BY MARTIN GARDINER, C.E. 89 



Suppose fll a a n+1 , ^ 6, i^, ^ c 2 .... CR+I , ^ <* a .... 



r/ &c., to be inscribed ft'gons, the successive sides of each of 

 which pass in order through the n points of the series. 



If we regard the point o l and its polar as vertex and axis to 

 homologic figures whose homological ratio is 1, and that we 

 look on the points a^ b^ c v d^ &c., as belonging to one figure, 

 then will 2 , b^ c g , d^ &c., be the corresponding points in the 

 other figure. 



Similarly, if we regard the point o 2 and its polar as vertex and 

 axis to homologic figures whose homological ratio is 1, and 

 that we look on the points 2 , b^, c y d^ &c., as belonging to one 

 figure, then will 3 , b^, c 3 , d^ &c., be corresponding points in the 

 other figure. 



We may thus proceed from the extremities of sides of the 

 n'gons of like subscript numbers to those of higher subscripts 

 until we arrive at the final extremities of the n'gons. And as 

 homologic figures are homographic, and that figures homo- 

 graphic with any figure are homographic with each other, there- 

 fore it is evident the first points a^ b^ c v d^ &c., of the inscribed 

 w'gons, and their final points n+1 , & n+1 , C B+I , &c., are correspond- 

 ing points of homographic figures. Moreover, it is evident that 

 the tangents to the cnrve S, at the corresponding points of these 

 homographiclfigures, are corresponding lines of the figures. 



2. By indicating tangents to the curve by capital letters of 

 like names and subscripts to the small letters indicating the 

 points of contact, we at once infer the following important 

 theorem : 



THEOREM I. 

 If in a curve of the second degree there be three n'gons a 1 a 2 



S that th * D 



