From Bowyers Company: "The word 'artillery' comes from the French 'arc tirer', to draw a bow, and the Artillery Company (later to become the Honourable Artillery Company) was originally a company of archers. Before 1650 it was headquartered in the Artillery Gardens in Spitalfields, near today's Artillery Lane and Artillery Passage; the 1558 map also shows musketry being practised there, with little puffs of smoke." "After the disruptions of the Civil War, the Honourable Artillery Company moved its headquarters from the Artillery Gardens in Spitalfields to its present Armoury House ground in Bunhill Fields".
From HAC: Henry VIII's "Fraternity or Guild of Artillery of Longbows, Crossbows and Handguns ... practised in the Artillery Ground in Bishopsgate (Spitalfields) until around the 1560s. During the years of the Armada threat in 1586-1588, this practice ground was again used by the officers who trained the City of London’s Trained Bands and who were also known as the ‘Captains of the Artillery Garden’. The word ‘artillery’ was used at this time to describe archery and other missile weapons, while bigger guns were known as ‘great artillery’. In 1611 practice was again revived in the Bishopsgate ground when the ‘society of arms’ also used the space for training. This 1611 group, who were performing the same training in the same space used by the 1537 fraternity, are the certain ancestors of today’s Honourable Artillery Company."
The map comes from the Bowyers site. They've yellow-highlighted the archery fields and the one over at the right is the Spitalfields Gardens - you can see the men drawing their bows.