364- Ml". J. Cockle's Analysis of the Theory of Equations, 



If we put the subject of actual and practical numerical so- 

 lution out of the question, the work of Dr. Hymers is a most 

 useful and excellent one. It devotes some space to the question 

 of the limits of the 7-oots, — an inquiry of importance, inasmuch 

 as it assists us in obtaining a first approximation. I observe 

 that you prefer the method of trying the factors of the 

 absolute term as an initial experiment (Hutton, p. 234) (/?). 

 Would it not be as well if the concluding chapter of Dr. 

 Hymers' work were removed to the end of that which treats of 

 algebraic (rigorous) solution, and the connexion between the 

 ordinary methods and that of Lagrange explicitly exhibited? 



(h). This is not quite the way in which I wished to be under- 

 stood : for I was then speaking of any proposed equation, that 

 might contain integer roots, from which it is always desirable in the 

 outset to clear it. When this is done, the method of superior and 

 inferior Umits will sometimes effect our purpose readily ; though it 

 is often so wide of tlie mark as to become practically inconvenient, 

 if not absolutely useless. I know of no infallible rule that can be 

 a])plied : and when I was a good deal accustomed to such com- 

 putations, I found it to be most convenient to try the effect of dimi- 

 nishing the roots by unit-transformations for a step or two ; and if 

 this did not much alter the coefficients, to employ a similar process 

 with tens or hundreds, as the case may seem to require. If the 

 changes of the coefficients with a unit-transformation were great, 

 the use of "1 or '01 as the quantities by which the roots were dimi- 

 nished, always led with tolerable facility to the desired information. 



When, however, the coefficients are large, especially " the abso- 

 lute term," the employment of the reciprocal equation for this 

 purpose will be convenient : and in all cases, attention to punc- 

 tuation (after the manner that numbers are " pointed off" for the 

 extraction of the square and cube roots) will greatly facilitate this 

 initial inquiry. It is, however, unnecessary to say more upon such 

 a question, where, after all, our mode of proceeding is essentially 

 a tentative one. 



So far as the question of numerical solution is concerned, 

 the work of our friend Professor Young is without a rival. 

 In it the method of Horner is fully developed, and lucid and 

 laboriously calculated examples serve to exhibit it in all its real 

 utility [i). In the subject of algebraic equations, Mr. Young 



(i). It should have been added, that the Very Reverend the Dean 

 of Ely has also given Horner's process in his elaborate and most 

 valuable treatise on algebra ; and that Dr. James Thompson of 

 Glasgow has also done the same in an elementary work on the same 

 subject. An improvement in the process of synthetic division has 

 also been given by the latter gentleman : and one very analogous 

 to it, but a little different in form, had been previously given by 

 Professor Christie in his Algebra, already mentioned. That promising 



