This tower, Nelson's Monument, is supposed to resemble an up-turned telescope. It commemorates the victory of Lord Nelson over the French and Spanish at the battle of Trafalgar in 1805. It was built in 1816 and has a stairway up to the top.
Before the invention of accurate timepieces it was very important for the ships at Edinburgh's port, Leith, to know the exact time. Indeed, this was a major function of the astronomers working at the City Observatory (just in the left of the picture). To this end, at exactly one o'clock in the afternoon a gun is fired at the castle. This is called, not surprisingly, the one o'clock gun. However, it takes time for sound to travel the 3km, or so, to Leith so, to ensure even better accuracy, a time-ball is dropped from the mast of Nelson's monument at exactly 1pm. The tower is directly visible from Leith and so the ships could set their chronometers accurately.