Under Batu Khan and Subotei using pretty much the same soldiers in Europe:
(1) Mongol Invasion of Rus: 20,000 Mongos vs 80,000 Russians
since you like wiki and I doubt you have a copy of the devils horsemen.
1226- 35,000 Mongols and 40,000 turks vs unknown Russians who numbered 80,000 in 1223 but suffered huge losses.
(2) Mongol Invasion of Poland: 10K vs 10K to 30K
The mongol records say 2 tumen that is 20,000 men
(3) Mongol Invasion of Hungary: 70K vs 80K (Mongols lost 1,000 men and the Europeans lost 40,000 men)
I didn't think that was ever in debate excpet by you who claimed the Mongols were always vastly outnumbered.
Notice how the same group of Mongol soldiers fighting on distant lands easily defeated many, different European soldiers on European turf. The Mongols broke apart and coalesced as necessary to defeat various, different European armies, but they belonged to same group under Batu and Subotei. The Mongols had minimal casualties while the Europeans suffered devastating losses. The East Europeans won a few small battles, but not enough to stop the Mongol onslaught. The Mongols preferred to and mostly fought from a range. All from wikipedia, which is sourced to books.
The Mongols used bow and crossbow technology similar to that found in the Qin and Han Dynasty.
You really need to look past the bows. If you want to know the secret to Mongol success it was command and control along with a huge herd of remounts. This let the Mongols dictate the terms of the fight in most cases. The Europeans suffered the same fate as the Chinese
From wikipedia to history books, ancient Romans were NOT known for using their mobile ranged weapons (i.e., bows, crossbows, and mobile artillery) as an effective or primary weapon when battling against Parthia. Parthia was known for their ranged weapons, which historian descriptions indicated the Parthian bow probably had inferior draw weights to Qin and Han dynasties' ranged weapons (70 lb vs at least 90 lb).
The parthian bows decend from the same technology as all recurves and Turkish examples have an average draw weight of 111lbs.
Wikipedia continues to claim that good iron weapons eventually replaced good bronze weapons by civilizations throughout Europe to the Middle East to Asia. Cherry pick your excuses, but this happened.
You continue to ignore the why.
Cherry pick all you want, but historical reality is against you. The Qin and Han Dynasties had the best bows, crossbows, and mobile artillery of their time, and were capable of effectively penetrating European armor.
Logical fallacy, as already demonstrated- during the mongol period the big killer was not the bow, but the inability to fight at the Mongols pace. The knights in chain mail had to be run to exhaustion and then finished off.