( 377 ) 
Art. XXIII . — A general Theorem relating to Regular Poly - 
gems. By William Wallace, F.R.S.E. Professor of Ma- 
thematics in the University of Edinburgh. 
Theorem. 
XjET there be any regular polygon of n sides, described about 
a circle whose radius is r , and let a denote the angle at the 
centre, subtended by a side of the polygon ; let the distance of 
any point within the polygon from the centre be v, and from that 
point let perpendiculars be drawn to the sides of the polygon : 
The area of the rectilineal figure formed by straight lines which 
join the bottoms of the adjacent perpendiculars, is equal to 
n{ \ r 2 sin ^ -j- J v 2 sin 2*}. 
A 
Taking a particular case, let A A' A" A'" A IV be a regular po- 
lygon of five sides : From P, any point within the polygon, let 
perpendiculars PB, PB', PB", PB'", PB IV be drawn to the sides, 
and let B B', B'B' 5 ', &c. be joined, so as to form the figure 
BB'B"B'"B IV , let * = angle of 72°, r = radius of inscribed 
