"By the ninth century instead of primitive schemes... tables... were common... [and] gave as a function of the sun's altitude the shadow lengths cast by it... Lengths were measured in units of a standard vertical ... These are tables of \mathrm{Cot} \theta \equiv R \cot \thetausually calculated for \theta = 1^{\circ}, 2^{\circ}, 3^{\circ}, \ldots , 90^{\circ} ."