BY MARTIN GARDINER, C.E. 81 



15. The problem can be investigated in the following 

 manner : 



Looking on the four successive sides p l p_^ p 9 p s , p 3 p^ p^ p & 

 let us see whether we could replace them by a less number of 

 sides passing through determinable points, and having their inter- 

 sections on determinable straight lines, and their extremities inp 1 

 and p.. 



Let ij be the point of intersection of the straight lines o o g 

 and o 3 . Then since o , o 2 , i , are in one straight line, it follows 

 that the intersection T I of the straight lines o j p 2 and ^ p 3 is 

 in a known straight line R r And since o 3 , o 4 , i^ are in one 

 straight line, it follows that the intersection s l of the straight 

 lines o p and ^ p 3 is in a known straight line S . 



Hence it is evident the solution of the problem is reduced to 

 that of describing a closed (n l)'gon p l r l s l P 5 P 6 P n Pi, 



whose sides p l r v r { s^ s l p., p^p & p n p i pass through the 



n 1 known points o^ i^ o^ o g , o n , and whose angular 



points jp 1? r 1? 8 V p 5 , . . . . p will rest on the n 1 known straight 

 lines L X , B , S 1? L 5 , .... L^. Similarly, by proceeding with the 

 first four sides of this closed (n l)'gon as with those of the 

 fz'gon, we can reduce the solution to that of describing a 

 closed (n 2)'gon p l r 2 s 2 p Q . . . . p n p l whose sides pass in order 

 through n 2 known points o^ i.^ o 5 . . . . o n , and whose angular 

 points p^ r 2 , s 2 , p 6 . . . . p n rest on the n 2 known straight lines 

 Lj, R 2 , S 2 , L 6 , . . . . L n . 



And thus, step by step, we can proceed until we make the 

 solution of the problem depend on that of describing a triangle 



Pl V3 S -3^1 WhOSG SideS Pl r n-3' T n-3 S n-V S n_3 PI> wlU P 338 



through known points 1? * _g, o , and whose angular points 

 Pv r n-3 s n-3 ? w ^^ rest on ^ nown straight lines L p B n _ 3 , S n _ 3> 



