Upvote:7
The Hermotybians and Calascirians mentioned by Heordotus appear to be simply two parts of the Egyptian army, each recruited from particular parts of the country. Indeed, Hordotus himself makes this explicit:
"Their warriors are called Calasiries or Hermotybies and they are of the following districts, for all Egypt is divided into districts.
The following are the districts of the Hermotybies:Busiris, Sais, Chemmis, Papremis, the island called Prosopitis, and the half of Natho. From these districts are the Hermotybies, being in number, when they are most numerous, a hundred and sixty thousand. None of these learn any mechanical art, but apply themselves wholly to military affairs.
These next are the districts of the Calasiries: Thebes, Bubastis, Aphthis, Tanis, Mendes, Sebennytus, Athribis, Pharbaethus, Thmuis, Onuphis, Anysis, and Myecphoris; this district is situated in an island opposite the city of Bubastis. These are the districts of the Calasiries, being in number, when they are most numerous, two hundred and fifty thousand men ..."
This division of the army into two parts was not something introduced by the Greeks, but is also attested in much earlier records.
For example, the Great Edict of Horemheb states that:
The two divisions of troops which are in the field, one in the southern region, the other in the northern region, stole hides in the whole land, not passing a year, without applying the brand [?] of the royal house to cattle which were not due to them, thereby increasing their number, and stealing that which was stamped from them. They went out from house to house, beating and plundering without leaving a hide for the people.
This division of the armies is hardly unsurprising, given the geography of Egypt. Dividing their forces into a northern and a southern contingent enabled Egypt to guard against any threat from Kushites to the south and also from whoever was the dominant Mesopotamian power of the day in the north.
These is some difficulty with the Nomes (or 'districts' or 'cantons' depending on one's preferences) listed by Herodotus, since many of the 42 Egyptian Nomes are not mentioned:
It is possible that some Nomes were exempted from providing troops or, perhaps more likely, that Herodotus was unfamiliar with the system of Nomes - having probably never actually visited Egypt - and so only listed the ones that he was aware of.
As for the Medjay, the last mention of the Medjay is during the Twentieth Dynasty (1189–1077 BCE). As the Wikipedia article you cited notes,
"It is unknown whether the Medjay as an occupation had been abolished or the name of the force had changed".
Herodotus makes no suggestion that the Medjay were in any way related to the Hermotybians and Calascirians (or Calasiries and Hermotybies, depending on the translation of 'Histories' you have available). This is hardly surprising, since Herodotus wasn't even born until 485 BCE, fully six centuries after the Medjay were last mentioned in the actual records of Ancient Egypt.