According to the local press at the time, the two men chose hot air balloons because they felt that they possessed higher intellectual properties than normal men. So even though they were trying to murder each other, they also wanted to do it in a way that was somehow insulting to the rest of the population.
So, on May 3, 1808, the duelists entered identical hot air balloons in front of a huge crowd. Each man was allowed a blunderbuss (a primitive shotgun) and a co-pilot to help him operate the balloon. In this case, though, at least one of these wingmen was about to end up like Goose.
Strangely, this incredibly inaccurate image isn't even as stupid as how it really went down.
Once they'd reached a height of about 2,000 feet above Paris, the world's first balloon dogfight commenced. De Pique fired first, but failed somehow to hit his opponent's enormous balloon with a shotgun. De Grandpre was more accurate, and so both de Pique and his co-pilot (who was, at this point, probably regretting every second of de Pique's friendship) plummeted to their deaths.