Published on behalf of The Mathematical Association
What follows is a condensed statement and classification of various proofs of the above well-known theorem.
For two positive quantities, a and b, the proof may be put as follows:
Huygens’ approximation to the length of a circular arc is 
, where l 0 is the chord of the arc and l 1 is the chord of half the arc. This result is most easily obtained from the series for sin 
and sin 
by neglecting powers of 
beyond the third, L being the true length of the arc and R the radius of the circle. For arcs less than 45° Huygens’ rule is very accurate, the error for an arc of 30° being less than 1 in 300,000.