3y0 PHILOSOPHICAL TRANSACTIONS. [anNO 1753. 



nor have they ever been so bad, as to prevent his acting in his business as a sailor, 

 till within a few weeks befoje he applied to Mr. W. 



XL. On Infinite Series and Logarithms. By Air. James Dodson. p. 273. 



The terms of one of the most simple series, for expressing the logarithm of a 

 given number, is composed of the powers of the excess of that number above 

 unity, divided by their respective indices; of which the 1st, Sd, 5th, &c. terms 

 are affirmative, and the 2d, 4th, 6th, &c. terms are negative ; and the differ- 

 ence between the sums of the affirmative and the negative terms, is the Neperian, 

 hyperbolic, or as some call it, the natural logarithm of the given number. 



Now a mathematician, who understands the nature and management of series, 

 (though wholly ignorant of fluxions, or what Dr. Halley, in his investigation 

 of this very series, published in N° 2l6 of the Philos. Trans, calls ratiunculae, 

 &c.) might arrive at the same conclusion, in the following manner : 



Since the logarithm of 1 is universally determined to be nothing ; that of 2, 

 3, 4, 10, or any other number, considered as a root, is 1; that of 4, Q, l6, 

 100, &c. considered as the square of that root, is 2, and so on ; it follows, that 

 in all cases the logarithm of a greater number will exceed that of its less; and each 

 logarithm will have some relation to the excess of its number above unity, the 

 number whose logarithm is nothing : the terms of the series therefore which 

 will represent the logarithm of any number, will consist of the powers of the 

 excess of that number, above 1, with some, yet unknown, but constant co- 

 efficients. 



That the logarithm of the square of any number is twice the logarithm of its 

 root, is a well-known property of those artificial numbers ; and therefore the 

 doubles of the particular terms of the assumed series will constitute a series ex- 

 pressing the logarithm of the square of the given number. But by prop. 4, 

 book 2 of Euclid, the square of any quantity is equal the sum of the squares of 

 its 2 parts, plus a double rectangle of those parts ; which, in this case (where 

 the given number has been assumed to consist of I and an excess) will be l plus 

 twice that excess, plus the square of it. 



If therefore the several powers of the compound quantity (twice the excess of 

 the given number above 1 plus its square) be multiplied by the above assumed co 

 efficients, and afterwards ranged under each other, according to the powers of 

 the said excess, their sums will again express the logarithm of the square of the 

 given number. 



Now since the logarithm of the square of the given number may be thus ex- 

 pressed by 2 infinite series, each constituted of its excess above 1 , and its powers ; 

 it follows, that the co-efficients of the like powers of that excess, in each series, 

 will be equal between themselves ; and consequently the values of the unknown 



