PROPERTIES of the CIRCLE. 3 $ 



when a regular figure circumfcribing it touches it in faid points, 

 gives Dr Stewart's 37th theorem, fince the fame reafoning and 

 mode of demonstration holds good in regard to half the amount 

 of this expreflion, whether the points p and q be in PQ^or in 

 PQ^produced. 



And univerfally if m have to / any ratio whatfoever, 



r + Qjj +r—Oi * r + Og 1 + r — Og 1 r+OPi + r — Oh ~ M & 



m w Ht 



r l r r l 



— inrs + f — j- -rxOi + Og + Oh + &c. + 7- -7- . — j~. 

 m — -d Joi' + Og + Ob + &c. , m m — / m — 2/ m — %l m — 4/ 



2. .X 2- + — . ; • — . -li- . jT- 



4/ r I I 3/ 4/ 5 l ' 



m-sl x • t . Pi' + Og 1 + Oh\ Sec. + & & 



ir ,xi r* 



This laft theorem or expremon is more general than any of 

 Dr Stewart's theorems, and from it may eafily be derived an 

 endlefs number of new and curious infinite feries, with their 

 fummations. 



It is almoft needlefs for me to obferve, that befides thefe theo- 

 rems of Dr Matthew Stewart, an unlimited number of other 

 theorems, respecting figures both regular and irregular, circum- 

 fcribing and inferibed in the circle, may eafily be derived from 

 the foregoing investigation, as well as a great number of geo- 

 metrical infinite feries, with their fummations. And as to theo- 

 rems refpecting given points, right lines and figures either re- 

 gular or irregular, given by pofition, and right lines interfecling 

 each other either in one point or in different points in angles 

 either equal or unequal, that are deducible from it, they are in- 

 numerable. 



Vol. VI.— P.I. E Now, 



