INVESTIGATION of PORISMS. 171 



the conditions by which that is brought about, they are of the 

 nature of problems. 



17. In the preceding definition alfo, and the inftances 

 from which it is deduced, we may trace that imperfect de- 

 fcription of Porifms which Pappus afcribes to the later geo- 

 meters, viz. " Porifma eft quod deficit hypothefi a theore- 

 " mate locali." Now, to underftand this, it muft be ob- 

 ferved, that if we take the converfe of one of the propofitions 

 called Loci, and make the conftruction of the figure a part of 

 the hypothefis, we have what was called by the ancients a Lo- 

 cal Theorem. And again, if, in enunciating this theorem, 

 that part of the hypothefis which contains the conftruction" be 

 fupprefTed, the proportion arifing from thence will be a Porifm ; 

 for it will enunciate a truth, and will alfo require, to the full 

 understanding and inveftigation of that truth, that fomething 

 fhould be found, viz. the circumftances in the conftruction, fup- 

 pofed to be omitted. 



Thus, when we fay; If from two given points E and 

 D, (fig. 4.) two lines E F and F D are inflected to a third point 

 F, fo as to be to one another in a given ratio, the point F is in 

 the circumference of a circle given in pofition : we have a Lo- 

 cus. 



But when converfely it is faid ; If a circle ABC, of 

 which the centre is O, be given in pofition, as alfo a point E, 

 and if D be taken in the line E O, fo that the rectangle EO.OD 

 be equal to the fquare of A O, the femidiameter of the circle ; 

 and if from E and D, the lines E F and D F be inflected to 

 any point whatever in the circumference ABC; the ratio of 

 E F to D F will be a given ratio, and the fame with that of E A 

 to A D : we have a local theorem. 



And, laftly, when it is faid; If a circle A B C be given 

 in pofition, and alfo a point E, a point D may be found, fuch, 

 that if «the two lines E F and F D be inflected from E and D to 

 any point whatever F, in the circumference, thefe lines fhall 



Y 2 » have 



