DIVISION Of A CIRCL1. gJJ 



Conflfu&ion. 



• Draw the radius AO, and the tangent BE; in A O take Diyifion of an 

 A I equal to the given line, u ; and making I V perpendicular * xh ofacircle 

 to A O, let it meet BE in V ; xlraw F V to bifecl: the angle having thwr **■ 

 E V I, and let it cut the given arch in F; then will AF, F B, flne8 > or cofines, 

 betherequiredarcl.es. £$6** 



Demonftration. 



Draw the tangent AG, which is parallel to I V, alfo make 

 F P, F R perpendicular to A G, B E, and let P F produced 

 meet I V in H. 



Then fince AG, IV, are parallels, and the angle H P A 

 is right, FHV is alfo a right angle, (by Euc. 29. I.) there- 

 fore it is equal to the angle F R V, (by conftruction.) 



But the angles R V F, HVF, are alfo equal, (by conltruc- 

 tion) ; confequently the triangles R F V, HFV, are equi- 

 angular ; and they have one fide common, namely the fide 

 V F ; therefore F R = F H, (Sue. 4. VI.) and PF-fFR 

 = PH=AI, (Euc. 34?. I.) =z u, (by conftruaion.) 



But the fum of the verfed fines of AF, FB, .is equal to 

 P F -f F R, (by Prop. I.) therefore this fum is equal to the 

 given line, u. Q. E. D. 



Limitation.— -If A I be greater Mian the verfed fine of the 

 whole arch A B, the point F will evidently fall in the oppo- 

 site fegment, and the confiruclion will be impoffible. 



Again, fince the angle I VE is equal to the angle A O B, 

 draw the radius O C, to bifect the angle A O B, and it will 

 evidently be perpendicular to VF; therefore L C, a tangent 

 at C, will be parallel to V F ; confequently if A I be fo taken* 

 that V may lie in B L produced, the conftru6tion will alfo be 

 impofiible; which will therefore happen when u is lefs than 

 twice the verfed fine of the arch A C, or B C. 



Carol. Since the fum of the verfed fines of two arches is 

 the fame with the difference of the diameter and the fum of 

 the cofines, if the latter fum be given the problem may be 

 conftrucled by the laft proportion. 



Proposition VII. Problem. 



To divide A F B, a given arch of a circle, (Fig. 7.) into 

 iwo parts, fo that the fum of their fines may be equal to a 

 given right line, iv f 



Conftruftion. 



