i.'bTV 
Ml 
BtO 
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 8 th power, the coefficients are 
1, 8, 28, 56, 70, 56, 28, 8, 1. 
In general, the coefficient of the re -j- 1 th 
term is 
re .re — 1 . re — 2 3 . 2. 1 
1.2.3 re — 2 . re — 1 .re 
The coefficient of the re th term is n ’ n 1 ' n 
1.2.3 
— 2 3.2. „ 
■ 1 — = re; of the re — 1 * term, 
re — 2 . re — 1 
re . re — 1 . re - 
..3. 
re . re — 1 
1.2.3 re- 
1 . 2 
, &c. 
The sum of the coefficients 1 + re + re . 
ft \ 
-I- &c. is 2*. 
2 ~ 
For if x — a = 1, then x-f-aT = 1 — (— ll” 
= — -re-j-re. n Q -- -|- &c. 
Since x -j- «|” = x” -J- re a x” “ 1 -|- re. 
ft 1 
— a 2 x" ~ 2 -J- &c. 
And x — a]" = x” — wax’ 1 — 1 -[-re . ? - i 
- &c. 
By addition, x-f-al'’ -f- x — w" — 2.x” 
~1 
2 . re 
: fl 2 2 _|_ 
Or 
x -j- «1” 
2 
•a 1 „ , re 
= x" -J- re . — 
2 
a 2 x”— 2 -|- &c. 
By subtracting one series from the other, 
x-| -al" — x — a]“ „ , i re — 1 
— — — — nax n — 1 4- re . . 
2 ' 2 
a 3 x n ~ 3 -j- &c. 
3 1 
The trinomial a -(- b -)- c may be raised to 
any power by considering two terms as one 
factor, and proceeding as before. 
Thus, a -j- 0 cl " — uP -\-n . b-\-c . a” — 1 
re — l 
+ » 
, a n — 2 -j- &c. and the 
powers of b -J- c may be determined by the 
binomial theorem. 
BIOGRAPHY, a very entertaining and 
instructive species of history, containing 
the life of some remarkable person or per- 
sons. 
Lord Bacon regrets, that the lives of 
eminent men are not more frequently writ- 
VOL. I. 
BIO 
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 his- 
tory, in many respects as useful and impor- 
tant as that of history itself; inasmuch as 
it represents great men more distinctly, un- 
encumbered with associates : and descend- 
ing into the detail of their actions and cha- 
racters, their virtues and failings, we obtain 
a more particular, and, of course, a more 
interesting acquaintance with individual* 
than general history allows. A writer of 
lives may, and ought, to descend to minute 
circumstances and familiar incidents. He 
is expected to give the private, as well as 
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 charac- 
ter. 
The subjects of biography are not only 
the lives of public or private persons, who 
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 warn- 
ings to others, by exhibiting the fatal con- 
sequences 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 princi- 
pally, 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 histoiy. There are instances 
both ancient and modem that may be ad- 
duced 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 
■pp Wrf 
' - ■ - . 
