83 



fore PF/j is equal to the sum of P'Fy and P'Cy, or to F¥'p' with 

 twice HC'Vj : but FP, FT', and C"K are all equal, and therefore 

 Pg^ is equal to P'q' and twice KA", or to P'q' and twice hi. Whence 

 the proposition is manifest. 



Schol. The angle E'CU is half the angle /"CP or f'C'P', and 

 therefore by lent. 4, the semidiameter CL is equal to the straight 

 line/' P orf'P'. This gives another and easier method of finding the 

 , point L. 



Hence every arc of a hyperbola may be found by means of 

 two elliptic arcs. This beautiful theorem was discovered by Landen, 

 and is now for the first time demonstrated geometrically ; but the 

 manner in which it is stated by him and by succeeding writers differs 

 from the above, and is much more complicated. The different forms 

 under which they have presented it may be easily deduced from the 

 preceding, by means of the theorem of Fagnani, of which a geome- 

 trical demonstration was first given in Vol. IX. of these Transactions, 

 by the Lord Bishop of Cloyne. 



Decmier 23, 1829. 



