B I N 



Bi jr 



clples, ill a more natural and oHviois way ; one of tlie firft 

 of thefe attempts being that of Landen, in his " Difcourfe 

 concerni-.ig tiie rcfidual analyfis," and the next that of E- 

 pinus, in tlie eighth voUimeofthe " New Ptterfbiirg Me- 

 moirs." But the legitimacy of the former may be objefted 

 to, as depending upon vanithing fracti(;ns, and other conC- 

 derations of too difficult and abltrail a nature to be regarded 

 as fuiiiciently conviiiciiig ; and the latter, tiioiigh very inge- 

 nious, is not Ids difficult and embarraffing ; at leaft, fuch is 

 the opinion of Eiiler, who having himlclf lirft given a de- 

 monftration of this theorem, in which, like Maclaurin, he 

 employed the dilTerential calculus, or metiiod of fluxions, 

 was afterwards led to deduce it frovi the principles of alge- 

 bra alone ; though he docs not appear to have been much 

 more fiiccefsful tlian either of the former. 



S. Lhuilier of Geneva, perceiving the defe£ls and obfcurity 

 of thefe methods, has made a new demonftration of this for- 

 mula in one of the preliminary articles of his excellent work, 

 entitled, " Principioruni calculi differentialis ct integralis, 

 &c." which is purely elementary ; and abating from its 

 length, and a fatiguir.g detail of particulars, which the na- 

 ture of the fubjcct does not fecm to req'.iire, he appears to 

 have accomplifhed his object ; at leaft as far as the method 

 he adopted would allow ; for it mud be confelTed, that nei- 

 ther this, nor any other inveftigation, that had hitherto ap- 

 peared, have been attended with the fimpllcity and ilridlnefs 

 which could be dcfired. 



The reafon of this, as Dr. Woodhoufe properly obferves, 

 in his " Principles of Analytical Calculation," feems to be, 

 that moft mathematicians appear to have fought forfome high 

 origin of this theorem, diilinct' from the fimple operations of 

 multiplication, divifion, extracting of roots, &c. : and in- 

 ftead of confidering the nature of the operations it was 

 known to comprehend, hoped to fuperfede them by deduc- 

 tions drawn from abilrufe and fine theories ; whereas, it is 

 clear, that whatever imperfedions thefe fundamental opera- 

 tions are attended with, are alfo attached to the binomial 

 theorem, which, in a certain fenfe, may be faid to be a me- 

 tl'.od of trial and conjedture. For, as this formula is only 

 meant to exprefs, in general terms, the algebraical rules 

 above mentioned, it cannot pofiefs a greater degree of cer- 

 tainty than is volTefled by the fimple operations them- 

 L-lves. 



To avoid entering into a too prolix inveftigation of the 

 well known and fimple elements upon which the general for- 

 mula depends, it is iiiirlclent to obfervc that it is clearly ma- 

 nifell f.om fome of t'le firil and moft com.mon rules of alge- 

 bra, that whatever is the operation which the index (in) in 

 fl-t-.vi" diredls to be performed upon the binomial a-\-x, 

 whether of continued multiplication, or elevation, or of di- 

 vifion, or of extraftion of roots, the terms of the refulting 

 feries will necefiarily arife by regular and wliole pofitive pow- 

 ers of x; and that the two firil terms of this feries will al- 

 ways be a"' -}- ma'"' ' x-, fo that the entire expanfion of it 

 may be reprefented under the form a''' + ma''~ '.v-j-/'.v-i-y.\-^ 

 +rx\ &c. 



For, omitting the practical part of the procefs, which is 

 taught by the above mentioned rules, it will conllantly be 

 found, by performing the operations at length in the ufual 

 wiy, that 



fi±x ' z= a" ± lux -1- «' 

 a±xY =; rt' + ^a'x -\- ^a'x' +.r' 

 <3 + .v ■• = a-';i; ^'x -j- Ca'x' ± 4(7.v' -f .-c* 



<J±.v 



Sec. &c. 



a~-\ + a^^x ' ^ «"'' .V , S:c. 

 6 



J "-(-2 a ^a: -1-30 "f x" -J- 4a ^ X ' , &.C. 



■ . , = a~5 -J- ■, „— 4 X -4- 6a~~-x^ ± ioa~^x , Us. 



&C. &C. 



a±x 



a±x) — at J; — a -x — — <3 i .V 3: — (J ^ -x' &C. 



2 



I _ 2 



a'±^' z=a\±-a 



3 9 



fl±.v =33-1 a X a 7x + — a 



-3 9 Si 



&c. &c. 



81 



2 _i 



a±.- 



-' _1_ — •» I 7:1 



3 



■^±^a~ix+Sa- 

 3 9 



81 



20 



&c. 



-'f-v'&c. 



■f .v-:F^«-V.v',&c. 



Sec. Sec. 



In all the inftances here given, it is apparent, that the 

 firll term of the feries, in each of them, is the fame as the 

 power or root of the firft term of the binomial quantity to 

 which it belongs ; and that the coefficient of x in the fccond 

 term is always had by multiplying the index of the firft term 

 into that term, having its index diminilhed by 1 ; and as 

 thefe cafes are of the fame kind with thofe that are defigned 

 to be exprefled, in univerfal terms, by the general formula, 

 it is in vain, as far as regards the two firft terms of the expan- 

 fion, to look for any other origin of them, than what may 

 be derived from thefe an d fim ilar operations. 



Affuming, therefore, a-f.v^'" = a" + ma"" ^' +/*'+?•■<' 

 -{-r.vS Sec. it only remains to determine the value of the co- 

 efficients p, q, r, &c. and to ftiew the law of their dependence 

 on the index (m) of the operation by which they are pro- 

 duced. 



For this purpofe, let m denote any number whatever, in- 

 teg;ral or fratlional, poiitive, or negative ; and let the co- 

 efficients of the 3d, 4th, 5th, Sec. terms of the mth power 

 of any binomial be denoted by/', q', r', &c. 



Then for x, in the above form, put y-\-z, and there will 



arife iz-j-j' 4-^1'" =<>-}-)• + c)°' = a-j-y + ~^'"; which are all 

 identical expreffions ; and when expanded accordmg to the 

 proper forms, mull be equal to each other. 



But « +>■ +2* "■ = 'J" + »'«" ~ '( y + z) +/ 0''-l-z;-.&c.) 

 -\- q{y^-^^y'z, 8iC.) SiC. (omitting to fct down the higher 

 powers of s, which are not wanted in the demonftration) = 

 a'" + ma-^- ' y+py'+gfSiC. +ma''~ ' z + 2J>y^+Sqy--z, &c. 



And a+y+z'^'" — a-\-y]"' +ni.a+yY'~ 'z. Sec. = J+^'' 

 + mz (a™- '-f OT- i.«'"- 'y+py'+q'y'ySec.) =a'"+rr,a" -'y 

 -^py^ + qy\ Sec. -\- ma"- 'z + ni.m- l.a^ "'yz +mp'y'z + 

 mq'y^z. Sec. Hence the two feries being identical, a" -{- 

 ma" ~'y+py''+qy' Sec. + ma"' ~'z + 2pyz-\-Jjy'z, Sec.=a"' + 

 ma" '"y-^■p^■'+qy' &c. +mj" ~'z +m.rr. - 1 ..'7'" ~'yz -\-mp'y' 

 j; -\-mq'yH', Sec. or, leaving out the terms common to each, 

 ^px'z-^-^qy'z Sec. ^= in.m— i.a"' ~'y'z,-\-pip'y'-z. Sec. 



And fincethe coefficients of the terms involving the fame 

 powers of the arbitrary quantities _j' and s muft be the fame. 



ftiall have 2p = m.m — I .<j" 



tn.m— 1 

 or/= 



and hence />' 



m— l.m — 2 



Alfo 3y = lap' =: 



