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The "what would I try to invent?" question one is fascinating. If I found myself transported back to Roman times, I'd aim to make a lens. The Romans had glass so the raw material is there. With a lens you can fairly quickly go a telescope, which has immediate military application, so would perhaps be widely adopted. Astronomy might then follow, heading off the whole "Earth is the centre of Universe" thing that held b…
I love this game. I think I would send the idea of plant breeding. They had this idea to some extent, but I don't think there was big public, long term effort to breed new crops or get much higher yields. If you get modern crops faster, you have larger populations, which leads to industrialization sooner. This isn't just limited to food crops, it's possible ancients could have bred plants and animals for many useful…
All that being said, assuming the chronophone will let me, I'd bring their attention to the idea of the printing press. It's easily made, the benefit is obvious, and the effect is dramatic. Kind of amazing it wasn't invented earlier.