124 GEOMETRICAL PORISMS, 



if fome condition, unconne6led with the hypothefis of the pro- 

 pofition, be added, there will be formed a problem perfedlly li- 

 mited in its nature. 



The method of applying the porlfms to the folution of many 

 problems is obvious enough ; and, as fome of thefe may be of 

 a very extenfive nature, and fuch as many others can be reduced 

 to, therefore the utility of the porifins will by this means be 

 greatly extended. The condition that may be joined to the hy- 

 pothefis of each porifmatic propofition, it is evident, may be 

 greatly varied : And, hence, it were eafy to form abundance of 

 problems, differing from any hitherto propofed: but this would 

 extend the paper to too great a length. We Ihall therefore on- 

 ly give a few examples, of which, let the firft be the Sedio Ra~ 

 tionis of the ancient geometers. 



PROP. XL PROBLEM, Fig. i6. PI. III. 



Two ftraight lines AB, AC are given by pofition, and two 

 points B, C are given in thefe lines. It is required to 

 draw a line through P, a given point, without them, to 

 meet them in D and E, fo that BD may have to CE the gi- 

 ven ratio of M to N. 



Because the ratio of BD to CE is given ; if a circle be de- 

 fcribed through the points A, B, C, there is given a point H in 

 the circumference, fuch, that the points A, H, D, E are in a 

 circle, (Prop, i.) therefore if HD, HA be joined, the angle 

 HDP is equal to HAE, that is, to a given angle ; now H and 

 P are given points, therefore D is in the circumference of a gi- 

 ven circle, but it is alfo in AB, a line given by pofition ; there- 

 fore D is a given point, and PE is given by pofition, wliich was 

 to be fournd. 



Con- 



