This should be put in the category of educated insults, since only those who have swallowed the dictionary or know Latin literature understand what it means. A thrasonical person is a braggart. The original was a former soldier named Thraso, a character in the play Eunuchus (The Eunuch), which was written in 161 BC and became the most popular of the six by the writer whom we know as Terence.
Thrasonical started to appear in English in the sixteenth century, in time for Shakespeare to put it into the mouth of Rosalind in As you Like It. She describes Julius Caesar’s famous assertion veni, vidi, vici (“I came, I saw, I conquered”) as a thrasonical brag.
These days, its most frequent appearances are in a widely-reproduced bit of advice to aspiring authors or public speakers:
Let your conversation possess a clarified conciseness, compacted comprehensibleness, coalescent consistency, and a concatenated cogency. Eschew all conglomerations of flatulent garrulity, jejune babblement, and asinine affectations. Let your extemporaneous descantings and unpremeditated expatiations have intelligibility, without rhodomontade or thrasonical bombast. Sedulously avoid all polysyllabical profundity, pompous prolixity, and ventriloquial vapidity. Shun double-entendre and prurient jocosity, whether obscure or apparent. In other words, speak truthfully, naturally, clearly, purely, but do not use large words.
Notes and Queries, 11 Feb. 1893.
In an idle moment, I set out to trace this to its origin. It turns out to be a hardy perennial, which became popular on both sides of the Atlantic from the 1880s on, appearing regularly in magazines and newspapers. The earliest unearthed so far is in the Pennsylvania School Journal of 1874. It is surely older still.
You may feel that both thrasonical brag and thrasonical bombast are tautological. I couldn’t possibly disagree.