225 DIVISION OF A CIRCLE. 



exhibiting them in one view, and, which is equally defirabTe* 

 of deriving them from a general principle, I have ventured to 

 offer it for infertion in your Journal. 



JOHN GOUGH. 

 Reeth, near Richmond, York/Jure, 

 Augujl 28. 



Proposition I. Theorem. 

 Divlfion of an Let A F be the arch of a circle, (See Fig. 1 , PI. XII.) A P 

 fmo^oVam 1 * a ta °gent at A; FP a perpendicular toAP, then A P is 

 having tfaeir equal to the fine of A F ; F P, the part of the perpendicular 

 ^ s, °^^^ intercepted by the tangent and the point F in the arch, is 

 given ratio. equal to its verfed fine ; and the fame line, P M, intercepted 

 again by the circle in M, is equal to the verfed fine of its fup- 

 plement. 



Demonftration. 



Draw the diameter AK, and the fine FS perpendicular 

 thereto ; alfo from the center O, draw O L at right angles to 

 PMj then, fince PA touches the circle in A, PAK is a 

 right angle, (Euc. 16. iij.) Aifo, the angles F PA, ASF, 

 are right, by conduction ; therefore A S F P is a parallelo- 

 gram, the oppofite fides of which are equal, namely, AP = 

 the fine S F, and P F = the verfed fine A S, (Euc. 34-. i.) 



Again, fince O L is perpendicular to P M, it is parallel to 

 A P and S F, therefore P L = A O, or OK; and F L = 

 SO, (Euc. 34. I.)— But FL = LM, (Euc. 3. III.) Con- 

 fequently P M =: S K, or the verfed fine of the fupplement 

 AF. Q. E. D. 



Proposition II. Theorem. 



If A F B be an arch of a circle, (See Fig. 2.) and A P, B R, 

 be tangents at A and B, from any point, F, in the circum- 

 ference, draw FP, F R, perpendicular to the two tangents, 

 and F Q alfo perpendicular to the chord A B, then will the 

 reftangle PF X FR = F~Ql*; and the rectangles AP x 

 B R, and A Q x Q B, will alfo be equal. 



g Demonstration, 



Join A F, F B, and the triangles P F A, Q F B, are equi- 

 angular, becaufe they are right-angled at P and Q, by con- 

 ftruaion ; and the angles P A F, Q B F, are equal, (Euc* 

 32, III.) 



therefore, 



