3-44: Mr. G. F. Becker on some 



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

 1738. It was discovered independently by Maclaurin, who 

 published it in 1742 *. The very essential discussion of the 

 remainder was left to Poisson, Jacobi and others. The 

 formula for mechanical quadratures commonly given in 

 textbooks, and ordinarily ascribed to Laplace, is merely 

 Euler's equation with the substitution 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 x , B 3 , B 5 



be Bernoulli's numbers ; for brevity let also 

 k (k) (k) 



v ="A<H/(*o), 



or the difference of the &th dermitive at the two limits. 

 Then Euler's equation may be written thus : 



•«)- 



B^V 

 2! 



, B 3 7iV" 

 + 4! 



x n <r 







.... + R 



Of course h = 



and h may be any integral factor of x n — x Q . The total 

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

 integrated into n parts each of width h. 



Euler's formula is capable of some transformations which 

 do not seem to have been noticed and a variety of special 

 forms can be deduced from it. For this purpose it is con- 

 venient to make certain changes in notation. The first term 

 of the second member is a polygon bounded by the axis, the 

 extreme ordinates and chords connecting the extremities of 

 all the ordinates. Let this polygon of chords be denoted by 



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

 Commentarii Acad. Sci. Imp. Petrop. vol. vi. ad annos 1732 et 1733 ; 

 1738, page 68 5 Maclaurin gives it in his 'Treatise of Fluxions,' 1742, 

 page 672. 



