i88 On the ORIGIN and 



mean proportional between the fum of the fquares of the fides 

 of the triangle, and the fum of the fquares of the above men- 

 tioned perpendiculars." 



29. But to return to the fubject of Porifms : It is evident 

 from what has now appeared, that in fome inftances at leafi, 

 there is a clofe connection between thefe propofitions and the 

 maxima or minima, and, of confequence, the impoffible cafes, 

 of problems. The nature of this connection requires to be 

 further inveftigated, and is the more interefting, that the tran- 

 sition from the indefinite, to the impoffible cafes of a problem 

 feems to be made with wonderful rapidity. Thus, in the firft 

 proportion, though there be not, properly fpeaking, an impof- 

 fible cafe, but only one where the point to be found goes off 

 ad infinitum, we may remark, that if the given point F be any 

 where out of the line HD, the problem of drawing GB equal 

 to GF is always poffible, and admits juft of one folution ; but 

 if F be in the line DH, the problem admits of no folution at 

 all, the point being then at an infinite diftance, and therefore 

 impoffible to be affigned. There is however this exception, 

 that if the given point be at K, in this fame line DH, determi- 

 ned by making DK equal to DL, then every point in the line 

 DE gives a folution, and may be taken for the point G. Here 

 therefore the cafe of innumerable folutions, and the cafe of no 

 folution, are as it were conterminal, and fo clofe to one ano- 

 ther, that if the given point be at K, the problem is indefinite, 

 but that if it remove ever fo little from K remaining at the 

 fame time in the line DH, the problem cannot be refolved. 



1 had obferved this remarkable affinity between cafes, which 

 in other refpects are diametrically oppofite, in a great variety of 

 inftances, before I perceived the reafon of it, and found, that 

 by attending to the origin which has been affigned to Porifms, 

 I ought to have difcovered it a priori. It is, as we have feen,. 

 a general principle, that a problem is converted into a Porifm, 

 when one, or when two, of the conditions of it, neceffarilv in- 

 volve in them fome one of the reft. Suppofe then that two of 



the 



