170 On the ORIGIN and 
be refolved : Two points D and E are given, (fig. 3.) from which 
two lines are to be inflected, and a circumference A BC,in which 
thefe lines are to meet, as alfo a ratio, which they are to haveto one 
another *. Now, thefe conditions are all independent of each 
other, fo that any one of them may be changed, without any 
change whatever in the reft. This at leaft is true in general ; 
but neverthelefs in one cafe, viz. when the given points are fo 
related to one another, that the re¢tangle under their diftances 
from the centre, is equal to the fquare of the radius of the cir- 
cle, it follows from the foregoing analyfis, that the ratio which 
the infleGted lines are to have to one another, is no longer a 
matter of choice, but is a neceflary confequence of this  difpofi- 
tion of the points. For if any other ratio were nowaffigned 
than that of AO to OD, or, which is the fame, of EA to AD, 
it would eafily be fhewn, that no lines having that ratio could. 
be infle¢ted from the points E and D, to any point in the circle 
ABC. Two of the conditions are therefore reduced into one ; 
and hence it is that the problem is indefinite. 
16. From this account of the origin of Porifms, it follows, 
that a Porifm may be defined, 4 propojition affirming the poffibi- 
lity of finding Juch conditions as will render a certain problem inde- 
terminate, or capable of innumerable folutions. 
To this definition, the different characters which Pappus has 
given will apply without difficulty. The propofitions defcribed 
in it, like thofe which he mentions, are, ftrictly {peaking, nei- 
ther theorems nor problems, but of an intermediate nature be- 
tween both; for they neither fimply enunciate a truth to be 
demonftrated, nor propofe a queftion to be refolved; but are 
affirmations of a truth, in which the determination of an un- 
known quantity is involved. In as far therefore as they affert, 
that a certain problem may become indeterminate, they are of 
the nature of theorems ; and in as far as they feek to difcover 
the 
* The given points, and the centre of the given circle, are underftood, throughout, to 
be in the fame ftraight Vine. 
