BY MARTIN GARDESTER, C.E. 79 



known closed (11 l)'gons. And, putting a n the point in which 

 the straight line a n _ 1 a 1 o n _ i cuts L n , it is evident that for the 

 w'gon a l a 2 . . . o^ a n a^ we must have any answerable point o n 

 situated on the straight line a an which is the last side of 



n 1 1 n 1 



the other known closed (?i l)'gon. 



And if Cj c a . . . . c e 1 be any closed rc'gon, having its first n 1 

 sides passing through the n 1 given points, and having its first 

 n 1 angular points on the n 1 given lines; it is evident that by 

 drawing a straight line from b through c , and by producing c^ c^ 

 to cut the line o n _ } ^ n _ l a 1 ; then will this point of intersection 

 and the line b c n be answerable positions for o n and L^ : (because 

 we have the three closed rz/'gons a l a .... a a 1? b 1 b^ . . . b n b^ 

 i C 2 '" C n c i' ^filling the conditions). 



Hence the following porism : 



PORISM 7. 



If a closed n'gon have its first n 1 angular points on n 1 

 given straight lines, and its first n 1 sides passing through n 1 

 given points : then tv:o straight lines and a point in each of tliem 

 can be found, such that if from either one of these determined points 

 (in a determined line) we draw a line L n through the n th angular 

 point of the n'gon, and chat we produce the n th side of the n'gon 

 to cut the other of the two determined lines in a point o n , and that 

 we regard th& line L and point o n as fixed : we can deform, the n'gon 

 so that its n sides v:ill continue through the n points composed of 

 the n 1 given ones and, the determined one o , and its angular 

 points move along the n straight lines composed of the n 1 given 

 ones and the determined one L B . 



13. Be-considering the problem, it is evident we can use the 

 following system of equations : 



