usertron2020
Donor
It's not called stirrup, it's called stapeda.
And just to disent, i'm going to go with the microscope. The microscope implies a degree of knowledge of lenses and the laws of optics. After someone having invented the microscope, someone else would be able to reverse-engineer it, and create telescopes, binoculars and stereoscopes, all of them of great militar utility.
The germ theory wouldn't take too long to develop if the physicians begin to exhaustively examine the "humors" of diseased people under the microscope, and compare them with "healthy humors". The humor theory itself would collapse very quickly.
To dissent from your dissent, I would suggest something even more important for the immediate survival of Rome, as opposed to the survival of Western civilization. After all, this is all about preventing the Dark Ages, right?
Discover the Theory of Chemistry. REAL chemistry, not what the Ancient Greeks believed. Then you ultimately get dynamite. Blast through all those mountain passes, allowing the making of Roman Roads that are curved (saving tremendous time, effort, and money), chemically discover macadam, aspirin (available in tree bark form since the Old Stone Age, but once in concentrated pill form...), etc, etc, etc.
All this will ultimately mean that Germania and Dacia (hell, the whole of the Balkans) become more militarily and economically possible to conquer.
Guns are simply a matter of developing an understanding of the potential propellant properties of explosives. It took twelve centuries of Chinese fireworks before a wise man sat down and came up with the idea. A wise man named Ghengis Khan...