342 THIRD REPORT 1833. 



Similar processes, also, have been investigated and applied 

 with remarkable ingenuity and success by Mr. Holdred *, Mr. 

 Horner f, and Mr. Nicholson J. The first of these writers, a 

 mathematician in humble life, who had formed his taste upon 

 the study of the older algebraical writers of this country, gave 

 very ingenious rules for finding the roots of numerical equa- 

 tions. The method proposed by Mr. Horner was founded upon 

 much more profound views of analysis and of the relation which 

 exists between the processes of algebra and arithmetic, and he 

 has not only succeeded in making a very near approximation to 

 the true principles upon which the limits of the roots of numerical 

 equations are assigned, but by considering the rules for extract- 

 ing the roots of numbers and of affected numerical equations 

 as founded upon common principles, he has reduced the rules 

 for these purposes to a form which admits of very rapid and ef- 

 fective, though not perhaps of very easy, application. Mr. Ni- 

 cholson, by a combination of the methods of Mr. Holdred and 

 Mr. Horner, has greatly simplified them both, and reduced them 

 to the form of practical rules, which are not much more compli- 

 cated than those which are commonly given for the extraction 

 of the cube and higher roots of numbers. 



The Newtonian method of approximation, which we have 

 hitherto considered, may be termed linear, in as much as the 

 equations of a straight line combined with the general equation 

 of the parabolic curve are competent to express all the circum- 

 stances which characterize it. But methods of approximation 

 of higher orders than the first, involving the second or higher 

 powers of the unknown quantity to be determined, have likewise 

 been considered by Fourier and other writers. That of the se- 

 cond order, viewed with reference to the properties of curve 

 lines, may be said to result from the contact of arcs of a conical 

 parabola. The superior and inferior limits, thus determined, 

 converge with great rapidity, the error cori-esponding to each 

 operation being the product of a constant factor with the cube 

 of the preceding error. Such methods, however, if viewed with 

 reference to the facility of their practical applications, are incom- 

 parably less useful than those which are founded upon linear 

 approximations ; but there is much which is instructive in their 

 theory, and particularly as furnishing the means of determining 

 immediately the nature of two roots of an equation included in 

 a given interval, which the application of the methods for the 



* This method is particularly noticed in Mr. Nicholson's Essay on Invobi. 

 Hon and Evolution. I have never seen the original tract published by Mr. 

 Holdred. 



+ Philosophical Transactio7is for 1819. 



J Essay on Involution and Evolution. 1820. 



