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 B C,in which 

 thefe lines are to meet, as alfo a ratio, which they are to have to 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 lead is true in general ; 

 but neverthelefs in one cafe, viz. when the given points are fo 

 related to one another, that the rectangle 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 inflected lines are to have to one another, is no longer a 

 matter of choice, but is a neceffary confequence of this difpofi- 

 tion of the points. For if any other ratio were now afligned 

 than that of A O to O D, or, which is the fame, of E A to A D, 

 it would eafily be fhewn, that no lines having that ratio could 

 be inflecled 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, A propofition affirming the pofifibi- 

 lity ofi finding finch conditions as will render a certain problem inde- 

 terminate, or capable of innumerable fiolutions. 



To this definition, the different characters which Pappus has 

 given will apply without difficulty. The proportions defcribed 

 in it, like thofe which he mentions, are, ftrictly fpeaking, 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 afTert, 

 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 understood, throughout, to 

 be in the fame flraight line. 



