I just took a look at Archimedes' treatise on spirals, and yes, something like polar coordinates is implicit in his definitions there:
Definitions.[...] 2. Let the extremity of the straight line which remains fixed while the straight line revolves be called the origin of the spiral. 3. And let the position of the line from which the straight line began to revolve be called the initial line in the revolution.
This is from the Heath translation; he admits taking a few liberties here, but these are essentially the notions of pole and polar axis, as used in polar coordinates today. The ideas are only implicit, but they're there.
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Date: 2005-12-06 08:26 pm (UTC)Definitions.[...]
2. Let the extremity of the straight line which remains fixed while the straight line revolves be called the origin of the spiral.
3. And let the position of the line from which the straight line began to revolve be called the initial line in the revolution.
This is from the Heath translation; he admits taking a few liberties here, but these are essentially the notions of pole and polar axis, as used in polar coordinates today. The ideas are only implicit, but they're there.