INVESTIGATION of PORISMS. 201 



37. This Porifm may alfo be extended to figures of any number 

 of fides, and may be enunciated more generally thus : " A rec- 

 tilineal figure of any number of fides, as m, being given and 

 three ftraight lines being alfo given in pofition, m — 3 ftraight 

 lines may be found given in pofition, fo that innumerable rec- 

 tilineal figures may be defcribed, fimilar to the given reclilineal 

 figure, and having their angles on the ftraight lines given in po- 

 fition." 



Hence alfo this theorem: " If any two reclilineal figures 

 be defcribed fimilar to one another, and if ftraight lines be 

 drawn, joining the equal angles of the two figures, innumera- 

 ble rectilineal figures may be defcribed, which will have their 

 angles on thefe lines, and will be fimilar to the given reclilineal 

 figures ; and the fegments of the lines given in pofition, inter- 

 cepted between any two of thefe figures, will have conftantly the 

 fame ratio to one another." 



As a Locus, the fame propofition admits of a very fimple 

 enunciation, and has a remarkable affinity to that with which 

 Euclid appears to have introduced his firft book of Porifms. 

 ** If three of the angles of a reclilineal figure, given in fpecies, 

 be upon three ftraight lines given in pofition, the remaining 

 angles of the figure will alfo be on ftraight lines, given in pofi- 

 tion." 



If the reclilineal figures here referred to be fuch ; as may be 

 infcribed in a circle, or in fimilar curves of any kind, agreeably 

 to the hypothefis of the problem *, by which rhefe latt Porifms 

 were fuggefted, we fhall have a number of other Porifms re- 

 fpecling ftraight lines given in pofition, which cut off, from 

 innumerable fuch curves, fegments that are given in fpecies. 

 A great field of geometrical inveftigation is, therefore, opened 

 by the two preceding propofitions, which, however, we muft 

 at prefent be content to have pointed out. 



38. A question nearly connecled with the origin of Po- 

 rifms ftill remains to be refolved, namely, from what caufe 



Vol. III. C c has 



* Prin. Math. lib. i. prop. 29. 



