Megasthenes (350–290 BC) was a Greek ambassador, geographer and traveller whose writings do not survive. Fortunately, his reference to some hanging gardens was quoted by Abydenus (c.105–35 BC) in his Concerning the Assyrians. This work is also lost, but it was quoted in turn by Eusebius, whose work is preserved.
Eusebius mentioned the Hanging Gardens twice—once in Part I (Chronographies) of his Chronicon (a universal history, written 303, revised c. 325 AD), and once in his Praeparatio Evangelica (written c. 314 AD).
Here is the text of the Chronographies, and here is the Praeparatio Evangelica. Or you can jump to the extracts pasted in brown below.
Eusebius's account of Babylon is mainly based on our next source, Berossus (fl.253–253 BC), whose work he knew only through Josephus and Alexander Polyhistor. But he tacitly rejected their claims about the gardens. Ending his summary of Josephus exactly before the part about the gardens, Eusebius turned instead to Abydenus, whose source was Megasthenes.
This group of writers clearly state that some hanging gardens were built by Nebuchadnezzar in the city of Teredon. Teredon was on the Gulf coast (about 200 miles from Babylon), according to Strabo.
From the Sacred Texts site.
ABYDENUS, in his history of the Assyrians, has preserved the following fragment of Megasthenes, who says ...
... And conceming the rebuilding of Babylon by Nabuchodonosor, he writes thus: ... and built the city of Teredon to check the incursions of the Arabs; and he adorned the palaces with trees, calling them hanging gardens.Euseb. Chron. 49. Euseb. Pręp. Evan. lib. 9.
From the Tertullian site
Thus far Josephus. (This section took the reader through Josephus ch 19 but stopped exactly before the part about fortifying the city and building the gardens.)
CHAPTER XLI
I [Eusebius] FOUND also the following statements concerning Nebuchadnezzar in the work of Abydenus Concerning the Assyrians:
[ABYDENUS] 'Now Megasthenes says ... Also concerning the building of Babylon by Nebuchadnezzar the same author writes thus: ...
'He [Nebuchadnezzar] also walled off the inundation of the Red Sea, and built the city Teredon at the place of the incursions of the Arabs. His palace too he adorned with trees, and gave it the name of the Hanging Gardens.'