Peter Nicholson’s *Carpenter’s New Guide* of 1803 contains an interesting technique:

To find a right line equal to any given Arch of a Circle. Divide the chord

abinto four equal parts, set one partbcon the arch fromatod, and drawdcwhich will be nearly equal to half the arch.

Apparently this was an item of carpentry lore in 1803. In the figure above, if arc *ad* = *bc*, then *cd* is approximately half of arc length *ab*.

Nicholson warns that this works best for relatively short arcs: “This method should not be used above a quarter of a circle, so that if you would find the circumference of a whole circle by this method, the fourth part must only be used, which will give one eighth part of the whole exceedingly near.”

But with that proviso it works pretty well — in 1981 University of Essex mathematician Ian Cook found that for arcs up to a quadrant of a circle, the results show a maximum percentage error of 0.6 percent, “which I suppose can be said to be ‘exceedingly near.'” He adds, “[I]t would be of interest to know who discovered this construction.”

(Ian Cook, “Geometry for a Carpenter in 1800,” *Mathematical Gazette* 65:433 [October 1981], 193-195.)