,80 On the ORIGIN and 



ratio of H G to M N, thefe lines muft: both become infinite at 

 the fame time, and in the fecond, that for the fame reafon they 

 muft both vanifhat the fame time, we might have been accufed 

 of departing from the ftricl: form of reafoning employed in 

 the ancient geometry. But when the thing is dated as above, and 

 it is proved, that when A E' does not meet K F, it is impoffiblc 

 for BE' to meet M L ; and again, that when AE" pafTes through 

 H, it is impoffible for B E" not to pafs through M, the air of 

 paradox is entirely removed, and the tracing of the law of 

 continuity is rendered perfectly confident with the utmoft fe- 

 verity of geometrical demonftration. 



Dr Si ms on has applied this Porifm very ingenioufly to the 

 folution of the fame problem from which it is here fuppofed to 

 be derived * ; and it is worthy of remark, that fuppofing the 

 points A and H, and the lines D E and F K to be as in the fi- 

 gure of this Porifm, if the third of the given lines be not pa- 

 rallel to BE', that problem can always be refolved, and ad- 

 mits of two folutions ; but if it be parallel to B E', the problem 

 either becomes impoffible, or a Porifm ; that is, it either admits 

 of no folution, or of an infinite number. We fhall foon have 

 occafion to extend the fame obfervation to other Porifms. 



Another general remark which I have to make on the ana- 

 lyfis of Porifms is, that it frequently happens, as in the laft 

 example, that the magnitudes required may all, or a part of 

 them, be found by confidering the extreme cafes ; but for the 

 difcovery of the relation between them, and the indefinite 

 magnitudes, or res bmumera y we muft have recourfe to the hy- 

 pothefis of the Porifm in its mod general, or indefinite form, 

 and muft endeavour fo to conduct the reafoning, that the inde- 

 finite magnitudes lhall at length wholly difappear, and leave a 

 propofition, containing only a relation of determinate magni- 

 tudes to one another. Now, in order to accompliih this Dr 



SlMSON 

 * Opera Reliqua. de Forifmatibus., prop. 2£ 



