376 



common to them all ; whether the more remote figures agree or disagree 

 cau be of no moment in reference to the object in view; since, agree or 

 not, they are confessedly useless. 



We submit that there is no difference of opinion as to Horner's 

 being the best method of computing the real roots of a numerical equa- 

 tion of an advanced degree by continuous approximation ; and with ap- 

 proximations only, in all those instances where a root has interminable 

 decimals, we must be content, even though those interminable decimals 

 may be but a very simple vulgar fraction in another form. As is usually 

 the case with general methods of computation, in whatever department 

 of practical mathematics they are proposed, there will always be parti- 

 cular examples that might be better treated by particular rules. The 

 present writer is not likely to be charged with undervaluing Horner's 

 method : he believes that its merits are such that, as a general method, 

 it will never be superseded. But, however high one's estimate of any 

 practical process may be, it is right fairly to state its inconveniences in 

 particular cases, as well as its general advantages ; and an inconvenience 

 in Horner's process, it certainly is — we think the only inconvenience — 

 that a fractional root has to be developed in decimals. 



Suppose, for example, one of the roots of an equation to be y : this 

 root, by Horner's method, would be determined in the approximate form 

 •142857 . . . . , and if the equation were of an advanced degree, a good 

 deal of numerical work would be required to obtain this approximative 

 value of ~. If the development were to be extended two or three 

 places further, the recurrence of the figures would, no doubt, suggest 

 the equivalent fraction; but fractions may readily be assigned the 

 equivalent decimal of each of which would not be seen to be a recurring 

 decimal till many more figures were computed. However, if it be of 

 no practical consequence in the inquiry before us that a root with in- 

 terminable decimals, and which is not the development of any finite 

 fraction, should be approximated to beyond, say, six places of decimals, 

 neither can it be of any practical consequence that '142857 should re- 

 place -f in that inquiry. 



Viewing the matter generally, in reference to incommensurable roots, 

 it would be more strictly accurate to regard our approximations — not as 

 approximations to the exact roots of the equation we are dealing with (for 

 it may not have exact roots — roots expressed in finite numbers), but to 

 consider each as the complete or exact value of a root of an approximate 

 equation — this approximate equation differing from the equation pro- 

 posed only in its final term or absolute number. The amount of the 

 difference may be made smaller than any assignable decimal ; for the 

 development of the incommensurable root may be carried to such an 

 extent, that the final term of the transformed equation, at which the 

 operation is stopped, may differ from zero by as small a quantity as we 

 please ; and the root thus far developed will be a complete root of that 

 approximate equation which would arise from merely correcting the 

 absolute term of the proposed equation by the small decimal alluded to. 

 What is here said as to a single incommensurable root applies, of course, 



