How did Ptolemy’s empire function?

Ptolemy synthesized Greek knowledge of the known Universe. His work enabled astronomers to make accurate predictions of planetary positions and solar and lunar eclipses, promoting acceptance of his view of the cosmos in the Byzantine and Islamic worlds and throughout Europe for over 1400 years. I don't know if that directly answers your question though.

c) troops march forward in close formation against newer, more accurate weapons

the foundation of greek city-states: following the fall of the mycenaean civilization in the 10th century bc, the area surrounding the empire went through its own mini dark ages. answer and explanation