lished a “ Biographical Chart," of which “By the several void spaces between such our plate is given as a specimen. This groups of great men, we have a clear idea chart represents the interval of time be- of the great revolutions of all kinds of tween the year 1200 before the Christian science, from the very origin of it; so that æra, and 1800 after Christ, divided by an the thin and void places in the chart are, in equal scale into centuries. It contains fact, no less instructive than the most crowdabout 2000 names of persons, the most dis- ed, in giving us an idea of the great intertinguished in the annals of fame, the length ruptions of science, and the intervals at of whose lives is represented by lines which it hath flourished. The state of all drawn in proportion to their real duration, the divisions appropriated to men of learnand terminated in such a manner as to cor ing, is, for many centuries before the revirespond to the dates of their births and val of letters in this western part of the deaths. These names are distinguished in world, exactly expressed by this following to several classes by parallel lines running line of Virgil: the whole length of the chart, the contents
Apparent rari nantes in gurgite rasto. of each division being expressed at the end of it. The chronology is noted in the mar But we see no void spaces in the division of gin, on the upper side, by the year before statesmen, heroes, and politicians. The and after Christ, and on the lower by the world hath never wanted competitors for same æra, and also by the succession of empire and power, and least of all in those such kings as were most distinguished in the periods in which the sciences and the arts whole period. See Plate BIOGRAPHY. have been the most neglected.
For a more full account we refer to Dr. “But the noblest prospect of this nature is Priestley's description which accompanies suggested by a view of the crowds of names the chart; from which we shall make a in the divisions appropriated to the arts and short extract, that cannot fail to entertain sciences in the two last centuries. Here the reader.
all the classes of renown, and, I may add, “ Laborious and tedious as the compilation of merit, are full; and a hundred times of this work has been (vastly more so than as many might have been admitted, of my first conceptions represented it to me), equal attainments in knowledge with their