BY MARTIN GARDINER, C.E. 69 



Assuming rectangular axes of reference, let 

 G 1 x + H x y + K L = 



H 2 y + ^ = 



G n * + H n ?/ -f K n = 0, 



be the equations of the n successive straight lines A^ K^ A g A 

 ...... A W A I? taken in order. Then putting x', y' t to represent 



the co-ordinates of any one of the positions of O, it is evident we 

 can express the equations of perpendiculars from this point to the 

 lines represented by the above equations, and that we can find 

 the co-ordinates of the feet B , B 2 , . . . B , of these perpendiculars 

 in terms of x', y', and known quantities. Hence it is obvious 

 that if we indicate the co-ordinates of B , B 2 , ..... B W , by 



(*!> 2/1) > Ov y a ) (*V ^ ^ fche ec i uatio11 



then by substituting the values of x l y^ x^, y 2 , &c., implicating 

 x'j y', and known quantities, we will have the equation of the 

 Circle which is the locus of O. 



And by expressing the equations of the sides of the given 

 figure A X A 2 . . . . A M Aj in terms of the co-ordinates of the points 

 A I? A 2 , . . . A^, we can arrive at the theorems already found, but 

 not so obviously as by the method already exposed. 



9. Theorems pertaining to all kinds of plane figures are very 

 limited in number, owing no doubt to the bias for investigations 

 concerning peculiar forms which the minds of geometers suffer 

 in learning the science of geometry. 



When theorems of such a general nature are discovered they 

 should not be passed unnoticed in elementary class-books ; for 

 they show to us that in geometry (as in nature) we may have 

 forms of the most irregular character adapted to fulfil definite 



