304 PHILOSOPHICAL TRANSACTIONS* [aNNO 1717 » 



no occasion for any of the transcendental methods, and is expeditious enough 

 for making the tables without much trouble. 



^ general Series for expressing the Root of any Quadratic Equation, — Any 

 quadratic equation being reduced to this form xx — mgx 4- wjy = O, the root 

 X will be expressed by this series of terms. 



^ = f + ^ X 4^ + ^ X a-^ + ^ X 6^ + ^ X 3rb ^^• 



Which must be thus interpreted. 



1. The capital letters a, b, c, &c. stand for the whole terms with their signs, 



preceding those wherein they are found, as b = a X — 5 . 



2 



y 

 1. The little letters a, h, c, &c. in the divisors, are equal to the whole divisors 



of the fraction in the terms immediately preceding; thus Z> = a^ — 2. 



For an example of this, let it be required to find y/l. Putting y/1 z= x -\- 1, 

 we have a?* + 2ar — 1 = O, which being compared with the general formula, 

 gives mq =z — 2, and my :=. — 1 : therefore for m taking — 1, we have 

 9 = 2, and ^ = 1, which values substituted in the series give a: = - — ^ 



^ ^ ^ &c. The 



2 X 6 X 34 2 X 6 X 34 X 1154 2 X 6 X 34 X 1154 x 1331714 



fractions here written down giving the root true to 23 places. 



A new Method of computing Logarithms. — This method is founded on these 

 considerations, 



1. That the sum of the logarithms of any two numbers is the logarithm of 

 the product of those two numbers multiplied together. 



2. That the logarithm of unit is nothing ; and consequently that the nearer 

 any number is to unit, the nearer will its logarithm be to O. 3dly, That the 

 product by multiplication of two numbers, whereof one is larger, and the 

 other less than unit, is nearer to unit than that of the two numbers which is on 

 the same side of unit with itself; for example the two numbers being ■§- and -f, 

 the product \ is less than unit, but nearer to it than -f^, which is also less than 

 unit. On these considerations, I found the present approximation ; which will 

 be best explained by an example. Let it therefore be proposed to find the re- 

 lation of the logarithms of 2 and 10. In order to this, I take two fractions 



128 8 2' 2' f 



•—- and — , viz. —5 and —-,. whose numerators are powers of 2, and their de-. 

 100 10' 10* 10" ^ 



nominators powers of 10; one of them being greater, and the other less than 1. 

 Having set these down in decimal fractions in the first column of the following 



