104 On the ORIGIN and 



D H, that is, to the fquares of DL and L H, the rectangle 

 KHK' is equal to the fquare of H B, fo that H B touches the 

 circle B K K'. But B G is at right-angles toHB; therefore the 

 centre of the circle BKR' is in the line BG; and it is alfo in 

 the line D E ; therefore G is the centre of the circle B K K', and 

 GB is equal to G K. 



Thus we have an inftance of a problem, and that too 

 a very fimple one, which is in general determinate, admitting 

 only of one folution, but which neverthelefs, in one particular 

 cafe, where a certain relation takes place among the things 

 given, becomes indefinite, and admits of innumerable folu- 

 tions. The proportion which refults from this cafe of 

 the problem is a Porifm, according to the remarks that were 

 made above, and in effect will be found to coincide with the 

 66th proportion inDrSiMSON's Reftoration. It may be thus 

 enunciated : " A circle ABC being given in pofition, and alfo a 

 ftraight line D E, which does not cut the circle, a point K may 

 be found, fuch that if G be any point whatever in the line D E, 

 the ftraight line drawn from G to the point K, fhall be equal to 

 the ftraight line drawn from G, touching the circle ABC' 



10. The following Porifm is alfo derived in the fame man- 

 ner from the folution of a very fimple problem : 



PROP. II. P R O B. Fig. 2. 



A triangle ABC being given, and alfo a point D, to 

 draw through D a ftraight line D G, fuch, that, perpendi- 

 culars being drawn to it from the three angles of the tri- 

 angle, viz. A E, B G, C F, the fum of the two perpendi- 

 culars on the fame fide of D G, fhall be equal to the re- 

 maining 



