Ctesias' description of the Hanging Gardens (pasted in brown below) appeared in the earliest sections of his 23-volume work, Persica (History of the Persians, now lost). As we have seen, it is probably the most reliable eye-witness description. Today, we know it only through Diodorus, whose text on the gardens was copied in detail from Ctesias.
Ctesias was a doctor from Cnidus in Caria. According to his own account, Ctesias was taken prisoner of war by the Persians and served as personal physician to the emperor and his family for seventeen years. At this time, the Persian court was at Persepolis, not Babylon.
Historically, Caria had political ties to Persia, apart from sixty-years' subjection to Athens (c.469/6–412 BC). Lendering suggests that Ctesias may have been taken prisoner just two years before Caria's allegiance returned to Persia. At all events, Ctesias enjoyed the confidence of the emperor's family—notably of Parysatis (the king's mother) and Statira (the king's wife), and these were the main sources for his work. Indeed, over half the Persica concerned Parysatis, her mother, or her daughter.
Ctesias is in the middle of describing Semiramis' improvements to Babylon. This extract comes from a college essay by Rusty Sietsema.
She [Semiramis] made its outer circuit-wall over seven miles long, and the high wall was built from baked bricks with no sparing of expense. Inside this she built a second circuit-wall. Before the bricks for it were baked, all sorts of wild animals were engraved on them, and these were so ingeniously colored that they seemed almost real. This wall was almost five miles long, 300 bricks wide and 300 feet high, and it had towers which were 420 feet high. Inside there was a third wall which enclosed a citadel with a circumference of two and a half miles. The height and width of this wall were even greater than those of the middle wall. On it, and on its towers, there were again wild beasts of every kind, cleverly drawn and realistically colored to represent a complete big-game hunt. These animals were more than 6 feet long, and Semiramis was portrayed among them, mounted and hurling a javelin at a leopard. By her side was her husband Ninus, dispatching a lion at close quarters with his spear.
Beside the citadel was the building known as the Hanging Garden. This wooded enclosure was square in shape with sides four hundred feet long, and sloped like a hillside with terrace built on terrace as they are in a theater. During the building of the terraces galleries were built underneath them which carried the entire weight of the gardens, each rising a little above the one before it on the ascent. The uppermost gallery, which was 75 feet high, supported the highest level of the garden, and this was the same height as the battlements of the city-wall. The walls of this structure, which cost a fortune to build, were 22 feet thick, and were separated by passages 10 feet wide. The galleries were roofed with stone beams 16 feet long and 4 feet wide. Above these beams there was first a layer of reeds set in great quantities of bitumen, then two courses of baked brick bounded with cement, and then a covering of lead so that moisture from the soil would not be able to sink through. On this was piled earth, deep enough to contain the roots of the largest trees, and when it was leveled over, the garden was planted with all sorts of trees which would appeal to those who saw them either by their great size or by the beauty of their appearance. Because of their arrangement the galleries were all open to the light, and contained royal apartments of all kinds. One gallery had shafts leading from the highest level and machinery for raising water in great quantities from the river and supplying it to the gardens. This machinery was entirely enclosed, and so could not be seen from the outside. (Macqueen, 1964, pp. 158) [Macqueen, James G. 1965. Babylon. New York, NY. Frederick A. Praeger, Inc., Publishers.]
Be sure to compare this description with the variants on Diodorus' page, some of which include the detail of the "arches" (i.e. vaults) below the stone beams supporting the garden terraces.