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The question is what developments could help the Roman Empire endure up to the present day, preferably with a much expanded territory. In other words, take your typical Roman Wank, and justify it. Ideally, these should all be developments that would make sense on their own, and within the confines of the technological and cultural limitations of their era.
For example, Heron inventing a proper reciprocating steam engine probably wouldn’t work, since there’s so many different developments needed between the aeliopile and a useful steam engine.
A few suggestions to get the ball rolling:
Spinning Wheel. I’ve got a pretty healthy thread going on this, but the gist is that it improves the economic value of marginal lands (including those beyond the Empire) and lowers the cost of textiles.
Paper. This is more practical after the development of a spinning wheel, as that increases the amount of textiles in the economy, which can then be used as pulp for paper after being discarded. Paper, of course, is cheaper than papyrus and parchment, and thus can lower the cost of the written word.
Movable Type Printing. Again, it is most workable after the development of paper. I think we can all agree on the advantages of printing.
Gunpowder. Cliche, but alchemy as we understand it (there’s an awkward sentence) dates back to the era - one of the first known alchemists was a woman named Maria living at some point in the first century, most likely. The formula for gunpowder could be stumbled upon by almost anyone, and, within a few centuries, you have some pretty formidable weaponry, perfectly suited to a highly organized society that has a preference for infantry.
Distillation. For both medical and recreational reasons, as long as the artisans in charge figure out not to use lead. This also increases the economic yield of farmland.
Windmills. The Romans were quite familiar with watermills, but vertical windmills did not take off quite so well, even though Heron (hi again) did figure them out.
Oh, and we can’t forget:
Double-entry bookkeeping and arabic numerals, likely developed in Patavium