INVESTIGATION of PORISMS. 169 
above, that on the hypothefis of that propofition, L F M (fig. 3.) 
is a right angle, and L and M given points. 
14. Hence alfo an example of the derivation of Porifms 
from one another. For the circle A BC, and the points E and 
D, remaining as in the laft conftruction, (fig. 4.) if through 
D we draw any line whatever HDB, meeting the circle in B 
and H, and if the lines EB, EH be alfo drawn, thefe lines will 
cut off equal circumferences B F and HG. Let FC be drawn, 
and it is plain from the foregoing analyfis, that the angles 
DFC, CFB are equal. Therefore if OG, OB be drawn, the 
angles BOC, COG are equal, and confequently the angles 
DOB, DOG. Inthe fame manner, by joining A B, the angle 
DBE being bifected by B A, it is evident, that the angle AOF 
is equal to the angle A OH, and therefore the angle FOB to 
the angle HOG, that is, the arch F B to the arch HG. 
Now, it is plain, that if the circle ABC, and one of the 
points D or E be given, the other point may be found; 
therefore we have this Porifm, which appears to have been 
the laft but one in the third book of Euctip’s Porifms™*. 
“ A point being given, either without or within a circle given 
in pofition, if there be drawn, any how through that point, a 
line cutting the circle in two points; another point may be 
found, fuch, that if two. lines be drawn from it to the points, 
in which the line already drawn cuts the circle, thefe two lines 
will cut off from the circle equal circumferences.” 
THERE are other Porifms that may be deduced from the fame 
original problem, (§ 12.) all conneéted, as many remarkable 
properties of the circle are, with the barmonical divifion of the 
diameter. 
15. THE preceding propofition alfo affords a good illuftra- 
tion of the general remark that was made above, concerning the 
conditions of a problem being involved in one another, in the 
Porifmatic, or indefinite cafe. Thus, feveral independent condi- 
tions are here laid down, by help of which the problem is to 
Vot. IIL x be 
* Simson De Porifmatibus, Prop. 53. 
