B I N 



411 



B I N 



Limuli form a marine genus, making a natural group of 

 different form and habits; Linnzeus's genus, Monoculus, 

 comprehends Apus, Limulus, and other crustaceans. Dr. 

 Leach has formed a genus (Lepidurus) of those species 

 which have a plate between the bristles of the tail, but, as 

 Cuvier thinks, unnecessarily. The species figured is Apus 

 productut, Latr. (Lepidurus productus, Leach ; Monoculus 

 Apus, Linn.) The genus occurs in England, France, and 

 Europe generally. 



[Apus produetua.] 



BINOMIAL, in algebra, means an expression which 

 contains two terms, such as 



a + b b ex a? x py 



Any expression may be considered and used as a binomial 

 in any sense in which it may be said to contain two terms : 

 thus, 



a + b + c x ex 



when put in the form 



(a + 6) + (e - e) x 



is a binomial, the terms of which are a + b and (c e) x. 



BINOMIAL THEOREM, by far the most important 

 theorem in common algebra, first announced by Newton, as 

 will presently appear. It is frequently called on the Conti- 

 nent the binonie de Newton, and is engraved on his tomb 

 in Westminster Abbey. In explaining this theorem, we 

 shall consider ourselves as writing for those who have already 

 such a knowledge of algebra as will enable them easily 

 to recognise the various expressions of which we make 

 use. 



The binomial theorem, coupled with those preceding 

 theorems from which it springs, is as follows : 



( 1 .) If a be denoted by a 1 , a a by c?, a a a by a 8 , &c., then 



g,n 



a* x a" = a""*" 1 = a 1 * (m > n). 

 a" 



(2.) The equations in (1 .) will hold good when the symbol 

 a" is considered, provided that a always signifies unity. 



(3.) The equations in (1.) will hold good when negative 

 exponents are employed, provided that 



' means - 

 a 



i 

 a means - , &c. 





(4.) The equations in (1.) will hold good when fractional 

 exponents are employed, provided that 



a' means the square root of a 

 a* cube root of a 

 a* fourth root of a &c. 

 and akso that 



lil:ins i 



the cube root of a' 



(fi (a") the seventh root of a* 

 ** i 

 a" ,, (a"T the nth root of a" 



(5.) Binomial Theorem. In all the preceding cases, that 

 is, whether n be whole or fractional, positive or negative, 



&c., 



2 3 4 



the preceding being a series of an infinite number of terms 

 in all cases, except only where n is a positive whole num- 

 ber. The pih term of the preceding expression is 



n-ln-2 n-p + 2 p _, 



~2 3~ p-l 



which expresses any term after the second. 



(6.) The preceding series is convergent, whatever may 

 be the value of n, whenever x is less than 1 . If a; be greater 

 than 1, it is always divergent ; but the series remaining after 

 any term may be expressed in a finite form, as follows : 

 Let V,, V a , V 3 , &c. represent the several terms of the pre- 

 ceding series, then all the terms after the jBth term arc an 

 algebraical development of a term of the form 



v [j 4- 1 \ i "T" y x) t 



where 9 is a function of a?, the arithmetical value of which 

 is less than unity ; so that 



a 4. ~,\n V -4- V -I- -t-V-t-Vi fl-J-0 i r\ n ~P 



T vt/^ V 1 ~ V g T^ ( I t T j) ~ V j)_^ 1 \ L I VtlltJ 



The preceding theorem, though theoretically necessary to 

 those who do not allow the use of divergent series, is of no 

 practical use in the determination of (1 + so)', since the de- 

 termination of 6 itself is the more difficult problem of the 

 two. 



We shall now give the early history of this theorem, with 

 some remarks upon its demonstration. 



Before the time of Vieta, no materials for its expression 

 were in the hands of algebraists. That writer first used 

 general symbols of determinate number : and in his works 

 \ve find the first rude cases of the binomial theorem, though 

 only in the results of simple multiplications, and without 

 the discovery of any law of connexion among the coefficients. 

 For instance, in his Ad logislicen speciosum note priores, 

 we find the following : 



' Sit latus unum A, alterum B. Dico A quad.-quadratum + 

 A cubo in B quater, +A quadrato in B quadratum sexies,+ 

 A in B cubum quater, + B quad.-quadrato, sequari A + B 

 quad-quadrato.' This we should now express thus : 

 (a +6/ = a' + 4 a 3 6 + 6a'2> 2 + 4afr' + b\ 



The coefficients of the binomial theorem, in the case of a 

 whole exponent, had long been derived from the method 

 employed in what Pascal called the Arithmetical Triangle, 

 and Briggs the abacus 7rayx(>rj<m>c . To trace the history of 

 this method would here lead us too far [see FIGURATK 

 NUMBERS] ; it must suffice to say that Lucas do Borgo, 

 Stifel, Stevinus, Vieta, and others, all had in their possession 

 something from which, if we did not know that such simple 

 relations were difficult to discover, we should say a little 

 attention would have enabled them to find the first glimpse 

 of the binomial theorem, which, as we shall proceed to state, 

 occurred to Briggs. 



The abacus of the last-mentioned writer above alluded to 

 is as follows (we have only reversed right and left) : 

 1 1 1 1 1 &c. 



In which each number is formed by adding that on the left 

 to that immediately above. On which (TVvMMflMfn'a 

 Britannica, 1633, preface, p. 22) Briggs remarks, that by 

 ascending obliquely, the coefficients of the several powers 

 are obtained ; for instance, that 4, 6, 4 are the coefficients 

 of the fourth power, 5, 10, 10, 5 of the fifth power, and so 

 on Brig^s therefore knew the dependence of these coem 

 cients on the preceding columns of figurate numbers, but 

 not the algebraical expression for the nth of each class. 



The next step was made by Wallis, in his Anthmettca 

 Inftnitorum, published in 1655. One of the great objects 



