The Triumph was first published in 1790 by Peter Thompson (See here also), for a 32-bar dance which differs from that known now. Campbell-Aird published a slightly different version around 1796 and that appears to the the basis of English versions thereafter. Gow published a variation in 1805 which became the Scottish version. The third part of the dance was introduced in an anonymous manuscript of 1818 and was quickly adopted in both Scotland and England.
For full information on all versions, see Christopher Walker 'The Triumph' in England, Scotland and the United States, Folk Music Journal Vol 8 Number 1 (2001) pp4-40. PB