INVESTIGATION of PORISMS. 161 
with one another entirely, and leave the queftion of confe- 
quence unrefolved. -But though this circumftance muft have 
created confiderable embarraflment 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 interfection 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 
mutt neceflarily determine. 
On enquiring farther into the peculiarity in the ftate of the 
data which had produced this unexpefed refult, it might like- 
__ wife be remarked, that the whole proceeded from one of the 
. conditions of the problem involving another, or neceflarily 
including it; fo that they both.together made in fact but one, 
and did not leave a fufficient 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 
q perceive, that thefe cafes of problems formed very curious pro- 
- pofitions, of an intermediate nature between problems and 
a theorems, and that they admitted of being enunciated fepa- 
_ rately, in a manner peculiarly elegant and concife. It was to 
— fach propofitions, fo enunciated, that the ancient geometers 
_ gave the name of Pori/ms. 
9g. Tuis deduction requires to be illuftrated by examples. 
_ Suppofe therefore that it is propofed' to refolve the following 
_ problem: ; 
Vou. IIL. “ite PROP, 
a 
