INVESTIGATION of PORISMS. 189 



the conditions are exaclly in that ftate, which determines the 

 third ; then, while they remain fixed or given, ihould that 

 third one be fuppofed to vary, or differ, ever fo little, from the' 

 ftate required by the other two, a contradiction will enfue. 

 Therefore if, in the hypothecs of a problem, the conditions be 

 fo related to one another as to render it indeterminate, a Porifm 

 is produced ; but if, of the conditions thus related to one ano- 

 ther, fome one be fuppofed to vary, while the others continue 

 the fame, an abfurdity follows, and the problem becomes im- 

 pofiible. Wherever therefore any problem admits both of an in- 

 determinate, and an impoffibh cafe, it is certain, that thefe cafes 

 are nearly related to one another, and that fome of the condi- 

 tions by which they are produced, are common to both. This affi- 

 nity, which feems to be one of the moft remarkable circum- 

 ftances refpecling Porifms will be more fully illuftrated, when 

 we treat of the algebraic inveitigation of thefe propofitions. 



30. It is fuppofed above, that two of the conditions of a 

 problem involve in them a third, and wherever that happens, 

 the conclufion which has been deduced will invariably take 

 place. But a Porifm may fometimes be fo fimple, as to arife 

 from the mere coincidence of one condition of a problem with 

 another, though in no cafe whatever, any inconfiftency can take 

 place between them. Thus, in the fecond of the foregoing 

 propofitions, the coincidence of the point given in the problem 

 with another point, viz. the centre of gravity of the given tri- 

 angle, renders the problem indeterminate; bat as there is no 

 relation of diftance, or pofition, between thefe points that may 

 not exift, fo the problem has no impofiible cafe belonging to 

 it. There are, however, comparatively but few Porifms fo fim- 

 ple in their origin as this, or that arife from problems in which 

 the conditions are fo little complicated ; for it ufually happens, 

 that a problem which can become indefinite, may alfo become 

 impofiible; and if fo, the connection between thefe cafes, 

 which has been already explained, never fails to take place. 



31, Another 



