1 30 On the Method of developing an Incommensurable Fraction. 



the present short paper to the readers of the Philosophical 

 Magazine. The extension spoken of is implied in this state- 

 ment, namely : if we develope ani/ fraction by division, what- 

 ever be the numerator or denominator, and arrive at a re- 

 mainder which is either a multiple or a submultiple of a former 

 remainder, we may discontinue the operation and thencefor- 

 ward proceed by Colson's rule. A single illustration will 

 suffice to confirm this principle. 



Let us take the fraction proposed by Metius for the ratio 



355 

 of the circumference of a circle to its diameter, namely r^j-r. 



In performing the division, the first remainder we get is 16, 

 and the ninth remainder is 4>, the quotient up to this remainder 

 being 3-14159292. Consequently 



But 



?|=3-U159292jf3. 



355 _ 16 

 113 ~ 113 

 16 4 



^ =03539823 , , 

 113 4 



1(JL) 



4\113/ 

 .: l(A-^ = 0088*955 75 l,{-±^ 



•••p(iT3)='"'^^'^««8"43(iTs)' 



and so on. Consequently 



355 1 / 4 \ 



rr— =3-14159292035398230088495575221238 9375 -nl-—) 

 113 4'^\113/ 



00055 &c. 



93805 &c. 



where it is to be observed, that the figures cut off are those 

 under which the decimals, arising from the supplemental 

 fraction, are to be placed whenever we choose to put a stop 

 to the process. In the present case these decimalsare00055&c. 

 The above form has been given to this process for the pur- 

 pose of preserving an analogy to Colson's method ; but, after 

 having got the first group of decimals '14159292, by common 

 division as above proposed, we may discover the following 

 groups, as fast as they can be written down, by continually 



