INVESTIGATION of PORISMS. 169 



above, that on the hypothefis of that proportion, LFM (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 ABC, and the points E and 

 D, remaining as in the laft connruclion, (fig. 4.) if through 

 D we draw any line whatever HDB, meeting the circle in B 

 and H, and if the lines E B, E H be alfo drawn, thefe lines will 

 cut off equal circumferences B F and H G. Let F C be drawn, 

 and it is plain from the foregoing analyfis, that the angles 

 D F C, C F B are equal. Therefore if O G, O B be drawn, the 

 angles B O C, COG are equal, and confequently the angles 

 D O B, D O G. In the fame manner, by joining A B, the angle 

 D B E being bifected by B A, it is evident, that the angle A O F 

 is equal to the angle A O H, and therefore the angle F O B to 

 the angle HOG, that is, the arch F B to the arch H G. 



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 Euclid'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, (§ i2\) all connected, as many remarkable 

 properties of the circle are, with the harmonical 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 



Vol. III. Y be 



* Simson De Porifmatibus, Prop. 53. 



