INVESTIGATION of PORISMS. 161 



with one another entirely, and leave the queftion of confe- 

 quence unrefolved. 'But though this circumftance muft have 

 created confiderable embarrafTment to the geometers who firft 

 obferved it, as being perhaps the only inftance in which the 

 language of their own fcience had yet appeared to them ambi- 

 guous or obfcure, it would not probably be long till they found 

 out the true interpretation to be put on it. After a little reflec- 

 tion, they would conclude, that fince, in the general problem, 

 the magnitude required was determined by the interfeclion of 

 the two lines above mentioned, that is to fay, by the points 

 common to them both ; fo, in the cafe of their coincidence, as 

 all their points were in common, every one of thefe points 

 muft afford a folution ; which folutions therefore muft be 

 infinite in number \ and alfo, though infinite in number, they; 

 muft all be related to one another, and to the things given, 

 by certain laws, which the pofition of the two coinciding lines 

 muft neceffarily determine. 



On enquiring farther into the peculiarity in the ftate of the 

 data which had produced this unexpected refult, it might like- 

 wife be remarked, that the whole proceeded from one of the 

 conditions of the problem involving another, or neceffarily 

 including it; fo that they both together made in fa£t but one, 

 and did not leave a fufBcient number of independent conditions, 

 to confine the problem to a fingle folution, or to any determi- 

 nate number of folutions. It was not difficult afterwards to 

 perceive, that thefe cafes of problems formed very curious pro- 

 portions, of an intermediate nature between problems and 

 theorems, and that they admitted of being enunciated fepa- 

 rately, in a manner peculiarly elegant and concife. It was to 

 fuch proportions, fo enunciated, that the ancient geometers 

 gave the name of Porifms. 



9. This deduction requires to be illuftrated by examples. 

 Suppofe therefore that it is propofed to refolve the following 

 problem : 



Vol. III. X PROP. 



