INTERPOLATION. 



tlio fame is obvioiifly true of any terms of wliich the differ- 

 ences vanith. 



But if the differences do not vanilli, then any interme- 

 diate term foinid by this method only approximate towards 

 the true refult, which is however fufficiently correct in a 

 number of cafes ; thus, in finding any lognrithm, of which 

 thofe confecutive to it are given, the above formula may be 

 fnccofsfuUy employed; for though, in facl, tlie differences of 

 logarithm never become zero, yet their fourth differences are 

 fo fmall, that by confidering them as o, the error will not 

 effeA the truth of the refult to 8 or 9 places of decimals. 



Exam. — Given the logarithms of 101, 102, 104, loj, 

 to find the logarithm of 103. 



Here, caUiijg the log. of ici = a, of io2 = b, 103 = c, 

 104 = d, and loj = e; and confidering the fourth diiTcr- 

 enccs of tliefe logs = o ; wc fhall have from the formula 

 a — 4^ + 6^ — 4(/+f = o 

 _ 4 (^ + ^) - (fl + f) 

 6 



'•■ice the foilo-.ving computation : 



log. loi = 2.0043214= a 

 li)g. I02 = 2.00S6C02 —b 

 lo;j. 104 = 2.0170333 = a 

 log. loj = 2.021 1893 =e 



4.0256333 = b -^ d 

 4 



16.1025340 = 4(^ + '') 

 fubtr. 4.0255107 = a -f- e 



6)12.0770233 



log. 103 = 2.0128372 as required. 



Tliis method of finding the intermediate logarithms 

 between others that are known, though of little importance 

 in the prefent ftate of the fciences, was of very effential 

 fervice to the original computers ; and to «'hom the invention 

 of it is due, or more p.roperly to Briggs alone, who feems 

 to have been the firll that entered upon the inveftigation of 

 this theory. 



This doftrine is applied with great fuccefs in various aflro- 

 nomical operations, and is the means of faving in many cafes 

 immenfe laborious calculations. Thus, for example, in 

 finding the places of fome of the planets, whofe motion is 

 not very rapid, it will be fufficiently accurate to find their 

 places by calculation for every fourth or fifth day, and then 

 by means of the method above defcribed, their places for all 

 the intermediate days may be found by interpolating between 

 the known terms, which method will give a refult much 

 nearer the truth, than by proportional parts, becaufe this fup- 

 pofes a uniformity both in motion and time, which is not 

 corretf. 



Again, in computing the moon'.'! place for any particular 

 hour, fuppofmg its place for every day at noon to be given, 

 the method of interpolations may be applied with great fuc- 

 cefs, the refults having fcarcely any fenfible difference from 

 thofe that arife from aftual computation, and we may thus 

 frequently avoid one of the moft laborious of ailronomical 

 calculations. 



By this means alfo, the place of a comet at any particular 

 time may be afccrtaincd, from obfervatioiir, made on it prior 

 to, and fubfequcnt to, that precife period, as alfo the times 

 of the equir-o.xcs snd foifticci, which aie determined rnucl* 



more accurately by this method than can be done by propor- 

 tional parts, for in this, we are obliged to fuppofe that the 

 fun'8 declinations increafe and decreafc proportionally to the 

 dillancc of this body from the equinodial point ; which i» 

 evidently a falfe hypothefis. In facf , adronomy 1ms derived- 

 more affiliance from this theory than any other of the ina» 

 thema'ical fciences, although it has been applied to other 

 purpofes with very great fuccefs ; but in order to render its 

 application thus general, a much more accur.tte and extended, 

 inveftigation of the theory than that wiiicli has been at pre- 

 fent mentioned becomes neceffary, and of which we propofe^ 

 to give a flight view in the fubfequent part of this article. 



The general problem to be refolved in all tl'.ofe cafes is this.i 

 Having given two feries of numbers, which correfpond with 

 one another, according to a certain law, and of which one 

 tnay be called the feries of roof;, and the otber the feries of 

 f'.uiftions, to find an intermediate number, between two 

 funftions, which fhall anfwer or correfpond to a given interi- 

 mediate number betw-een two roots. T-liis problem is treated 

 of in all its generahty by Newton, in his " Metliodus Djf- 

 ferentialis," and by Stirhng in his " Traftatus de Summa- 

 tione, &c. ;" fee alfo Lacaiiles Aftronomy, and a paper by, 

 Mayer in the .'\cla. Petrop. torn. ii. page 108 ; alfo an ex- 

 cellent mei^pir by Lalande, tranfmitted to the Ac-ademy oi. 

 Sciences of Paris for 1761; in which he has fhewn that in. 

 calculating the moon's place, and much more that of any 

 other of the heavenly bodies, whofe motions r.re more re? 

 gular ; it will always be fufficient to confider only the third, 

 differences, and in many cafes only the fecond differences wilL 

 be requifite, and fometimes only the firft ; in fa£l he has, 

 fhewn how, and to ^^•hat degree of accuracy, the method of, 

 interpolation may be employed in moil cafes. The fame au- 

 thor has alfo a very complete article on this fubjecl in the 

 Encyclopcdie Methodique where, in explaining the nature 

 of altronomical in;orpolation, he fays, fuppofe a feries of 

 numbers o, 1,3, 6, 10, 15, 21, &c. of whicli the differences 

 are unequal, but follow a certain law, being in facl in arith- 

 metical progrefTion as i. 2, 3, 4, 5,6, &c. fo that the fecond 

 differences are conilant, and equal to unity. Now if we 

 take the alternate terms of this feries-, as for inllance o, 3, 

 10, 21, &c. the firft differences will be 3, 7, II, &c. and the 

 differences of thefe differences will be conftant, and equal 

 to 4 ; that is four times greater than before ; becaufe ia 

 doubhng the interval, we ha\'e for the firft difference the 

 fum of I and 2, and in the other the fum of 3 and 4; hence 

 it follows that the fecond difference increafe in the ratio of the 

 differences between 2 and 3, and i and 4: the increafe, 

 therefore, being equal to three times the firft, this fecond dif- 

 ference will be equal to four times that ; and, if we took 

 every third number, the fecond difference would be 9, and" 

 fo on. 



Thus, in general, the fecond differences increafe as the 

 fquare of the intervals of the numbers ; and hence we have 

 the following rule for fupplying the intermediate terms of a - 

 feries of numbers of which the law of, their generation is 

 uniform. 



For example, let there be four numbers Tvhich may, 

 reprefent longitudes, obfcrved at the interval of 12 hours 

 each; and of which the three differences are 78., 222, and 

 366, fo that their fecond differences will be conilant, and*. 

 equal to 144 ; according to the following tablet. 



Hours 



Nos 



3o:> 

 066 



ift Diff. 



2 2A. 



2d Diff: 



Such. 



