Properties of the Circle, S^S 



B ; and tliey will all terminate in the points at which the cir- 



cumferettce was last equally divided ; since m, 2m, &:c. are all 



-p 

 whole numbers ; and _ = one of the parts, into which the cir- 



n 

 cumferenee was last divided. 



Since m, and n, are prime to each other, and the several 

 numbers 1, 2, 3, &tc., until we come to n, are each less than n; 



it follows that -, ^', ^, ike, until we come to —, cannot 



n n n n 



either of them be a whole number. Therefore, no one of the 



«^^K«^ '^*P 2//iP ^ .., , nmV ^ . , 



arches — , ,. *i.c., untiJ we come to , can termmate at 



n n n 



the point B. ft follows also, that these arches all terminate in. 

 different points ; for in order that more than one of them should 

 terminate at the same point, some one q[ them must have ter- 

 minated at B ; when the remaining arches would go on to ter- 

 minate in the same points, at which those already taken had 

 terminated respectively : (since they increase from B, by the 



common difference ^—.) Therefore, since the arches o, —^ 

 n n 



, &c., and -~^, _5l-' , —JLJ , he,, the former se- 

 re n n n 

 ries reckoned from B, and the latter from A, do respectively 

 terminate in the same points ; and since the arches of the for- 

 mer series do respectively terminate at different points until we 



come to the arch —~ ; and since they each terminate in some 



one of the points B, C, D, &;c. ; and since there are as many 

 arches as points; it follows, that they divide the circumference 

 into as many, and the same parts, as the points B, C, D, ike. 

 divide it. But these points divide it by construction into n equal 

 parts ; theref»)re each series of arches, divides the circumference 

 into n equal parts. 



Case II. Let m and n have some common measure, and let 



t denote their greatest common measure ; let — . be reduced to 



n 



its lowest terms, by dividing m snd n by t ; and let _ denote 



the fraction thus reduced. Then will the arches '^15; 



n 



-Z-SJ l^Z;-__, fee, be equal respectively to -y, —^ , 



ti ' n S S 



