382 Mr. A. J. Ellis. Improved' Bimodular Method of [Feb. 3, 



the familiar proposition that, when the difference of two numbers 

 is small, the difference of their logarithms is nearly equal to the 

 bimodulus multiplied by the difference and divided by the sum of the 

 numbers themselves. The improvement here for the first time effected, 

 consists in prefixing a brief preparation, which makes the method uni- 

 versally applicable, and subjoining an easy correction depending on 

 the transformation of a well-known series proceeding by the odd 

 powers of the difference divided by the sum of two numbers, whereby 

 the number of places obtained is greatly increased. This method is 

 here applied for finding the natural and tabular logarithms of any 

 number to twelve places of figures by means of a table of two pages 

 for each kind of logarithm, and to sixteen places by help of a seven- 

 place table of tabular or Briggs's logarithms. An extremely simple 

 rule, which, so far as I know, was never before imagined, enables us 

 to pass from the logarithm to the number, that is, to find anti- 

 logarithms from the same tables. Although the method is applicable 

 to any system of logarithms, and was actually first applied by me to 

 the direct calculation of musical logarithms to the bases 2 (octave), 

 13 a/2 (equal semitone), and 81-T-80 (comma), and appropriate tables 

 have been constructed, I confine myself for brevity to natural and 

 tabular logarithms. The tables are constructed from existing mate- 

 rials, but the method is capable of constructing them independently. 



Section II. — Principles op the Bimodular Method and its 

 Improvement. 



Fundamental Relations. — Let n and d be any whole numbers of 

 which d is the smaller, and let p = d-+-n, a proper fraction. Let 



nat. log (l+p)=y, and log (l+p)=M.y . . . (1) 



where M is the modulus, and hence 2M the bimodulus to any un- 

 specified system of logarithms marked by log. Let 



o4ri=o?-=2> 2 1 =x > 2M 2= M *= Z • • • • (2). 

 In + a 2 + p 



In future n and n + d will often be called " the numbers," n "the 

 tabular number," d "the difference," 2Mcl "the dividend," 2n + d 

 " the sum " or " divisor," and ZM.d-i-(2n + d) " the quotient." 



Now it is familiarly known that 



y=p-ip*+if-iP*+ (3), 



=2(<Z+i2 3 +i!Z 5 + • • •) (4). 



Putting in (4) the values of q in terms of x and z from (2) we have 



