i68 On the ORIGIN and 



13. We are therefore to enquire, in what circumftances the 

 point L may coincide with the point A, and the point M with 

 with the point C, and of confequence the circumference LFM 

 with the circumference A B C. 



On the fuppofition that they coincide, E A is -to A D, and 

 alfo E C to C D, as « to (3 \ and therefore E A is to E C as A D 

 to C D, or, by converfion, E A to A C as A D to the excefs of 

 C D above A D, or to twice D O, O being the centre of the cir- 

 cle ABC. Therefore alfo, E A is to A O, or the half of A C, 

 as AD to DO, and E A together with A O, to A O, as A D 

 together with DO, to D O ; that is, E O to A O as A O to D O, 

 and fo the reclangle E O. O D equal to the fquare of A O. 



Hence, if the fituation of the given points E and D, (fig. 4.) 

 in refpecl of the circle ABC, be fuch, that the reclangle 

 E O. O D is equal to the fquare of A O, the femidiameter of 

 the circle j and if, at the fame time, the given ratio of a to (2 

 be the fame with that of E A to A D, or of E C to C D, the 

 problem admits of innumerable folutions ; and as it is mani- 

 feft, that if the circle ABC, and one of the points D or E be 

 given, the other point, and alfo the ratio which is required to 

 render the problem indefinite, may be found, therefore we have 

 this Porifm : " A circle ABC being given, and alfo a point D, 

 a point E may be found, fuch, that the two lines inflecled 

 from thefe points to any point whatever in the circumference 

 A B C, fhall have to one another a given ratio, which ratio is 

 alfo to be found." 



This Porifm is the fecond in the treatife De Porifmatibus^ 

 where Dr Simson gives it, not as one of Euclid's propofitions, 

 but as an illuflration of his own definition. It anfwers equally well 

 for the purpofe I have here in view, the explaining the origin 

 of Porifms ; and I have been the more willing to introduce it, 

 that it has afforded me an opportunity of giving what feems 

 to be the fimpleft invefligation of the fecond proportion in the 

 fecond book of the Loci Plani, by proving, as has been done 



above, 



