BIN 



BIN 



Different kintls of games played at bil- 

 liards. Besides the common winning- 

 game, which is twelve up, there are seve- 

 ral other kinds of game, viz. the losing 

 game, the winning and losing, choice of 

 balls, bricole, carambole, Russian caram- 

 bole, the barhole, the one-hole, the four- 

 game, and hazards: but on these it is not 

 necessary to enlarge. 



BINARY arithmetic, that wherein unity, 

 or 1 and 0, are only used. This was the 

 invention of Mr. Liebnitz, who shows it 

 to be very expeditious in discovering the 

 properties of numbers, and in construct- 

 ing tables; and Mr. Uangecourt, in the 

 " History of the Royal Academy of Scien- 

 ces," gives a specimen of it concerning 

 arithmetical professionals ; where he 

 shews that, because in binary arithmetic 

 only two characters are used, therefore 

 the laws of progression may be more ea- 

 sily discovered by it than by common 

 arithmetic. All the characters used in 

 binary arithmetic are Oand 1, and the cy- 

 pher multiplies 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 

 recommend this method for common use, 

 because of the great number of figures 

 required 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 expeditious. 



BINDwm/. See CONVOLVULUS. 



BINOMIAL, in algebra, a root consist- 

 ing of two members, connected by the 

 sign + or . Thus a -f- b and 8 3 are 

 binomials; consisting of the sums and 

 differences of these quantities. 



The powers of any binomial are found 

 by a continual multiplication of it by it- 

 self. For example the cube or third pow- 

 er of a-r-6, will be found by multiplica- 

 tion to be 3-t-3 a*b+3ab 1 -{-b3 ; and if 

 the powers of a b are required, they will 

 be found the same as the preceding, only 

 the terms in \vhichtheexponentof6 is an 

 odd number willbe found negative. Thus 

 the cube of a b will be found to be a3 

 Sa^+Sa^ 63, where the second and 

 fourth terms are negative, the exponent 

 of b being an odd number in these terms. 

 In general the terms of any power of a b 

 are positive and negative by turns. 



It is tobe obsei^vedthatin the first term 

 of any power of a+6, the quantity a has 

 tlic exponent of the power required, that 

 in the following terms the exponents of a 



decrease gradually by 'the same differen- 

 ces, -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 se- 

 cond term is unit ; in the third term its 

 exponent is 2 ; and thus its exponent in- 

 creases, till in the last term it becomes 

 equal to the exponent of the power re- 

 quired. 



As the exponents of a thus decrease, 

 and at the same time those of b 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+6, viz. a 6 -{-6a5 6+15 a* 6H-20 a3 6> 

 H-15 a- 64 -j- 6 a 6 6 + 65, the exponents of 

 a decrease in this order 6, 5, 4, 3, 2, 1, 0; 

 and those of 6 increase in the contrary or- 

 der 0, 1, 2, 3, 4, 5, 6. And the sum of 

 their exponents in any term is always 6. 



In general, therefore, if a-}- 6 is to be 

 raised to any power m, the terms without 

 their coefficients willbe a^a" 1 '6, a m 

 6 a , urn 363, am 46-)-, a m $65, &c. conti- 

 nued till the exponent of 6 become equal 

 to m. 



The coefficients of the respective terms 



will be 1 ; m ; m X -^ m X 



m 2 



m 1 m 

 m X X "3 



2 

 m 3 



m 1 

 X-^ X 



4 



2 m 3 in 4 

 X r X 



&c. 



2 3 4 ,~ 5 



continued until you have one coefficient 

 more than there are units in m. 



It follows therefore by these rules, that 



TO 1 , 



a +6w=onJ + m am 6 + w + o f- 



a m 



m 1 m 2 



- X TT- X 



ml 



m 3 



W + mX -X ^ ^ 



a m-464 +, &c. which is the binomial or 

 general theorem for raising a quantity 

 consisting of two terms to any power m. 



The same general theorem will also 

 serve for the evolution of binomials, be- 

 cause to extract any root of a given quan- 

 tity 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 + 6, is to raise a + 6 

 to a power whose exponent is . Now 

 a + 6'i m being found as above ; suppos- 

 ing m = |, you will find a+6 = a + 



