INVESTIGATION of PORISMS. 177 



extent, and purfuing Porifms to thofe extreme cafes, where 

 the indeterminate magnitudes increafe ad infinitum; into which 

 flate Dr Sim son probably did not think it fafe to follow 

 them, and was thereby deprived of no inconsiderable help to- 

 ward the Amplifying of his constructions. If therefore it can 

 be (hewn, that this help may be obtained without any facrifice 

 of geometrical accuracy, it will be fome improvement in this 

 branch of the analyfls. 



The Porifm juft mentioned may be confidered as having 

 occurred in the folution of a problem. Suppofe it were requi- 

 red ; two points A and B, (fig. 5.) and alfo three ftraight lines 

 D E, F K, K L, being given in pofition, together with two 

 points H and M, in two of thefe lines, to inflect from A and B 

 to a point in the third line, two lines that fhall cut off from 

 K F and K L two fegments, adjacent to the given points H and 

 M, having to one another the given ratio of a, to /3. 



Now, in order to find whether there be any Porifm connect- 

 ed with this problem, fuppofe that there is, and that the fol- 

 lowing propoiition is true. 



P R O P. IV. PORISM. Fig. 5. 



22. Two points A and B, and two ftraight lines D E and 

 F K., being given in poiition, and alfo a point H in one of 

 them, a line LK may be found, and alfo a point in it M, 

 both given in pofition, fuch, that A E and B E, inflected 

 from the points A and B to any point whatfoever of the 

 line D E, fhall cut off from the other lines F K and L K, 

 fegments, H G and M N, adjacent to the given points H 

 and M, having to one another the given ratio of a, to j2. 



First, let A E', BE' be inflected to the point E', fo that 



A E' may be parallel to F K, then fhall E' B be parallel to K L, 



Vol. III. Z the 



