I'm not a developer, but this returned what you are asking for I think:
/rest/greenhopper/1.0/sprintquery/1?includeHistoricSprints=true&includeFutureSprints=true (200)
Where the number after /sprintquery/ is your Rapid Board ID. You can find the Rapid Board ID by viewing the board and looking at the URL for "rapidView=31" - in this case 31 is the ID. That 200 at the end of the above URL is the max number of sprints to return.
That gives you the full list with:
"id": 21,
"name": "1.270.1",
"state": "CLOSED"
for each sprint. The three states are: Closed, Future, and Active
Let me know if this isn't what you need, happy to help if I can.
Does anybody know how to retrieve the issues in a sprint, and how to add / delete issues to a sprint? The "sprint" endpoint seems to have disappeared, too. To make things worse, my RAB doesn't have a "Show only public APIs" option, so I have no way to at least see the API endpoints.