INTERPOLATION. 



?3rthsr tlian to eight decimal place.?. Snppofing, therefore, 

 the cube root of any two confeciitive numbers greater than 

 loao to be known, that is, a^j' — r, and \fx^ =^ x ; the 

 firft difference is afcertained by fimple fubtradlion, and the 



fecond difference is found bv the formula ; and with 



tliefe data, we may calculate eight or ten of the fubfe- 

 quent roots, when a new computation becomes necef- 

 farA". 



This is not, however, the fimplefl. manner of employing 

 the formulx that have been deduced, becaufe it requires the 

 knowledge of two confecutive roots, whereas it- willbe fuffi- 

 cient to huve the roots of numbers equidillant from each 

 other, and by means of which the fird and fecond differ- 

 ences may be readily obtained, and the fucc-ifivc extrac- 

 tions will thus become fo many different checks upon each 

 Other. Suppofe, for example, that we know the cube roots 

 of 2000 and 2008 ; that is, ,/2CCO = a, and ^' 2008 = I; 

 and it w^re required to find the roots of all th? intermediate 

 numbers j calling the lirfl; difference = d, and tiie fecond 



Cube root 6:54 



tS.45J49»QSj 



978802 difference 



62S5 ^ 18.454477787 



97S697 difference 



6286 = 18.43-456484 



978592 diffwence 



difference =: 



then the cube roots will flacd thus : 



9<^ 



4.' 2 coo : 



4,'2GC2 = a -r 2 1/ — ; 

 ^20 3 = iJ -h 3 c/ — f, f 



^'2004 — a -\- \d — d r. 

 ^2005 = a + 5 r/ — IOC 

 V3006 = a ->s- (i d — 15 f 



■'^zco'] =a + 'jd— lie 

 '^-2Co8 -a + Sd— zSc = L 



And Gnce this lafl root ought to be equal to the known 

 root i, we have a fecure check upon all the preceding part 

 of the operation ; or more properly, this condition nviy be 

 employed for finding tlie firfl difference d, and the check 

 be made upon the fubfequent difference ; for fmce a + Sd — 



28 c = i, we have </ = — '■ — -- — ■ — . This operation 



will be better underflood from the following work at 

 length. 



■ Given the cube root of 6280 = 18 449582727, and of 

 6288= 18.45741356, to find all the intermediate roots be- 

 tween thofe numbers. 



By logaritlims we find — =r 000000105 = c, 



9 / 6 2 So 

 I) — a + zS c 

 and hence ^ = .00979222 = rf; lience,agau), 



the following operation. 



Cube root 6280 = 18449582727 



979222 difference 



62Si = iS.450561949 



979117 difference 



6282 = 18.451 541066 



979012 difference 



62^7 = 18.456435076 



9784S7 difference 



6288 = 18.457413563 

 And thus may the extra£tion be carried on at pleafure, by 

 fimple addition and fubtraftion ; the cube roots: of cenain 

 periodical terms being firfl kiiou h : it is unneceffary to ob- 

 fervethat the above fuccefGve differences are found by fub- 

 tractingalA'ays .000CO0105 from tl:e fori going one. 



At prefent v,e have Oiily conHden-d tie rrerc elementary- 

 principles on which the theory of interpolntion is founded ; :t 

 will be proper, however, before we cdhc ude this article, to 

 give fotne ide^ of the general formali and r^fuits obiained 

 from thefe fi.mple and obvious truth", by Newton in his 

 "Method s Difierentialis ;"andby Stirling, in his tr.tt enti- 

 tled " Truflatus de Summatione et Interlopatione Sericrtm 

 Infir.itarum,'' publifhed in 1730, both of which works ore 

 worthy of their celebrated authors, 



Let'us imagine a right Hne given in petition, on vhich let 

 there be erefted any number of ordinal es parallel tp one 

 another, and at equidiltances, which ordinates may be con- 

 ceived to reprefent the terms of a regular feries, continually 

 increafing or dxreafing ; and one and the farr.e curve will pafs 

 through ail their extremities, the equation of which will be 

 determined from tlie given equation to the fei-ies ; that is, 

 from the equation cxpreffmg generally the relation between 

 any two or more fuccefGve ordinates : and if this equation 

 be finite, the equation of the c.irve will alfo be finite, and 

 confequently an abfolute interpolation of the feries may be 

 effeded ; for this is nothing more than afligning to every 

 particular abfcifs, its correfponding ordinate, and this will 

 alivays be the cafe if any order of the differences become 

 equal to zero ; thus, in all kinds of figurate numbers, as 

 trsangulars, /quarts, psnlagonah, &c. the curve paffmg through 

 their fummits will be the common conical parabola. But if, 

 as is frequently the cafe, no order of di.^erences become 

 zero, then the algebraic equation of the curve cannot be 

 found, and confequently no interpolation can then be made, 

 except by means of an infinite feries, the quadrature of fome 

 curve, or fome other approximation. 



Let there be propofed, for example, the feries r, 8, 27, 

 64, &c. the general interpolation of which is required. This 

 feries is no other than that of the natural cubes, and there- 

 fore the methodahat we are about to follow is wholly unne- 

 ce.ffary in this cafe, but the example is chofen in order to 

 illuftrate the general method, which is applicable in ail cafes, 

 even when the terms of the feries appear to fol'ow no cenain 

 law. We firft write the terms of tlie propofed fenes in one 

 line, then the differences of thefe in the next, and then again 

 the differences of thefe, and fo on, till the lafl order of dif- 

 ferences become all equal, or fucli that no fcniible error will 

 nrifc in confiderir.g them as fuch ; then confidering z as any 

 abfcifs, the lirfl terms of thefe fucceffive differences will be 

 t!ie co-efficients of the feries of terms 



Vol. XIX. 



62S3 = 18,452,520078 



978907 difference 



^U 



_0 ^(=^i)(.-2) ^^ 

 • 1.2.3' 



O o which 



