Quotienh of Unity by prime Numhers. 4I05 



The quotient of an unit, divided by the prime number 17', will confift of 16 places of 

 figures, forming a complete circulating decimal. If the numbers, 2, 3, 4, &c. to 16, be 

 divided by the fame prime number, each refpeftive quotient will ftill confift of 16 peaces oT 

 circulating decimals. Thus far the property of like diviGons has been afcertained by 

 various writers oa decimal arithmetic, &c. 



But at leaft one very curious, concifc, and ufeful property attached to fimilar divjfions 

 in general, yet remains to be unfolded. It is this — that the quotient arifing from the firft 

 divifion, virtually reprefents the quotient of every other divifion above-mentioned. 



FIRST QUOTIENT. 

 Dividends. Divifors. 



For I" ^ 17* r= 'o 5 8 8 2 3 (^ 2 9 4 1 i 7 <?' 4 7 



* V ^ ~l o 5 8 8 235294 



, ^ 17 _ f - - - - i 7 6 4 7 



3 • ' ~ l o $ 8 8 2352941 



' * '^ =( ° '5 '«"^" ^ 3 ^ ^ 9 -^ ^ ^ 7 6 4 7 



And in like manner the quotient arifing from the divifion of an unit by every other 

 whole number, lefs than the divifor, will commence with a different figure In tlie firft 

 quotient, and will circulate to that figure again. 



And thus the complete quotient arifing from the divifion of each whole number, lefs 

 than the divifor, in that divifion may be expreffed on the firft quotient, by placing the 

 refpedtive dividends over their firft quotient figure. 



Thus C .1 1° 15 M 4 6 9 5 16 7 2 3 13 II 8 12 dividends. 

 C •05882352941 17647 quotients. 



This difpofition of the dividend and firft quotient enables us to find, by infpe£tion 

 the complete decimal quotient or expreffion for a vulgar fraftlon, whofe numerator or 

 dividend is any given whole number, between i and 17, and whofe denominator or divifor 

 Is 17. The quotient of tt» iV' an^ tt* 'S feen in the firft elucidating arrangement 

 above, and perfeftly coincides with this laft. By this, if I want the complete decimal* 

 quotient of -ff, I have only to fearch for the number 16 in the line of dividends, and under ' 

 it is the firft figure of the circulating decimal that will comprize complete quotient of j-, 

 viz. '9411764705882352, and the fame of the other dividends. 



But what authorifes the above property to be termed curious, concife, and ufeful, is 

 that it does not attach to the prime number 17' only, but under certain laws is equally 

 applicable to all prime and multiples of prime numbers whatever. 



3 F 2 Conjlruaion 



