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Extension of Huygens’ Approximation to a Circular arc

Published online by Cambridge University Press:  03 November 2016

E. M. Milne
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Extract

Huygens’ approximation to the length of a circular arc is , where l0 is the chord of the arc and l1 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.

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Other
Information
The Mathematical Gazette , Volume 2 , Issue 40 , July 1903 , pp. 309 - 311
Copyright
Copyright © Mathematical Association 1903

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References

* Huygens published his approximation in 1654 in his “De Circuli Magnitudine Inventa” in the following form: “Any are less than a semicircle is greater than its subtense together with one third of the difference, by which the subtense exceeds the sine, and less than the subtense together with a quantity, which is to the said third, as four times the subtense added to the sine is to twice the subtense with three times the sine.” If we apply the former part of this proposition to half the are we obtain the rule as generally stated.