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I have an old topographic map that I need to georeference for my data, but I'm not familiar with its projection. The information it has on the map is this, but I don't have any idea how to use this information in ArcMap or QGIS.

What should I do?

this

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It appears to be related to the NEIEZ zones that are listed in the EPSG registry, but the projection and parameters don't match. Depending on where in Indonesia the map is representing, the geographic coordinate reference system (datum) could be Makassar, Batavia, or Segara. They have different areas of use, so take a look at them in EPSG.

It looks like you'll have to define a custom projected coordinate reference system.

Lambert conical orthomorphic = Lambert 1SP
false easting: 550000.0 m
false northing: 400000.0 m
longitude of natural origin: 110 E
latitude of natural origin: 8 S (or is it -5? did I read that correctly? My eyesight's not the best)
scale factor at natural origin: 0.9997

Disclosure: I work for Esri. I am on the subcommittee that maintains the EPSG registry.

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