I don't think this is a possible world in the first place. The knowledge of steam engines and intricate mechanical machines wasn't "lost" by accident: it was lost because there was no good reason to develop it further. The Greek steam engine was a cool demo, neat little piece of stage magic for the temple whose doors it powered, but there weren't any practical applications in sight. I bet it also was a hell of a maintenance burden.

The unfortunate reality of our lives seems to be that technology only propagates as long as there's enough funding for it. That means at least several people have to need it very much to support a single inventor. Many more people if there's a need for extra labor, or expensive resources (such as metals). That's as true today as it was 2000 years ago.

What I'm saying is: the reason the Greek steam engine was forgotten is that there was no demand for it back then, and that's because of a combination of factors, such as:

- It could do only so much, because other important technologies - such as precision manufacturing - weren't in place;

- The things it could do could be served better and cheaper by other means - such as hired, slave[0] or animal labor;

Learning history on my own[1], past the school curriculum, I was surprised to discover that a lot of technology was always there in the background. Medieval times weren't just a long period of stagnation - it's when the building blocks for industrial evolution were slowly invented. A lot of important developments happened in mining, where increasingly powerful and reliable pumps allowed people to dig deeper. The progress here was sustained, because it was paying for itself.

The industrial revolution happened when things aligned just right. The steam engines could be manufactured cheaply and robustly enough, and put to important enough work, that they paid for themselves - which financed their improvements, which opened more net-profitable applications, and so on - creating a positive feedback loop that run so long, it triggered more feedback loops, and turned humanity into proper technological civilization.

Note that there's a lot hiding in the "put to important enough work". The overall political and economic situation of the era was what created the "jobs" for steam engines, and that situation is downstream from technologies developed for sailing (progress in astronomy, healthcare and shipbuilding), communications (the printing press), and many others.

Point being: inventions aren't islands; they can't sustain their existence if there isn't a need for them. The question "where would we be if we didn't forget ${technology that we 'forgot' and 'rediscovered' centuries later}" doesn't seem valid to me.

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[0] - There's an argument I see raised every now and then that historically, technology made jumps when labor was expensive; conversely, cheap labor meant it was cheaper to just throw more bodies at any problem.

[1] - I'd like to say "studying history", but what I really mean is just occasionally watching some high-quality documentaries and reading blog posts by people who (claim to) know this stuff.