REPORT ON CERTAIN BRANCHES OF ANALYSIS. 341 



late number * immediately greater than this quotient, and n the 

 order of the articulate or subarticulate number which is not less 

 than the difference of the limits b — a, then if we divide f{b) 

 by/' {b), and continue the operation as far as the {2n + kf^ 

 decimal, and increase the last digit by 1, the quotient which 

 arises being subtracted from or added to, b, according as f{b) 

 and/' (b) have the same or different signs, will give a result 

 which will differ from the true value of the root by a quantity 



less than ( tt^ ) • And if the same operations be repeated, 



forming successively new limits by means of the results thus 

 obtained, we shall obtain a series of limits which are correct as 

 far as the {4>n + 3 kj^, the {8n + 7 kf^, &c., decimal place f. 



The processes of approximation which have been described 

 above, as well as those which belong to all other methods, re- 

 quire divisions and other operations with numbers which are 

 sometimes beyond the reach of logarithmic tables, and which it 

 is extremely important to abbreviate as much as possible, con- 

 sistently with the determination of the accurate digits of the 

 results which are required to be found. Such processes were 

 taught by Oughtred and other algebraists of the seventeenth 

 century, but both their theory and applications have been 

 greatly and, perhaps, undeservedly, neglected in later times. 

 The consideration, however, of such methods has been partially 

 revived by Fourier and some other writers, the first of whom 

 has given examples of what he terms ordinate division {division 

 ordonn^e,) the principle of which is to conduct the division by the 

 employment of a small number of the first digits of the divisor 

 only, and to correct the successive remainders, augmented by 

 the successive digits of the original dividend, in such a manner 

 as to bring into operation the successive digits of the divisor 

 when they are required for the determination of the correct 

 digit of the quotient, and not before. Such processes, however, 

 are incapable of being briefly described, and we can only refer to 

 the original work| for the developement of the rule and for ex- 

 amples of its application. 



* An articulate number is one of the series 1,10, 200, 7000, &c., where 

 the first digit is followed by zeros only. A subarticulate number is one of 

 the series '1, '02, '003, &c., and the number which designates the place of 

 the first significant digit is supposed to be negative, 



t The course of the approximation, in order to be perfectly regular and rapid, 

 would require that 2 n + h should be greater than n, or that n should be 

 greater than — ^, a circumstance which might occur if A: or w was negative. 

 In such a case it will be necessary, or rather expedient, to subdivide the in- 

 terval b — a, until the difference of the two limits does not exceed ( — - ) , 



where « is equal to, or greater than, 1 — k. 



I Analyse des Equations determinecs, livr. ii. p. 188. 



