V- 
?+/ • • I 
BIN 
stand in the place where it was stopped. 
24. He who plays without a foot upon the 
floor, and holds his adversary’s ball, gets 
nothing for it, but loses the lead. 25. He 
who leaves the game before it is ended loses 
it. 26. Any person may change his stick 
in play. 27. If any difference arises be- 
tween players, he who marks the game, or 
the majority of the company must decide 
it. 28. Those who do not play must stand 
from the table and make room for the 
players. 29. If any person lays any wager 
and does not play, he shall not give advice 
to the players upon the game. 
Different kinds of games played at bil- 
liards. — Besides the common winning game, 
which is twelve up, there are several other 
kinds of games, viz. the losing game, the 
winning and losing, choice of balls, bricole, 
carambole, Russian carambole, the bar- 
hole, the one-hole, the four-game, and ha- 
zards : but on these it is not necessary to 
enlarge. 
BINARY arithmetic, that wherein unity, 
or 1 and 0, are only used. This was the inven- 
tion of Mr. Leibnitz, who shews it to be 
very expeditious in discovering the proper- 
ties of numbers, and in constructing tables ; 
and Mr. Dangecourt, in the “ History of the 
Royal Academy of Sciences,” gives a speci- 
men of it concerning arithmetical progres- 
sionals ; where he shews that, because in 
binary arithmetic only two characters are 
used, therefore the laws of progression may 
be more easily discovered by it than by 
common arithmetic. All the characters 
used in binary arithmetic are 0 and 1, and 
the cypher multiples every thing by 2, as in 
the common arithmetic by 10. Thus, 1 is 
one; 10, two; 11, three; 100, four; 101, 
five; 110, six; 111, seven; 1000, eight; 
1001, nine ; 1010, ten ; which is built on 
the same principles with common arith- 
metic. The author, however, does not re- 
commend this method for common use, be- 
cause of the great number of figures requir- 
ed to express a number; and adds, that if 
the common progression were from 12 to 12, 
or from 16 to 16, it would be still more ex- 
peditious. 
BIND-iemZ. See Convolvulus. 
BINOMIAL, in algebra, a root consist- 
ing of two members connected by the sign 
-(-or — . Thus, b and 8 — 3 are bi- 
nomials, consisting of the sums and differ- 
ences of these quantities. 
The powers of any binomial are found by 
a.continual multiplication of it by itself. For 
example, the cube or third power of a -(- b, 
BIN 
will be found by multiplication to be a 3 + 3 
a 2 6 + 3aft 2 + ft 3 ; and if the powers of a — b 
are required they will be f uund the same as 
the preceding, only the terms in which the 
exponent of b is an odd number will be 
found negative. Thus, the cube of a — b 
will be found to be a 3 — 3 a 2 b -(- 3 a b 2 — ft 3 , 
where the second and fourth terms are ne- 
gative, the exponent of b being an odd num- 
ber in these terms. In general, the terms 
of any power of a — b are positive and ne- 
gative by turns. 
It is to be observed that in the first term 
of any power of u-j-o, the quantity a has 
the exponent of the power required, that in 
the following terms the exponents of a de- 
crease gradually by the same differences, 
viz. unit, and that in the last terms it is 
never found. The powers of b are in the 
contrary order ; it is never found in the first 
term, but its exponent in the second term is 
unit; in the third term -its exponent is 2 ; 
and thus its exponent increases till in the 
last term it becomes equal to the exponent 
of the power required. 
As the exponents of a thus decrease, and 
at the same time those of ft increase, the 
sum of their exponents is always the same, 
and is equal to the exponent of the power 
required. Thus, in the sixth power of a -(- ft, 
viz. a 6 + 6<t 3 ft + 15tt 4 ft 2 + 20a 3 ft 3 +15a 2 
ft 4 + 6 a ft' + ft 6 , the exponents of a decrease 
in this order 6, 5, 4, 3, 2, 1, 0 ; and those of 
6 increase in the contrary order 0,1,2, 3,4, 
5, 6, And the sum of their exponents in any 
term is always 6. 
In general, therefore, if a + ft is to be 
raised to any power m, the terms without 
their coefficients will be a'", a m ~' ft, o’" — 1 ft 2 , 
a"‘ — 3 ft 3 , a m — 4 ft 4 , a m ~ 6 ft 3 , &c. continued till 
the exponent of ft become equal to m. 
The coefficients of the respective terms 
will be 1 ; m ; m X — - — ; m,X — - — X 
; m x 
m — 1 
2 
X 
m — 1 m - 
X — — X 
X 
-2 m — 3 
- x ~ 
, &c. 
2 3 " 4 
continued until you have one coefficient 
more than there are units in m. 
It follows therefore by these rules, that 
a + ft ' 4 
' + m a m — 1 ft + m X 
m — 1 
X 
2 ft 2 + m x 
ft 3 + m X 
m — 1 
X 
2 3 4 
a— 4 ft 4 +, &e, which is the binomial or 
- - 
