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The first "blowing up" of anything is the blowing up of bombs. This did happen in China in the 12th century (Science and civilisation in China, Volume 5, Part 7 By Joseph Needham, Gwei-Djen Lu, Ling Wang). But that's not really what you mean, you mean using gunpowder to blow up things that are not weapons. The damage from these early bombs is more from shrapnel than from the explosive power of the bomb itself, I suspect.
Needham also mentions an "enourmous bomb" used to repel the Mongols in 1277. This is probably similar to what the 14th century manual "Huolongjing" (Fire Dragon Manual) describes in various land and sea mines, but without the trigger mechanisms that they used by the 14th century.
I think the Helms Deep explosion could reasonably be seen as something similar to this 1277 "enourmous bomb". The Chinese knew about making small bombs that were thrown with trebuchets. That somebody comes up with the idea of super-sizing it is not surprising, and the same would go for the Tolkien Universe.
The first case I know of blowing things up (or trying to) in Europe with gunpowder is of course the 1605 gunpowder plot. There may be others.
Upvote:2
Henry of Navarre first made use of the petard in 1580. This predates rock blasting at quarries in Hungary, which is otherwise the first use of gunpowder to blow something up in Europe. (Fawkes failed, if you recall.)
While it does require a metal "bell" to direct the blast, the purpose of the petard is to knock a hole in a structure rather than to shoot a projectile, so I think it counts by your criteria.
Upvote:3
(For a mobile bomb see Swamp Yankee's answer)
The first reliable documented use of engineering explosives was at the Siege of Serezenella (1487) by the Genoese. The chief engineer of the Genoese, Pietro Navarro, a Spaniard, was the designer. He also mined the Castle of the Egg (Castel dell' Ova) at Naples (1503), when he was serving with the Spanish. Reportedly the early use was not decisive, but at the Castle of the Egg, an explosive mine was the key factor in the success of a siege for the first time. For this reason explosive mining is normally dated as beginning in that year, 1503.
Upvote:6
Also, Henry Stuart, Lord Darnley had his house blown up by his wife, Mary Queen of Scotts - after he stabbed her handsome continental boy-toy in front of her -
"She brought him to Kirk O'Field with the [stated] intention of incorporating him into the court system again. Darnley stayed in Edinburgh while Mary attended the wedding of one of her friends. Around 2 am on the night of February 10, 1567, while Mary was away, two explosions rocked the foundation of Kirk O'Field. These explosions were later attributed to two barrels of gun powder that had been placed in the small room under Lord Darnley's sleeping quarters."
Since that didn't do the trick, her conspirators strangled him, dumped him in the woods, and framed Blackadder for it.