From the Wikipedia page 
Originally the outer precinct of the Priory and Hospital of St Mary Spital, it was converted to an Artillery Ground in 1538, under Henry VIII, for the use of 'The Fraternity or Guild of Artillery of Longbows, Crossbows and Handguns', also known as 'The Fraternity of St George'. This group were later known as the Honourable Artillery Company and used the Ground in conjunction with the Gunners of the Tower.
In 1658 the Honourable Artillery Company moved to a new Artillery Ground in Bunhill Fields, leaving the Gunners of the Tower in possession of the area until 1682, when it was sold off to speculative builders. These latter developed the area for housing, designating the streets with their present names of Fort Street, Gun Street, Artillery Passage and Artillery Lane which recall its former use.
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