16 Mr. J. R. Young on a Property of 



under those above : we thus have 



i=-142S57U2857142857142 8 

 7 



42857142857142857142 8 



' Cut offtveo places, and multiply by 2 : 



•14285714285714285714 28 

 285714285714285714 28 



Cut off t/iree places, and multiply by 6 : 



•1428571428571428571 428 

 8571428571428571 428 



III like manner, cut off Jour places, and multiply by 4 ; ^ve 

 places, and multiply by 5 ; six places, and multiply by 8 ; 

 seven places, and multiply by 3 ; eig/ii places, and multiply 

 by 9, or by 2 ; 7nne places, and multiply by 6. And in all these 

 cases the decimals will recur as above, to whatever extent the 

 development be carried. 



Similar circumstances have place for all incommensurable 

 fractions, the multipliers being different for different fractions. 



Thus, take — . 

 13 



■^ =-076923076923076923076. 

 la 



If two figures be cut off, the proper multiplier will be 9 ; if 

 three be cut off, it will be 12 ; if four be cut off, it will be 3 ; 

 ii[five, 4, &c. 



I proceed to show what I venture to think to be a useful 

 application of the above property. 



In finding the circumference of a circle from the diameter, 

 the usual process is to multiply the diameter by the number 

 3*1416; and, in determining the diameter from the circumfe- 

 rence, to divide by the same number. When the diameter 

 consists of several figures, the multiplication is rather long; 

 and there is a proportionate liability to error, especially in the 

 addition part of the work. And when the circumference is 

 given, whether it be a large number or a small one, the divi- 

 sion becomes tedious if several decimals are required. For 



22 

 rough purposes, the ratio — of Archimedes is employed, as 



involving simpler operations. I propose here to show how 

 the rule of Archimedes may be perfected by aid of the above 

 property. 



