Daily entries from the 17th century London diary
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Cockpit, against the Privy Garden, W. It is a part of the Royal Palace of Whitehall, standing in a Street situated betwixt two Gateways; the one fronting Charingcross, and the other Westminster Abby. Here the Principle Secretaries of State generally keep their Offices.
W. Stow 1722
1746 Map
The 1746 map of London includes the Cockpit at Whitehall:
http://www.motco.com/Map/81002/SeriesSearchPlatesFulla.asp?mode=query&artist=384&other=361&x=11&y=11
The Royal Cockpit
The Cockpit is the octagonal building, near the bottom right corner of this 1658 map.
http://www.oldlondonmaps.com/newcourtpages/newcourt18.html