River Lune the West Coast Main Railway Line run. Below the gorge, the valley broadens out into Lonsdale. Bridges over the Lune include the Devil's Bridge near Kirkby Lonsdale and the Lune Millennium Bridge in Lancaster. At Caton, about upstream from Lancaster, there is a cluster of three bridges (one stone road bridge and two disused iron rail bridges now used as foot/cyclepaths) at the "Crook o' Lune". Here in a 180-degree right-hand bend the Lune turns back on itself; this is followed by a 90-degree left-hand bend forming the shape of a shepherd's crook and creating a beauty spot which was
Lune of Hippocrates which the arc "AEB" lies. Consequently, the smaller circle has half the area of the larger circle, and therefore the quarter circle AFBOA is equal in area to the semicircle AEBDA. Subtracting the crescent-shaped area AFBDA from the quarter circle gives triangle ABO and subtracting the same crescent from the semicircle gives the lune. Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area. Using a similar proof to the one above, Alhazen showed that two lunes, formed on the two sides of a right triangle, whose outer boundaries