CONVERSION OE NUMBERS FROM ONE SCALE TO ANOTHER. 1 33 



notation; particularly, as the number 10, though not the 

 best that misj^ht have been adopted, is very proper for the 

 purpose; and as the advantages possessed by the other are 

 not such as can lead us to exryect, or even to wish, that it 

 should ever be substituted for that, which long established 

 custom has rendered so familiar to all our ideas of numbers. 

 I shall therefore, in what follows, limit myself entirely to the 

 method of transforming a number from, one scale to an- 

 other, and showing its application to the rule of duodeci- 

 mals, which, so far as relates to the converting of a number 

 from our own scale to any other, and its application to the 

 rule above mentioned, I conceive to be new, and more sim- 

 ple than any other that I am at present acquainted with. 



Let r be the radix of any system, and a, b, c, d, &c., the Method of 



digits, by which any number (N) is expressed in that sys- transforming 



«,•','' ^ ' ^ ^ a number from 



tern, then we have one scale 10 



N = r-"a + ;"-'i + t'^-^'c + r"-'J, &c. + 9- '*"°'^'^'' 



Thus 1746 in the decimal scale may be expressed by 

 lO'.l + 10*.7 + 10*. 4+ 6, and, in the duodenary scale, 

 8467 may be represented by 12^8 + 12*.4 + 12'.6 + 7. 



And so on for others; where it will be readily observed, that .. , - 



*' . ' Number of 



the number of characters, including the 0, in any scale of characters 



notation, will never exceed the number that expresses the ^\^^^ ^^"*^ , 



. ^ the root of the 



radix of that system. Thus, m the bmary scale only two system. 



characters are wanted, namely 1 and 0; in the senary, six ; 

 in the decimal, ten; and in the duodenary, twelve; two ad- 

 ditional characters being necessary for expressing 10 and 

 11, those I shall represent thus 10 — ^, and 11 :=: y,so that 

 the digits of this system are as follows. 



0, 1, 2, 3, 4, 5, 6, 7, 8, 9, <P> y. 



Duodenary 



Now, r being the radix of any system, and a, h, c, d, &c. '-'^^'^'^*^^'^'" 

 the digits of a number in that system ; also r' the radix, 

 and a, 6', c', d\ &c. the digits in any other system : then, 

 in case of equality, we shall have r'a + r""*6 + r^'^c + 

 r'^-^d, &c. r= r ""«' + r'"'-*6' + r'"'"\', &c. ; and, therefore, 

 in converting a number from one scale to another, this equa- 

 lity must be established, and we must solve the above equa- 

 tion, the value of all the numbers being given on one side, but 



on 



