TIE 
INVESTIGATION of PORISMS. 201 
37. Tuts 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 flraight 
lines may be found given in pofition, fo that innumerable rec- 
tilineal figures may be defcribed, fimilar to the given re@ilineal 
pene, and having their angles on the gaa lines given in po- 
“fition.”’ 
Hence alfo this theorem: “ If any two reétilineal 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 re¢tilineal 
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 
Euciip appears to have introduced his firft book of Porifms. 
“ If three of the angles of a reGiilineal figure, given in fpecies, 
be upon three ftraight lines given in pofition, the remaining 
angie of the boi will alfo be on ftraight lines, given in pofi- 
tion.’ 
Ir the rectilineal 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 thefe laii Porifms 
were fuggefted, we fhall have a number of other Porifms re- 
fpefting ftraight lines given in pofition, which cut off, from 
inAumerable fuch curves, fegments that are given in {pecies. 
A great field of geometrical inveftigation is, therefore, opened 
by the two preceeding propofitions, which, however, we muft 
at prefent be'content to have pointed out. 
38. A QUESTION nearly connected with the origin of Po- 
rifms ftill remains to’ be refolved, namely, from what caufe 
Vor. II. Cc has 
* Prin. Math. lib. 1. prop..29. 
