L18 G. F. Becker— X<tr Mechanical Quatlrattiro:. 



ratarea obtained by integration of a differential equation and 

 thus do not differ essentially from integrations by series. The 

 expressions for quadrature to which Taylor's theorem leads 

 are in some cases Bemi-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 

 by Euler in 1732-3, but his paper was not printed until 1738. 

 It was discovered independently by Maclaurin who published it 

 in 1712.* The very essential discussion of the remainder was 

 left to Foisson, Jacobi and others. The formula for mechan- 

 ical quadratures commonly given in text-books, and ordinarily 

 ascribed to Laplace, is merely Eider's equation with the substi- 

 tution of finite differences for derivatives. 



It is easy so to transform Taylor's series as to express an 

 integral in terms of a sum of the ordinates and sums of the 

 successive derivatives. The function to be integrated and its 

 derivatives can be similarly expressed. From the system of 

 equations thus developed the sums of the derivatives can be 

 eliminated and the result taken between limits is a definite 

 integral expressed in terms of the sum of the ordinates, 

 together with the derivatives at the limits. This is Euler's 

 quadrature. 



Let h be the constant increment of x and B l5 B 3 , B 6 be 



Bernoulli's numbers; for brevity let also 



or the difference of the Jct\\ derivative at the two limits. 

 Then Euler's equation may be written thus : 



B.AV B„AV" 



r * i./Vo V„\ BAV 



J ydx=h {^ +y x + y 2 + . . . . + ^J - -A_ + 



4! 



-...+R. 



3Cq 



Of course 



7 "^n •*-<> 



and h may be any integral factor of x a —x . The total number 

 of ordinates is n + 1 and they divide the area to be integrated 

 into n parts each of width A. 



* Euler's formula, based on Taylor's theorem, will be found in Commen- 

 tarii Acad. Sci. Imp. Petrop., vol. vi, ad annos 1732 et 1733 ; 1738, page 68, 

 Maclaurin gives it in his Treatise of Fluxions, 1742, page 672. 



