JS^eic Mechanical Quadratures. 343 



be found. These depend upon the assumptions that the 

 increments of the abscissa are constant and that the differ- 

 ences above a certain order are negligible. It is said that if 

 the attempt is made to include in quadratures of this species 

 differences above the sixth, the formulae become unmanage- 

 able, but this I have not tested. It is possible to base 

 quadratures on Stirling's and Bessel's interpolation formulas, 

 but as these are not applicable at the beginning or the end 

 of a series of values their usefulness is limited, and since 

 these modes of interpolation are founded on Newton's, the 

 quadratures are not essentially different. 



Gauss used as the basis of his method of quadratures 

 Lagrange's interpolation formula. When after integration 

 the increment of the abscissa is assumed to be constant, 

 Cotes's numbers result, but Gauss showed that, by a proper 

 selection of unequal increments of the independent variable, 

 formulae can be deduced whose convergence is very rapid. 

 Unfortunately most of the increments are surds, rendering 

 the application of the method very laborious in spite of its 

 elegance. 



Interpolation by means of Taylor's series has several 

 advantages over methods depending upon finite differences, 

 and its limitation to continuous functions is rarely of any 

 moment. In dealing with known functions, the methods of 

 infinitesimal calculus are habitually employed excepting for 

 interpolation, while finite differences constitute a distinct 

 algorithm. Since Taylor's series is the very foundation of 

 analysis, its application to interpolation is both more con- 

 sistent and more elegant than that of finite differences, 

 while, if needful, the final results can be expressed in terms 

 of finite differences without the least trouble. Similarly, 

 so-called mechanical quadratures founded on Taylors series 

 in its application to interpolation are quadratures obtained 

 by integration of a differential equation, and thus do not 

 differ essentially from integrations by series. The expres- 

 sions for quadrature to which Taylor's theorem leads are in 

 some cases semi-convergent series, yet the error involved 

 may be reduced ad libitum. Although convergent series 

 would yield results of absolute accuracy were an infinite 

 number of terms to be computed, this accuracy is purely 

 theoretical and computation terminates when the error 

 becomes negligible. The distinction between convergent 

 and semi-convergent series is clear, yet there is no difference 

 between the results obtainable by their use in effecting 

 quadratures. 



The quadrature founded on Taylor's series was first given 



