Gunpowder
Gunpowder began as a by-product of the search for an elixir of life. Tang-era Daoist texts give the earliest evidence, from 808 and about fifty years later, and the earliest chemical formula for gunpowder appears in the Wujing Zongyao, compiled by Zeng Gongliang between 1040 and 1044.
Westward, the dates bunch up in the 13th century, the age of the Mongol conquests. The Syrian writer Hasan al-Rammah recorded recipes and methods for purifying saltpeter between about 1240 and 1280, and Roger Bacon's descriptions of 1267 are the earliest in the West. The Mongols are the usual suspects as carriers, but no clear transmission route is documented, so the "how" remains an open question even though the Chinese priority is not.
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