BY MARTIN GARDINER, C.E. (53 



PAPER No. 1. 



Researches concerning figures peculiarly derived from other figures 

 by MARTIN GARDINER, C.E. 



[Read 9th July, 1862.] 



Given a closed n'gon A X A 2 ............. A W A X , to find the 



locus of a point O, such, that if we join the successive points 

 Bj , B 2 , ..... B M , B X , of the feet of perpendiculars from it on the 



respective sides A^A^, A^ A^ , . . . . A n A I , of the given n'gon, 



then will the rectare of the new n'gon B X B 2 ---- B n B X , thus 



derived (which we will call the derived figure) be of a given 

 magnitude, . 



PROCESS OF INVESTIGATION. 



Let a lt a 2 ,...., a n , be the centres of the circles circum- 

 scribing the quadrilaterals 



Now, paying attention to the formations of magnitudes, and 

 looking on the quadrilateral B n A X B X , we at once perceive that 

 J rectare (OB^ +0^6^ = rectare (OB^) -rectare (a^B^Bj). 



And from the quadrilateral B X A 2 B 2 , we have 

 f rectare (CKB^ + O'AgBg) = rectare (OB^)- rectare (a^BJ. 



And in the like manner we get the following equations from 

 the other n 2 quadrilaterals : 



i rectare (CKB^ A, + OA n B n ) = rectare (OB^BJ- 



rectare (a n < B B _ 1 BJ. 



Therefore as the sum of the first sides of these n equations is 

 obviously equal half the known rectare of the given n'gon 



