INVESTIGATION of PORISMS. 189 
the conditions are exactly in that ftate, which determines the 
third; then, while they remain fixed or given, fhould 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 hypothefis 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- 
poflible. Wherever therefore any problem admits both «f an in- 
— determinate, and an impoffible 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 refpecting Porifms, will be more fully illuftrated, when 
we treat of the algebraic invettigation of thefe propofitions. 
30. Ir is fuppofed above, that ¢wo 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. Buta Porifm may fometimes be fo fimple, as to arife- 
from the mere coincidence of ove 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, vz. the centre of gravity of the given tri- 
angle, renders the problem indeterminate; but as there is no. 
- relation of diftance, or pofition, between thefe points, that may 
not exift, fo the problem has no impoflible 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 
impoffible; and if fo, the connection between thefe cafes, 
which has been already explained, never fails to take place. 
31. ANOTHER 
