350 Proceedings of Royal Society of Edinhurgh. [june 4, 
this inversion of the problem both the geometrical investigation 
and the arithmetical work are greatly simplified ; so much so that 
the subject may be 
presented to a mere 
beginner in the study 
of geometry. Not 
having seen it noticed 
in any treatise, I pro- 
pose, while describing 
it to the Society, to 
give some hints for 
expediting even this 
rapid process. 
If, in the isosceles 
triangle B05, the angle 
at the vertex be an 
aliquot part of the 
entire revolution, the 
triangle itself is part 
of a regular polygon 
on the base B5, having 
0 for its centre, OA 
for the radius of the 
inscribed circle, OB 
for the radius of the 
circumscribed one. 
The line AO being 
continued indefinitely, 
let OP be measured 
equal to OB, and let 
BP, 5P be drawn ; 
then it is clear that 
the angle BP5 is half of B05, wherefore the triangle BP5 may 
be repeated round the vertex P, twice as often as B06 was 
repeated round 0; that is to say P is the centre and BP5 a por- 
tion of a regular polygon having twice the former number of sides ; 
its perimeter, therefore, would be double of the proposed perimeter. 
Let us then bisect AP in C and draw DCc? parallel to BA6 ; then 
