INTERPOLATION. 



that is not abroUitely neceflary to the eonftru^lion, varioos 

 readings are found that differ materialt)^ from each other, we 

 have reafon to fufpeft its authenticity j and that all the read- 

 ings are interpolations of tranfcribers, who have attempted 

 by ilifLrent methods to fupply the feeming deficiency of the 

 original. An interpolation is fometimes betrayed by the 

 cireiimftance of its being delivered in the language of a later 

 church. To this purpofe Michaelis obferves, that in the 

 time of the apoftles the word " Chriit"' was never ufed as 

 the proper name of a perfon, but as an epithet expreffive of 

 the miniftry of .lefus, and v.as frequently apphed as fyno- 

 nymous to " Son of God." The expreffion, therefore, 

 " Chria is the Soft of God," (Afts, viii. 37 ) is a kind 

 of tautology, and is almoit: as abfurd as to fay Chriil is the 

 Me.Tiah, that is, the Anointed is the Anointed. But the 

 -word being ufed in later ages as a proper name, this impro- 

 priety was not perceived by the perfon who obtruded the 

 paflage on the text. If one or more words, that may be 

 confidered as an addition to a paffage, are found only in 

 MSS., but in none of the mod ancient verlions, nor in the 

 quotations of the early fathers, we have reafon to fufpeft an 

 interpolation. Ads, viii. 39. ■znvfix [k-^ioV srsn-ES-iv ".ti -rov ivya- 



■ X«'> aw^^o: ^0 K^fls rps-oo-s tot ii>.iT.in-j, is an inftance of this 



■ kind, where the words between the crotchets are probably 

 fpurious. Interpolations of confiderable length are occa- 

 fioned fometimes in the following manner: The owner of a 

 MS. makes a note in the margin, either explanatory of fome 

 narrative in the text, or containing an account of fome event 

 that was handed down by tradition ; which MS., being af- 

 terwards trnnfcribed, the copyill writes text and notes with- 



' oi.t diftinftion in thebody of his work. " I am perfuaded," 



fays Michaelis, " that .John, v. 4., a ver^' fufpicious paflage, 



and omitted in a very great number of MSS. has been in- 



' truded in this manner into our prefent test, and that this 



' fcholion v.'as written originally not in Greek, but in fome 



■ oriental language." The difputed paflage in i John, v. ;. 

 may probably be a fpecimcn of this kind of interpolation. 

 Its'fpurioufnefs has been rtiewn by fir Ifaac Newton, in a 

 letter to Le Clerc, firft publifticd in London in 1754, and 

 more correftly by Dr. Horfley in 1785, from the author's 



' origin al copy. ( See his edition of Newton's Works, vol. v. 

 p. 495^531-) This letter, fays Dr. Marfh, is lefs known 



• than it deferves, as the immortal author has difplayed in it 

 Tis much critical knowledge, as penetration in his mathema- 



■ tical inquiries. The queltion has been likewife examined, 



■ •ind with great impartiality, by Bengel, in his " Apparatus 

 •■Criticus," p. 458 — 482. 2d ed. ; and the difpute has been 



• Ifatlsfaftorily terminated by the eminently learned Porfon, in 

 ""^s " Letters to Travis," -publifhed in 1791. See Various 



Rkadixgs. 



'{■sriinFOi-ATioy, in A/g^cira, is ufed for the finding an 

 intCTmediate term of a feries, its place in the feries being 

 given ; and the method of doing this is called the method of 

 ' tnUrpalatlons. 



When the algebraic eqiration of the feries is given, the 



' 'ter^ required, whether it be a primary or intermediate 



term, may be found by the refolution of affefted equations; 



but when this equation is not given, as it often happens, the 



■ value of the term fought mud be exhibited by a converging 

 ' ferie.s. or by the quadrature of curves. 



When the firll, fccond, and other fucceffive differences of 

 the t<n-ms of a feri.-s become at laft equal, the interpolation 

 "of any term of fuch a feries may be found by fir Ifaac New- 

 ton's differential method. 



The method of interpolation was firft mvented by Mr. 



■ "Briggs, Savilian profcffor of Geometry at Oxford, and ap- 



piicd by him to the calculation of logarithms. His prin- 



ciples were followed by Dr. Wallis, *ho made feveral inge- 

 nious applications of this theory ; and by Reginald and 

 Monton, in France. Sir Ifaac Newton, in lemma 5. lib. iii. 

 Phil. Priiicip. Mathem. gave a moit elegant folution of the 

 problem for drawing a curve line through the extremitici 

 of anv number of given ordinates ; and in the fubicquent 

 propofition applied the folution of this problem to that of 

 finding fro.m certain obferved places of a comet, the place of 

 it at any given intermediate time. Dr. Waring, who adds, 

 that a folution ftill more elegant, on fome accounts, has been 

 Gnce difcovered by Menr.<!. Nicholi and Sluling, has alfo re- 

 folved the lame problem, and rendered it more general, with- 

 out having recourfe to finding the fucceflive differences. 



The theory of interpolation is of very extenfive ufe, rot 

 only in pure analyfes and geometry, but in various other 

 fubjefts of mathematical inquiry and computation, and par- 

 ticularly in aftronomy ; we fliall therefore endeavour to ex- 

 plain the principles upon which it is founded, and fliew its 

 application in a few cafes to praftical operations. 



Firfl:, then, let a, b, c, d, e,f, &c. reprefent any feries of 

 fimilar quantities, and let tlie difference between the firft and 

 fecond, the fccond and third, the third and fourrh, &c. 

 terras, be taken ; and thefe feveral remainders will form 

 what is called the firfl: order of difference ; then again, let 

 the differences of thefe differences be taken in the fame way; 

 and the differences of thefe laft again the fame, and fo on, 

 which will give the following refult, obferving, that for the 

 convenience of exhibiting the operation, we have only re- 

 tained the firfl remainders in each fucceflive fubtraflion. 



feries 

 III diff. 

 2d diff. 

 3d diff. 

 4th dift". 

 ,-th diff. 

 6ih diff. 



/> &c. 



-/ 



6/+.' 



a b c d e 



a- b 



a ~ 2 b + c 



a- ib + $€ - d 



a ~ 4b + 6e — ^d+e 



a— ^ b + IOC— led + 



a~6b+iyc — 2cd + 



&c. &c. &c. 

 Now the co-efficients of thefe terras are refpeflively the fame 

 as thofe of the co-efficients of the binomial, and the order 

 of their generation evidently follows the faaie laR-, and 

 therefore we may conclude with equal certainty, that the 

 «th difference of any feries of quantities will be expreffed by 

 the formula 



1.2 1.2-3 



1.2.3.4 



Now it is obvious, that if the given quantities be fuch, 

 that any order of their differences become equal to o, that 

 any one of thofe quantities may be accurately expreffed in 

 fnnclions of the others ; thus, for example, fuppofe the 

 fourth difference to become zero, that is 

 a — 4i-f6^ — 4^-t-« = o 



... — a + 4* -L 4(/— « 



then will c = — • 



6 



and it is obvious that any other of thefe quantities might be 

 expreffed in a fimilar manner ; ai'.d therefore, if all thofe 

 quantities but one be known, that one may be afcertained. 

 Thus, by way of illuftration, fuppofe we had the tlu-ee 

 fquares 10'=; 100, 8' = 64, and 7 ■ = 4, and the fquare 

 of 9 was required ; fince tlie third difl'crences of fquares 



'CO + a. 64 - 49 „ J 

 ^ — -ii= 81, and 



equal 



fliosld have 9' 



