B I N 



head lafge afcl fufous ; with two tubercles on the pe- 

 tiole. 



BINOMINAL, or BiNOMiAi,, from Ms, twice, and 

 nomen, namf, in Algtbra., a quantity confiding of two terms 

 or members, connected by the fign •\- plus, or — m'lntis. 



Tlius a-{ b and 5 — 3 are binomials, conilfting of the fum 

 or difTerence of thofe quantities ; though the latter is often 

 called refulual, and by Euclid, apolomc. 



The terms binomial and refidual are faid to have been firil 

 introduced by Robert Recorde. See Algebra. 



The powers of a binomial are found by a continual multi- 

 plication of it by itfclf, as often as an unit is contained in the 

 index of the power required. Thofe of a refidual, a — ^, are 

 obtained in the fame manner, only with this diiTerence in the 

 refult, that the terms in which the exponent of i is an odd 

 number, will be negative. 



If a root have three parts, as a-\-h-\-c, it is called a trino- 

 mial ; if more, a multinomial. 



Binomial, impojillc, or imaginary, in Algchra, is ufed 

 for a binomial, one of the terms of which is an impoffible 

 or imaginary quantity : as a4: >/ —lb is an impolTible bi- 

 nomial. 



Dr. Mafl<elyne, the aftronomer ra)'al, has given (in his 

 Introduction to "Taylor's 7\bles of Logarithms," p. 56.) 

 the fallowing method of finding any power of an impoffible 

 hiiio.-nial, by another fimilar binomial. The logarithms of a 

 and b being given, it is required to find the power of the 



impoffihk binomial n + ,/ — b' whofe index is — , that is 

 to find (a ± ^/^^y' by another impoffible binomial ; and 



the^ice the value of (,7 -f ^ — b) " + {" — \/ — l'')' ' 

 w'.:ich is always poffible, whether a or i be the greater of 

 ihe two. 



b 

 Sjiuti'jn. Put — = tana:, s. Then 



(i7+-v/ — b'-J " ( = a- + l>')"' X (cof. — X. ± v/ — iin.-™- »)• 

 Hence (a+y'-b'] " + {a - ^—b ) ~ = (a' -j- 3=) " x 2 



cof. — 2. = axfec. z," X 2 cofm. — ^ = (^Xcofec.^.) -■ 

 n « 



X 2 cofin. - 2, where the firft or fecond of thefe two lad 



« 

 expreffions is to be ufed, according as z is an extreme or 



mean arc ; or rather, becaufe— is not only the tangent of z 



a 



but alfo of z + 360°, z + 720°, &c. ; therefore the faftor 



in the anfwer will have feveral values, viz. 



2 cof. — z ; 2 cof.— (2-1-360^) ; 2 cof. - (z -\- 720'') ; 



n n » 



&c. ; the number of which, if ;« and n be whole numbers, 



and thefraftion — be in its leall terms, will be equal to the 

 n 



denominator n ; otherwife infinite. 



By Logarithms. Put log. ^-f-10 — log. n = log. tan. z. 



Then log. («+ V'~=T)"'' + {a - ^' ^T) " ) = ^ 



X (1. a + IO — 1. cof. z) -f 1. 2 +1. cof.— Z— 10 = — 



.X (1. h + 10 — 1. fin. z) + 1. 2 -f 1. cof.— z — 10 ; where 



tlie firfl or ftcond cxprcffion is to be ufed, according as z is 

 Vol. IV. 



B I N 



an extreme or mean arc. Moreover, by taking facceflively, 



I. cof. -z ; 1. cof. - (z+36o'')i 1. cof. - (s+720°), &c. 

 V. n n 



there will arife feveral diflinft anfwers to thequeftion, agree« 



ably to the remark above. 



Binomial furd, is ufed for a binomial, the terms of 



w^hich are fnrds ; as ,^a-^,Jh, or a -\-b\ if m and n be 



fractions. The term binomial furd is alfo applied to any 



quantity having a rational part and a furd part, as 25-!-^ 



968. Euclid enumerates fix kinds of binomial lines or furds in 



the loth book of his " Elements," which are exaftly fimilar 



to the fix refiduals or apotomes, of which he has alfo tr.'ated 



in the fame place. See Apotome. Thcfe apotomes become 



binomials by merely changing the fign of the latter term from 



viinits to phis, and they are as follow : lil. 3+v''5 ; 2d. 



6th. v'6 + v/a- 



For the extraftion of roots of binomial furds, fee New- 

 ton's Arithmctica Univerfalis ; St. Gravefande's Commen- 

 tary ; and Mac Laurin's Algebra, p. 114 — 130. SeeSfRD. 



Binomial Curie, is ufed for a c>;rve, the ordinate of 

 which is expreffed by a binomial. Thus, if the ordinate of 



X 



a curve be of this form x-{-e-\-fs'^, the curve 13 called a 

 binomial curve. Stirling. Method. Di(T. p. 58. 



Binomial Theorem, is a general algebraical exprtfnon,or 

 formula, by which any power or root of a quantity, confiil- 

 ing of two terms, is expanded into a feries. 



It is alfo frequently called the Newtonian theorem, or 

 Newton's binomial theorem, on account of his being com- 

 monly confidered as the inventor of it, as he undoubtedly 

 was, at leall in the cafe of fractional indices, which includes 

 all the other particular cafes of powers, divifions, &c. 



This celebrated theorem, as propofed in its '.noil general 

 form, may be exhibited in a manner nearly fimilar to that of 

 Newton, as. follows : 



X: 



I + — 



m — 2(x\] 

 3" ^" 



+i7:;)+:.'iz.''(i)- + 



n \ a/ n zn ^ a 

 -I-&C. Or, ' " 



a -\- x\" = a 



» ^ a' 5» \ a 



'-3" 





&c. 



Where a, x, are the two terms of the binomial, — the in- 



n 



dex, and A,B,C,D,&c. each preceding term, including their 

 figns + or — , the terms of the feries being all pofitive 

 when X is pofitive, and alternately poGtive and negative 

 when X is negative, independently however of the effect of 

 the coefEcients made up of m and n, which may be any num- 

 bers whatever, pofitive or negative. 



A few eafy examples, in the extraction of roots, will be 

 fufHcient to ftiew the application of the theorem in all fimilar 

 cafes. For this purpofe, let it be required to find the 



fquare root of a + b, or a + ii\'' and the cube root of a — b, 



in the firft of which — = — and in the fe- 

 n 2 



or a — b I 



cond — = — . 



Then7T^^ = .^[,+i(i)-^^(-^)- 

 \a J 2.4.6.S \a J ' 



&c. 

 3C 



.+.6 



And 



