His first memoir on the foundations of mathematics was published there in 1810 but, after it received less than good reviews from Lacroix and Lagrange, Wronski broke off relations with the Institute in Paris.
Among other things he did was design caterpillar vehicles to compete with the railways. However they were never manufactured.
His main work involved applying philosophy to mathematics, the philosophy taking precedence over rigorous mathematical proofs. He criticised Lagrange's use of infinite series and introduced his own ideas for series expansions of a function. The coefficients in this series are determinants now known as Wronskians (so named by Muir in 1882).
In 1812 he published a work claiming to show that every equation had an algebraic solution, contradicting Ruffini's results which were already published. Wronski's work here, although of course wrong, nevertheless still has important applications.
Wronski spent the years 1819 to 1822 in London. He came to England to try to obtain an award from the Board of Longitude but his instruments were detained by the Customs as he entered the country. He found himself in severe financial difficulties but, after his instruments had been returned to him, he was able to address the Board of Longitude. His address On the Longitude only contained generalities and did not impress.
His book Introduction to a course in mathematics was published in London in 1821.
For many years Wronski's work was dismissed as rubbish. However a closer examination of the work in more recent times shows that, although some is wrong and he has an incredibly high opinion of himself and his ideas, there is also some mathematical insights of great depth and brilliance hidden within the papers.