i^96 Demonstrations of Steivart''s 



Now, since the fraction — is in its lowest terras, k and g are 

 g 

 of course prime to each other. Therefore, taking the arch AaB 



= — >, and supposing the circumference to be divided from B, 



mto g equal parts ; it may be shewn, as in Case I, that the arch- 

 es -3'^ -3tl-L — , _fy~ , &,c. will divide the circumfer- 



g g g, . 



ence at the supposed points, into as many equal parts as there 



are units in g ; when we have taken as many of them as there 

 are units in g : and then the remaining arches will go on to 

 terminate in the same points respectively, as those already ta- 

 ken, until as many are taken, in all, as there are units in 2g ; 

 then the remainder will go on as before, until the number taken 

 is equal to 3g* ; and so on, until the number becomes equal to 

 tg, or 11. Q. E. D. 



Cor. It appears from the demonstration, that when the 

 numbers m and n are prime to each other, the multiple arches 

 divide the circumference into as many equal parts as there are 

 units in n: and that, when m and n are not prime to each other, 



if the fraction -^, reduced to its lowest terms, be denominated 

 n 



k 

 by — , then the multiple arches divide the circumference into 



g 

 as many equal parts as there are units in g. And if t denote 



the greatest common measure of the terms of the fraction —, 



n 

 the number of arches which terminate in each point of division, 

 will be equal to t. 



Lemma II. 



IF, in the c ire ii inference of any circle, there be taken 

 three arches, which have equal differences ; it will be, 

 As the diameter, is to the supplementary chord of their 

 common difference ; so is the chord of the mean arch;, to 

 half the sum of the chords of the extreme arches. 



DEMOjYS tra tiojy. 



Let O (Plate III. Fig. 2.) be any circle ; and in its circumfer- 

 ence let there be taken any three equidifferent arches, as DK, 

 EH, and FG, so situated that DE = KH, and EF= HG. Draw 

 the chords DK, EH, and FG. From E draw the diameter FN, 



