INTERPOLATION
, in the modern Algebra, is used for finding an intermediate term of a series, its place in the series being given.
The Method of Interpolation was first invented by Mr. Briggs, and applied by him to the calculation of logarithms, &c, in his Arithmetica Logarithmica, and his Trigonometria Britannica; where he explains, and fully applies the method of Interpolation by differences. His principles were followed by Reginal and Mouton in France, and by Cotes and others in England. Wallis made use of the method of Interpolation in various parts of his works; as his Arithmetic of Infinites, and his Algebra, for quadratures, &c. The same was also happily applied by Newton in various ways: by it he investigated his binomial theorem, and quadratures of the circle, ellipse, and hyperbola: see Wallis's Algebra, chap. 85, &c. Newton also, in lemma 5, lib. 3 Princip. gave a most elegant solution of the problem for drawing a curve line through the extremities of any number of given ordinates; and in the subsequent proposition, applied the solution of this problem to that of finding from certain observed places of a comet, its place at any given intermediate time. And Dr. Waring, who adds, that a solution still more elegant, on some accounts, has been since discovered by Mess. Nichol and Stirling, has also resolved the same problem, and rendered it more general, without having recourse to finding the successive differences. Philos. Trans. vol. 69, part 1, art. 7.
Mr. Stirling indeed pursued this branch as a distinct science, in a separate treatise, viz, Tractatus de Summatione et Interpolatione Serierum Infinitarum, in the year 1730.
When the 1st, 2d, or other successive differences of the terms of a series become at last equal, the Interpolation of any term of such a series may be found by Newton's Differential Method.
When the Algebraic equation of a series is given, the term required, whether it be a primary or intermediate one, may be found by the resolution of affected equations; but when that equation is not given, as it often happens, the value of the term sought must be exhibited by a converging series, or by the quadrature of curves. See Stirling, ut supra, p. 86. Meyer, in Act. Petr. tom. 2, p. 180.
A general theorem for Interpolating any term is as follows: Let A denote any term of an equidistant series of terms, and a, b, c, &c, the first of the 1st, 2d, 3d, &c orders of differences; then the term z, whose| distance from A is expressed by x, will be this, viz, Theorem 1,
Hence, if any of the orders of differences become equal to one another, or = 0, this series for the interpolated term will break off, and terminate, otherwise it will run out in an insinite series.
Ex. To find the 20th term of the series of cubes 1, 8, 27, 64, 125, &c, or 13, 23, 33, 43, 53, &c.
Set down the series in a column, and take their continual differences as here annexed, where the 4th differences, and all after it become = 0, also A = 1, a = 7, b = 12, c = 6, and x = 19; therefore the 20th term sought is barely
Theor. 2. In any series of equidistant terms, a, b, c, d, &c, whose first differences are small; to find any term wanting in that series, having any number of terms given. Take the equation which stands against the number of given terms, in the following Table; and by reducing the equation, that term will be found. where it is evident that the coefficients in any equation, are the unciæ of a binomial 1 + 1 raised to the power denoted by the number of the equation.
Ex. Given the logarithms of 101, 102, 104, and 105; to find the log. of 103.
Here are 4 quantities given; therefore we must take the 4th equation , in which it is the middle quantity or term c that is to be found, because 103 is in the middle among the numbers 101, 102, 104, 105; then that equation gives the value of c as follows, viz .
Now the logs. of the given numbers will be thus:
Theor. 3. When the terms a, b, c, d, &c, are at unequal distances from each other; to find any intermediate one of these terms, the rest being given.
Let p, q, r, s, &c, be the several distances of those terms from each other; then let &c &c &c Then the term z, whose distance from the beginning is x, will be to be continued to as many terms as there are terms in the given series.
By this series may be found the place of a comet, or the sun, or any other object at a given time; by knowing the places of the same for several other given times.
Other methods of Interpolation may be found in the Philos. Trans. number 362; or Stirling's Summation and Interpolation of Series.
