BY MARTIN GARDINER, C.E. 97 



Now let i be the intersection of the lines r, s and o 



|-1 i-2 $n-2 n 1 



On ; and let ^^ be the point in which i 9 & n _ l again cuts the 

 conic. Since a^ 5 a ^3^-2 a i * s an ^^"bed 4'gon, it follows 

 that the point (see theorem 8) r 1 in which a b t cuts r s. 



Ji-*- 1 at*-! fi-2 -511-2 



is known. And since a n _ l ^^ a^ a n a^^ is an inscribed 4'gon, 

 the point s, in which a. b , cuts o o is known. Hence, 



ftt-l 1 4n-l n 1 n 



as the points r^^ and s are known, the points a in which the 

 straight, line r^ ^^ cuts the conic are known. 



Ejp I may also remark that porism 1 is prominently evident 

 from this method of solution for (from the well-known case of 

 theorem 8 in which n = 4) the straight line r s^ is such that 



*>MT- J 3 -5*1 2 



if we assume one of the points o n _ l5 <> n anywhere therein, we can 



find a corresponding position for the other one in the same line 

 which will render porismatic the inscription of the closed 4'gon, 

 and .'. also that of the closed w'gon. 



Second method of solution. 



Analysis. Let a a^ . . . . a^ a^ be a closed w'gon inscribed in 

 the desired manner (n being regarded as an even number). 

 Suppose that through 1 we draw the chord t b. 2 parallel to Q I a^. 

 Then it is evident (from the well-known particular case of 

 theorem 8 in which n = 4) that the point p^ in which b^ a. 3 cuts 



o, o is known. 



i - 



And if we suppose the chord b^ b 3 parallel to p 2 o 3 , and that 

 we draw & 3 4 to cut p 2 o 3 in p^ then for like reasons the point 

 j 3 is known. Similarly, if we draw the chord b^ b^ parallel to 

 p 4 , and that we draw b a. to cut p 3 o , in jj , then will the 

 point p i be known. 



And proceeding thus, it is evident we at length arrive at the 

 known point p n in which the chord b n 1 cuts the straight 

 line p o . 



1 n 1 n 



