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 circle. He proved that the lune bounded by the arcs labeled "E" and "F" in the figure has the same area as triangle "ABO". This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles. Heath concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter. Hippocrates' book on geometry in which this result appears, "Elements", has been lost, but may have formed the model for Euclid's "Elements". Hippocrates' proof was preserved through the "History