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 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