Ul 



WMiMIAL THKoKKM 



I'.ISV.MIAI. THKDUKM. 



It* 



property ot all the fundamental thoraiu of every branch of mntlio- 

 nutic* to be incapable of establishment iu manner in which all shall 

 agree, though the theorem, themselves are held indisputable. Among 

 thaw demonstration* are those of James Bernoulli, MacUurin, Landen, 

 Kpinus, Stewart, Euler, Lagrange, L'Huilier, Manning, Woodhouse, 

 Mutton, Boonycactle, Knight, Kobertson, Crenrell, Swinburne, and 

 Tylacota. We ahall not ducusa the various objections, because they 

 apply a* much to the general doctrine of infinite series as to the 

 binomial theorem in particular; and we must refer the reader t<> 

 TAYLOR'S THEOREM. We ahall however allude to the principal objec- 

 tions after we have given what appeal* to iu a sufficient proof of tin- 

 theorem, or rather after we have indicated the steps of such n proof. 

 



/Arfm'/io. By (1 +x)* we mean (w and being whole numbers) a 

 quantity which, multiplied 1 times by itself, gives (1 + f) m ; and by 







the exiKUision of (1 +je)*, we mean an algebraical series of powers of .r 

 (punitive or negative, whole or fractional) which h:is all the alp 



i . _ 



properties of (1 + r) m , and which, when it is convergent, has (1 +*)" 

 for its arithmetical limit or sum. 



Tkeorrm 1. The well-known proof of the expansion of (l+r) m r , 

 where m u a positive whole number, giving 



This theorem is not absolutely necessary, as we shall see. 



Theorem 2. If there be any function of a, namely <f> a, which 

 satisfies the condition 



* (a) X* ()>(+{), 



ihen if (a) must be , where c is any quantity independent of a. 



K..I the ron.lition gives 



<t> (a)x<t> (6 + r) = <f> (a + 5 + r) 

 or <f> (<t) x<t> (4)x if> (r) 



which leads (supposing b, e, tic., to be severally equal to n) to the 

 equations 



(<t> a)' =f> (no) ($> ')" = *("''), *c- 



where R and m are any whole numbers, and n, a', Ac., any quantities 

 whatsoever. Let us suppose mo'= , which gives 



f (W)=* () <* (<t> ') = (* )' 



or* (-!! 



Again, the supposed universality of the first equation gives 



$ (d)x <f> (n\ = <t> (a + 0) = $ (a) 

 r $ <0) = 1 : and also 



* (a) x0 (-) = * (n-a) = l 



whence *( n) = L d ( ia)= 

 <t>a <t>(ia) 



= fT). = < * a) ~" 



so that the equation * (n a) = (4> a)* is true for all values of n; if a 

 lie = 1, this gives *={*(!)}, and * (1) is not a function of the 

 general symbol n : let * (1 ) = c, which gives the theorem asserted. 



This theorem is the fundamental part of Killer's proof of the 

 binomial theorem. 



Thenrrm 8. If the values of a and 6 may be made as near to 

 equality as we please, then the limit of the fraction 



a* If . 



_ is no*-* 



a b 



In the case where H is a whole number, this is evident by the well- 

 known theorem 



(a*-P) -=- (a-6)= 

 (a-i<) -i- (a-ft) = a' + a*6 + a6' + i Ac. 



I*t * be a positive fraction, for instance, ; and let a = o 3 , h = ff. 

 Then a* = a*, l* = (f and 



6 



ef ff 



tlie limit of which, when a approaches to 6, is 2 a-f-3 a 1 or ? a ~ ' 

 i.rfa~*or-a*~. In the same way any other case may bo proved. 



n O 



Now let M be negative, ay it is t, where I is positive. Then 



o -* _ a-' b-' 1 of -V 



of b' a-b 



(t being positive), in 



a-6 a-b 



f.f which the limit, '>r the two preceding 



-a-* x /a'-' or fa-*- 1 ora"-'. 



Tkevrtm 4. If (1 +..> atlmit of lieiug exjianded inaserieaof wli.lf 

 powers of x, then th.it series must be 



I,' 



(!!.> = 



2 3 

 ',:r> + Ac. 



x-y 



which two sides being always equal, the limits to which they a|.|.i,. . li . 

 as x approaches to y, are equal ; or 



Multiply both sides by 1 +;r, which gives 



lint by the original assumption 



(l+x)' = N <+/,* -mf, a* + Ac. and 

 therefore , = < 





2 S 



Ac. Ao. 



But, making x in the original series, we find '=(!) =1. Whence 

 follows the theorem. 



Tluartm 5. The value of (1 +*) is in all cases the series above 

 investigated. 



Consider that series OK a function of . Or let 



i 



() = 1 + H .< + - :**+ Ac. 



2 



~2~~ 

 Actual multiplication will be found to give 



+ 1 . 



or if n x fn = <f> (n +m). 



Or we may dispense with this multiplication by remembering that 

 since $n in (1+jc)* and $m is (1 +.')", when u and i are whole 

 numbers, we must have, ! that cage (Theorem 1.), 



but the result of a multiplication does not depend upon the values of 

 the letters; if therefore <f>m and <pn give $>(m + ?0 when HI and n are 

 any whole numbers, they give the same result when m and n are frac- 

 tional or negative. But we do not yet know that <f> i in the latter 

 cases represents (1 + .T)". But by theorem (2.) it follows H..IH 

 <t>,n x<t>n = it>(m-t-ti) that 4>n is {*(!)}", or 



(, Ac. 



The greater part of the preceding proof is a concession to the 

 analytical taste of the age, which requires that synthetical denurn- 

 stration shall not appear in algebra. The theorem is demonstrated 

 rigorously so soon as it shall be proved that from *mx 0w = <f>(m -m). 

 it necessarily follows that <p m \R {$(!)}, and that the series above- 

 mentioned satisfies the equation just named. And in reading the 

 objections which have been made against the various proofs of the 

 binomial theorem, the student must bear in mind that there is one 

 class of objections against the the actual logic of the processes, and 

 another arising out of the conventions already alluded to. Against 

 the demonstration of Kuler, which consists in theorems 2. and 6. of 

 the preceding, one says that it is " tentative " (ti/nthttirai would have 

 been the proper word) ; another that it is not "algebraical," meaning 

 analytical, and assuming that algebra must be analysis. To all of 

 which we should reply by another question Is it logical deduction 

 from self -evident premises I 



The last attempt to produce an unanswerable demonstration of i !,. 

 binomial theorem was made by Messrs. Swinburne and Tylecote of 

 St. John's College, Cambridge (Deighton, 1827). The details are much 

 too complicated to describe, but the general result is the expansion 

 of (!+./)" to any number of terms, with a finite expression for Die 

 remainder. This expression is however so complicated and long, that 

 it can be of no use, except as proved that the remainder can be assigned 

 by the ordinary operations of algebra. The proof is certainly, if the 

 details be correct, of a logical character, but it is far above the student. 

 The remarks on other demonstrations in the preface, though dissenting 

 entirely from many of them, we should recommend to the attention of 

 the advanced student, as an exercise in the roniilerati"n of objections. 

 At the same time we may recommcn.l the leinarkn in \V. ...II, .,,.,- 

 An ilytical Calculations.' 



