BY MARTIN GARDINER, C.E. 99 



n'gons, the n sides of each of which will meet with the entities 

 taken in order, in such a manner as that the sides meeting with 

 the entities which are conies will be tangents of prescribed 

 rotatives thereunto. (Remembering that when we fix on the 

 rotative of any particular numbered sides of the w'gons whose 

 first points lie on one side of the line of contact of the entity 

 which they touch, we must have them of opposite rotative when 

 the first points fall on the other side of the line of contact.) 



Fourth method of solution (for particular case). 



When the given conic is a circle, the problem of the inscrip- 

 tion of the closed n'gons whose sides pass in order through the 

 n given points o , 2 , . . . . o^ can be investigated in the following 

 manner (which is worthy of particular attention as an illustration 

 of the importance of conceiving the methods of rotatives of 

 segments of lines in respect to particular points, and by such 

 means eliminating uncertainty as to which of two straight lines 

 is the answerable one to the object in view.) Method of investi- 

 gation : Suppose p l p 2 P n P l an inscribed closed rc'gon whose 

 sides pass in order through the n points. 



Let a l a 2 .... n+1 and ^ & 2 .... b be two inscribed w'gons, 

 formed at random, the sides of each of which pass in order 

 through the n points. From the properties of similar triangles 

 we have 



a p a . . p 



n r 711 * 



. . 



71+1 



. , . 

 n"n n-j-1 *\ 



holding in signs when the rotatives of the involved lines are 



