-S94 Demonsfratiom of Stewarfs 



around the circle in the same direction, be multipliecl 

 by any whole number, less than the number of the parts 

 into which the circumference is divided ; then, the ex- 

 tremities of these multiple arches will divide the circum- 

 ference into equal parts, 



jyEMOKSTRATIOJsr. 



Let O (Plate III. Fig. 1 .) be any circle, the eircumference of 

 which isdivided into any number of equal parts, at the points 

 <it,h, c, &.C. ; and let any point, at A, be taken in the circumfer- 

 •€nce, thus diviJed. Let P denote the circumference, and «> 



P 

 the number of the parts into which it is divided : then — = 



n 

 any one of the parts. 



Since Act is such a part of — , as may be expressed by sora€ frac- 



n 

 r 



tion, let — denote that fraction in its lowest terms : then Aa ~ 

 ' s 

 ■pjf ."p /~\ 



1 Let — be denoted by Q, : then, the arch Aa = -~- ; the 



■■■■su s n 



■arch Aab = — — — ; the arch Aahc = -5l — __, &c. ; since the 



n n 



• • P Q 



;iarches increase by the addition of — Let the arches -^ 



n n 



^_— , ^~ — , &c. be multiplied by any number m, less than 

 n n 



« : th^n will they be represented by _^, _li_' ,^ _i.J , 



7}, n n 



Case I. Let the numbers m and n, have no common mea- 

 snre. 



Set off from A, towards h, the arch AaB, equal to — -t; and 



n 



from B, divide the circumfereuce into as many equal parts as it 

 \vas divided into at first. 



Now, sin«e the arch AaB, or its equal — — , belongs to all the 



n 



dies — — , — -^^ 5 ■—^- , &c. ; ii it be taken away 



n n n 



from each of them, the remainders, o, — _, , &c. will 



n n 



express the distances at which tlie aforesaid arches will respec- 

 tively terminate, from B. Whence the first arch terminates at 



