Born in Strasbourg, Greuter worked in France, in Avignon and Lyon.
Apparently to escape the "strong intellectual and commercial pressure of Dutch cartographic publishing",[1] in 1606, he went to Rome where he produced works for Cardinal Scipione Borghese, Pope Paul V, for the Accademia dei Lincei, and Pope Urban VIII.
The inclusion of Hokkaido indicates that he updated the design to take account of new discoveries, as the island was not known to Blaeu, but was documented by another map maker in Rome.
[1] He made a "celestial globe" depicting the constellations in 1635, also based on Blaeu, who had used the data of Tycho Brahe.
[2] His satirical print of "Doctor Wurmbrandt" purging human follies was later adapted by Martin Droeshout.