Something to keep in mind is that portolan chart navigation maps like Piri Reis don't a have coordinate system, rather they are several different local coordinate systems munged together -- each one of the rosettas or compass rose like symbols is a projection source of origin, loosely speaking.
So you can't just georeference the image in the usual GIS way by assigning coordinate pairs to the four corners or a half a dozen points within, slap rectify and then overlay it on modern data which does have a coordinate system. Well you can, as Mapperz' answer clearly shows, and the comparison is interesting and possibly informative, just be careful about the conclusions you draw from it!
An interesting exercise for balance would be to drop other popular modern projections as georeferenced images on top of Google Earth and see how they fare.
A question about how to most properly georeference chart maps like Piri Reis would be an excellent addition to GIS Stack Exchange.