Middleton-on-Lune railway station Middleton-on-Lune railway station Middleton-on-Lune railway station was located in Westmorland (now Cumbria), serving the hamlet and rural locale of Middleton on the Ingleton Branch Line. It was opened as Middleton in 1861 and renamed Middleton-on-Lune on 19 July 1926, closing in 1931. The Lancaster and Carlisle Railway built the Ingleton Branch Line from the existing Ingleton Station to . By the time the branch was completed in 1861, the L&CR was operated by the London and North Western Railway (L&NWR). After formal closure the line was still on occasions used for weekend excursions and to transport pupils to and from
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