62 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



C + -T^ + 7-^) ^^ - (A X "-^^ B X V-1 + -7^-,> + A 



Wp ' )r(jr — p) ' p(p — w^)' ^ Tp • »»■. C'" — P) f . (p — »»■)' 



= 0: for write O, tt, and p, respectively for ar in the equation, and there will 

 result the quantities a, b, and c. He then gives some general approximating 

 equated values of the roots of equations, which he says will nearly be the same as 

 found, where a near approximate is given, from the method given by Vieta, 

 Harriot, Oughtred, Newton, De Lagny, Halley, &c. 



Sir Isaac Newton found the sum a, of the 2n*^ power of each of the roots of 

 a given equation, and then extracted the 2n}^ root of a, viz. "^^a, for an approxi- 

 mate value of the greatest root of the equation ; and further added some similar 

 rules on the same principle. In the Miscell. Analyt. and Meditationes the same 

 principle is applied in different rules for finding approximates to the greatest and 

 other roots of the given equation; and also limits of the ratios of the approxi- 

 mate values of the roots found by these rules to the roots themselves are given. 

 It is observed in the Meditationes, that from these rules in general to find the 

 greatest root, it is often necessary that the greatest possible root be greater than 

 the sum of the quantities contained in the possible and impossible part of any 

 impossible root of the given equation: for example, \f a -\- b\/ — 1 be an im- 

 possible root of the given equation, then it is necessary that the greatest pos- 

 sible root be greater than a -\- b. It may further be observed, that in equations 

 of high dimensions, unless purposely made, it is probable that the number of 

 impossible will greatly exceed the number of possible roots; and consequently 

 these rules most commonly fail. 



M. Bernoulli assumed a fraction whose numerator is a rational function of the 

 unknown quantity, and denominator the quantity which constitutes the equation; 

 and reduced the fraction into a series, whose terms proceed according to the di- 

 mensions of the unknown quantity ; and thence found an approximate value of 

 the greatest or least root of the given equation or its reciprocal, by dividing the 

 co-efficient of any term of the series resulting by the co-efficient of the preceding 

 or subsequent term. 



The rule of false has been found very useful in finding approximates to the 

 two unknown quantities contained in two given equations, and has been applied 

 to n equations having n different unknown quantities : for example, it has been 

 observed, that if 2 or more m values of an unknown quantity x are nearly equal 

 to each other, and to its given approximate value x^, the unknown quantity v = 

 X — >r' will ascend to 2 or more m dimensions in one of the resulting equations; 

 or in more equations than one will be contained such powers of the quantity v, 

 that if the more equations were reduced to one whose unknown quantity is v, the 

 resulting equation will contain m dimensions of the quantity v. Hence it appears, 

 that in this case alsc^ the con vergency of the approximate values found will depend 

 ©n the given approximate being much more near to one root than to any other. 



