- 188 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. Bur to return to the fubject of Porifms: It is evident 
from what has now appeared, that in fome inftances at leaft, 
there is a clofe connection between thefe propofitions and the: 
maxima or minima, and, of confequence, the impoflible cafes, 
-of problems. The nature of this conneétion requires. to be: 
further inveftigated, and is the more interefting, that the tran- 
fition from the indefinite, to the impoflible cafes of a problem. 
feems to be made with wonderful rapidity. Thus, in the firft 
propofition, though there be not, properly {peaking, an impof- 
Gible 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- — 
impoflible 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. 
I wap 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, neceffarily in- 
volve in them fome one of the reft.. Suppofe then that two of — 
the 
