2(8

NEW METHOD FOR

TRACT 11.

hence 1022 1015, or 146 : 145

7 7 145

4 ; 4 X 146

= (V) T nearly.

2

Consequently the square of this, or (V) T will be =

sought more nearly, being true in the last figure.

A NEW METHOD OF FINDING, IN FINITE AND GENERALTERMS, NEAR VALUES OF THE ROOTS OF EQUATIONS OFTHIS FORM, X n — pX n ~ x + qx"~ z — &C = 0 ; NAMELY,HAVING THE TERMS. ALTERNATELY FLUS AND MINUS.

1. The following is. one method more, to be added to themany we are already possessed of, for determining the rootsof the higher equations. By means of it we readily find aroot, which is sometimes accurate ; and when not so, it is atleast near the truth, and that by an easy finite formula, whichis general for all equations of the above form, and of the samedimension, provided that root be a real one. This is of usefor depressing the equation down to lower dimensions, andthence for finding all the roots, one after another, when theformula gives the root sufficiently exact; and when not, itserves as a ready means of obtaining a near value of a root,by which to commence an approximation still nearer, by thepreviously known methods of Newton, or Halley, or others.This method is further useful in elucidating the nature ofequations, and certain properties of numbers; as will appearin some of the following articles. We have already easy me-thods for finding the roots of simple and quadratic equations,

TRACT XI.