Jamieson, Fausset and Brown on Daniel 2:39

39. That Medo-Persia is the second kingdom appears from Da 5:28 and Da 8:20. Compare 2Ch 36:20; Is 21:2.

inferior--"The kings of Persia were the worst race of men that ever governed an empire" [Prideaux]. Politically (which is the main point of view here) the power of the central government in which the nobles shared with the king, being weakened by the growing independence of the provinces, was inferior to that of Nebuchadnezzar, whose sole word was law throughout his empire.

brass--The Greeks (the third empire, Da 8:21; 10:20; 11:2-4) were celebrated for the brazen armor of their warriors. Jerome fancifully thinks that the brass, as being a clear-sounding metal, refers to the eloquence for which Greece was famed. The "belly," in Da 2:32, may refer to the drunkenness of Alexander and the luxury of the Ptolemies [Tirinus].

over all the earth--Alexander commanded that he should be called "king of all the world" [Justin, 12. sec. 16.9; Arrian, Campaigns of Alexander, 7. sec. 15]. The four successors (diadochi) who divided Alexander's dominions at his death, of whom the Seleucidæ in Syria and the Lagidæ in Egypt were chief, held the same empire.