The ancient tradition of fly fishing is wet fly fishing.
It was wet fly fishing only for many centuries.
Dry flies, nymphs, and streamers all came much later.
In Britannia.
Englishman Frederic M. Halford, aka Detached Badger, published "Floating Flies and How to Dress Them" in 1886. He is rightly credited with perfecting the modern version of upstream dry fly fishing. Note that this was made possible only after the refinement of split cane (bamboo) fly rods. Prior to split cane, fly rods specifically and fishing rods in general were largely incapable of casting.
Meanwhile, the dry fly, fished from short rod with a length of line (probably horse tail hair) is first described by Claudius Aelianus, known to modern scholars (and presumably his fishing buddies) as Aelian, in his work "On the Nature of Animals," which dates to roughly the Year of Our Lord 200.
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Aelian wrote:
I have heard and can tell of a way of catching fish in Macedonia, and it is this. Between Beroea and Thessalonica there flows a river called the Astraeus. Now there are in it fishes of a speckled hue, but what the natives call them, it is better to enquire of the Macedonians. Now these fish feed upon the flies of the country which flit about the river and which are quite unlike flies elsewhere; they do not look like wasps, nor could one fairly describe this creature as comparable in shape with what are called Anthēdones {bumble-bees}, nor even with actual honey-bees, although they possess a distinctive feature of each of the aforesaid insects. Thus, they have the audacity of the fly; you might say they are the size of a bumble-bee, but their colour imitates that of a wasp, and they buzz like a honeybee. All the natives call them Hippūrus
[Kennedy adds a translation here for PaFF readers, “soldier flies”]. These flies settle on the stream and seek the food that they like; they cannot however escape the observation of the fishes that swim below. So when a fish observes a Hippurus on the surface it swims up noiselessly under water for fear of disturbing the surface and to avoid scaring its prey. Then when close at hand in the fly's shadow it opens its jaws and swallows the fly, just as a wolf snatches a sheep from the flock, or as an eagle seizes a goose from the farmyard. Having done this it plunges beneath the ripple. Now although fishermen know of these happenings, they do not in fact make any use of these flies as baits for fish, because if the human hand touches them it destroys the natural bloom; their wings wither and the fish refuse to eat them, and for that reason will not go near them, because by some mysterious instinct they detest flies that have been caught. And so with the skill of anglers the men circumvent the fish by the following artful contrivance. They wrap the hook in scarlet wool, and to the wool they attach two feathers that grow beneath a cock's wattles and are the colour of wax. The fishing-rod is six feet long, and so is the line. So they let down this lure, and the fish attracted and excited by the colour, comes to meet it, and fancying from the beauty of the sight that he is going to have a wonderful banquet, opens wide his mouth, is entangled with the hook, and gains a bitter feast, for he is caught. (original Greek at:
https://www.perseus.tufts.edu/hopper/text?doc=Ael.+NA+15.1)
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The use of wool in modern fly tying would be athematic to the construction of a dry fly, however in AD 200, the natural lanolin would likely still be part of woolen yarn. Being a natural animal oil, it would result in a floating fly. I mention this because in some fly fishing nerd circles, the use of wool precludes the Macedonian soldier fly pattern from being a dry fly. The description of technique always struck me as evidence enough that the fly is a floater (dry), but, like I say, wool body material has suggested to some a sinking (wet) fly. Technique and lanolin, in my mind, make it dry fly, but we'll never know for sure.
The Claudius Aelianus description likely recounts a fishing technique that had been in use for quite some time prior to AD 200 in ancient Macedonia, and there's every reason to believe these same anglers used sinking flies (and, of course, bait) when the fish were not rising for soldier flies.
Also, there's every reason to believe that fly fishing, as described by Aelian, was practiced by rod anglers far and wide throughout the ancient world.
Clearly Aelian did not "invent" fly fishing, nor was he writing about the latest fishing fad for a Macedonian version of Modern Angler Magazine. He was simply writing down what was common practice in his day. Similarly, my contention is that dry fly fishing did not go away in the centuries between 200 and Mr. Halford's 1886 work on floating flies.
On the subject of the preference for upstream fishing (in all instances) when using a fly, it is correct to say this was something strongly advocated by Mr. Halford. As he aged, he became increasingly convinced that dry flies cast upstream was the only true form of fly fishing. (I exaggerate a bit, but only a bit. There was an article in Forbes several years back in which the writer said of Halford's dry fly commandments: "Halford turned that method into a sort of cult with himself as high priest, dispensing not only revolutionary technical information on the dressing and employment of dry flies but also a body of on-stream etiquette that is still adhered to today on the Test and, to one degree of purity or another, on many of the other chalk streams as well."
Our sport is endlessly fascinating, and because so many of us are fishing nerds, there's a lot of writing about fishing history, opinion and combinations thereof, there's a lot to do during the off season.