INTERPOLATION. 



169 



N ' , INTERPOLATION" is a branch of mathematical 

 ~ ' analysis, that treats of the methods by which, when a 

 series of quantities succeeding each other, and form- 

 ed all according to some determinate law, are given, 

 others subject to the same law may be interposed be- 

 tween them. 



Thus, to take a simple instance, the given quantities 

 may be the cube roots of the numbers 2, 3, 5, 6, 7, and 

 it may be required to find from these, by interpolation, 

 the cube root of the number 4. Again, the given quan- 

 tities may be the logarithms of any series of numbers, 

 2, 4, 6, 8, 10, Sec., and it may be required to find the 

 logarithms of the intermediate numbers 3, 5, 7, 9, &c. 

 Or the given quantities may be the place of some ce- 

 lestial phenomenon, a comet for instance, at midnight, 

 on four successive days, to find its place at any pro- 

 posed time on any one of these days. Or, taking one 

 example more, having given the expansive force of steam 

 at certain temperatures ; bv interpolation, its expansive 

 force at any temperatures intermediate between these, 

 or a little greater or less may be found. 



From these examples, it will easily be understood 

 that the theory of -interpolation must be of great prac- 

 tical importance, not only in pure mathematics, but 

 also in astronomy and every branch of physics, in which 

 the result of a series of experiments can be expressed 

 by number*. 



The method of interpolation is altogether a branch 

 of modern mathematics. It appear* to have been for 

 the first time employed by Henry Briggs, the ingenious 

 improver of logarithms in the construction of these 

 numbers, (Mutton's Math. Tablet. In trod. p. 69;) 

 thii was before the year 1624. Dr. Wallii, by interpo- 

 lation, and a most happy application of the method of 

 induction, found a remarkable expression for the are* 

 of circle, (see his Arithmetic* hfinitomm, 1656), and 

 soon afterwards, Gabriel Mouton, an astronomer of 

 Lyons, applied the theory to the construction of astro- 

 nomical tmnles ; be had, however, proposed to himiatf 

 a problem which he could not resolve, and he gives 

 the honour of the solution to hi* friend fUgnault. 

 Mouton ' labours appeared in book of observations 

 on the diameters of the sun and moon, published in 

 1670. 



The first general solution of the problem of interpo- 

 lation was given by Sir Isaac Newtun, but without de- 

 monstration, hi hit Priieifia. lib. iii. lem. 5. He af- 

 terwards gave solution, with a demonstration, in his 

 Method** Di/erentialu, a small tract, which was pub- 

 Kshed along with others, by W. Jones, the author of 

 Synojuii Palmariomm Matheteo*. The problem has 

 since been discussed by every wrilef on the vkcortj of 

 differences', and series, and particularly by Mr. J. Stir- 

 ling in his Method** D*jfere*li*lu, MM Tract alia de 

 S>.mm*tio*r ct tmteraolalione Seriemm Imjimilomm, pub- 

 li-hed in 1753. The principal works on this branch 

 of mathematics, we believe, are contained in the follow- 

 ing list. 



iiriggt, Arithmetic* LafarUhmica. Load. 164. 



Trifonometria Rritannica, Good*-, 1&J3. 



Wallis, Arithmetic* Infimitonm, (hum. 1656. 



Mouton, Otuervationet Dimmttronm Sola et Luna 

 Apfartniimm, ttc. Lugd. 1670. 



Newton, Principia, lib. iii. lem. 5. Loud. 1687. 



Mfthod*i Diftremtialit (Ofmtcni*,) 1723. 



s, (M*o*otech*ia t lirt Covtruttio Tab. per Dtf. 

 irmiia*. 



VOL. xu. PA'RT i. 



Herman, P/ioronomia, (Appendix,) Amst. !?](). Histoiv. 



Craig, De Calculo Fluentium, I.ond. 1718. " T- 



Stirling, Mel/todut Dijferentialis, Loud. 1 753. 



Euler, Opuscula Analytim, t. 1. p. 157. 



-- Instil. Calculi Diff. dp. Hi. 



Walmesley, Mem. del' /lead, de Berlin. Ann. 1758. 



Charles, Encyc. Mcthodiqite, Did. de Math. Art. In- 

 terpolation. 



Charles, Mem. de FAcad. Par. Ann. 1788. 



Goldbach, Comm. Acad. Petrop. torn. 3. 



Delambre, Mem. de tAcad. de Turin, 1790-1, p. 143. 



Lagrange, Journal det Seances de FEcole Normale, 

 t4. p. H7. 



Prony, Etiai Experimental et Analytiqur .u/r la lois 

 de la dtiatibilite detjluides elattiqnet in Journ. dc I'Ecule 

 Polytechnique, Cahier, 1. p. 28. 



Emerson, The Differential Method, Land. 1~6'7. 



La Place, Mem. de tAcad. Par. 177l. 



- Theorte Analytique det Probability, (p. 13.) 

 1814. 



Lacroix, Trtule dft Diff. ft des Series faisant suite n 

 Traitc du C*Jc. Diff. et du Cole. Int. 



Tut THEORY or INTERPOLATION. 







Let x and y be two variable quantities, having some Theory of 

 determinate relation to each other, so that i/ may be I"" 

 what is called ^function of x. (See FLUXIONS, Art. '.!. tion - 

 also the word FUNCTION.) If the form of the fun. 

 be known, we can find the value of y, corresponding to 

 any value of x whatever. For example, let the rela- 

 tion between t and y be such, that 



then, when t = 0, we have y = 3, when x = },y (>, 

 when i = *, y 15, and so on, as is shewn in the two 

 following sets of corresponding values. 



x = 0, 1, 2, 3, 4, 5, Arc. 

 y = 3, 6, 13, 24, 39, 58, &c. 



The values of y being thus formed all according to 

 the same law, they constitute a regular urien, and the 

 corresponding values of x are the indices of the terms, 

 that is, they ihew their place, or distance from the be- 

 ginning of the series. U ben a series is formed in this 

 manner, by substituting successive values of x in i/, 

 some function of i, that function is called the general 

 term of the scries, it is analogous to the equation of a 

 curve, and k serves to characterize the series, and to 

 liMtiaguiili it from all others. Thu>, as from the gene- 



ral term jr= ' 



ing values of i and y, 



*=0, 1,2, 3, 4, 

 J= 1, 3,6, 10, 15, &c. 



10, on the other hand, when this last series is put under 

 this form 



' - we get these correspond- 



1.2 S.S 3.4 4.5 6.6 

 '~'~'''* 



we may infer that its general term is 



(*+!)(* + 2) 



and that it can be no other expression. 



It is easj to conceive that between every two adjoin- 

 ing terms of a series formed in this way, there may be 

 any number of Others interposed, forming, upon the 



T 



