a INVESTIGATION of PORISMS. 174 
extent, and purfuing Porifms to thofe extreme cafes, where 
the indeterminate magnitudes increafe ad infinitum; into which 
_  ftate Dr Stmson probably did not think it fafe to follow 
them, and was thereby deprived of no inconfiderable help to- 
‘ward the fimplifying of his conftructions. If therefore it can 
be fhewn, that this help may be obtained without any facrifice 
_ of geometrical accuracy, it will be fome improvement in this 
branch of the analyfis. 
t Tue 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 
% DE, FK, KL, being given in pofition, together with two 
points H and M, in two of thefe lines, to infle€t 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 @ to . 
Now, in order to find whether there be any Porifm conneé- 
ed with this problem, fuppofe that there is, and that the fol- 
_ lowing propofition is true. 
° 
PROP. IV. PORISM. Fie. 5. 
22. Two points A and B, and two ftraight lines DE and 
F K, being given in pofition, 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 AE and BE, infleéted 
from the points A and B to any point whatfoever of the 
line DE, fhall cut off from the other lines F K and L K, 
fegments, H G and MN, adjacent to the given points H 
and M, having to one another the given ratio of a to 6. 
SPERST, let AP, BE’ ‘be infleGed to the point E’, fo that 
AE may be parallel to FK, then fhall E’ B be parallel to K L, 
Wot. lil: Z, the 
