No. XXV. 



DEMONSTRATIONS 



OF 



STEWART'S PROPERTIES 



OF THE 

 1 



By THEODORE STRONG, 



Professor of Malhtmalics and ATatural Philosophy in Hamilton College. 



THE following propositions are to be found in Dr. Rees' Cyclope- 

 dia, under the article '■^Circle." Thej' were proposed to me for solu- 

 tion. Having examined and found tlicm to he very curious, and con- 

 nected by one general principle, I have, with a view of contributing 

 my mite to the advancement of science, thought proper to communi- 

 cate the following demonstrations of them to the Academy. I have 

 succeeded in demonstrating them in three ways, which are different 

 from the one here exhibited ; one of which, being founded upon the 

 principle of finding the sum of any powers of the chords and versed 

 sines of arches of the Circle increasing arithmetically ; together with 

 many curious inferences deducible from the following demonstrations ; 

 I may hereafter take the liberty of communicating to the Academy. 

 My only object, in the present communication, has been to demon- 

 strate the propositions, as they were proposed in the aforesaid work. 



THEODORE STRONG. 



Hamilton College, Sept. 2.0th, 1814. 



Lemma I. 



XF the circumference of any circle be divided into 

 any number of equal parts, and any point be taken in 

 it ; and if the several arches intercepted between the 

 assumed point, and the points of division, reckoned 



