Josef Hoëné Wronski was named Josef Hoëné after his birth in the town of Wolsztyn about 60 km south west of Poznan. His parents, from Czech families that had settled in western Poland, were Elzbieta Pernicka and Antoni Hoehne who was the municipal architect of Poznan. Josef Hoëné was educated in Poznan and Warsaw but in 1794 he became involved in the Kosciuszko Uprising.
Despite battles for independence, Russia and Prussia partitioned Poland in 1793. General Tadeusz Kosciuszko, a hero of the American War of Independence, returned to Poland in 1794 and gathered an army of Poles to again fight for independence. Hoëné Wronski (we use this name although he did not adopt Wronski until around 1810 just after he married) joined Kosciuszko as second lieutenant in the artillery. After winning the battle of Raclawice and capturing Warsaw and Wilno, Kosciuszko's army was defeated by Russian and Prussian forces after a siege of Warsaw. Kosciuszko was wounded and taken prisoner by the Russian army which carried out a massacre of the population in the Warsaw. Hoëné Wronski was also taken prisoner, and made to serve in the Russian army which he did until 1797 when he was released having reached the rank of lieutenant colonel.
By this time Hoëné Wronski's father had died and left him money which allowed him to spend the next years in Germany studying philosophy at a number of different universities. He enlisted in the Polish Legion at Marseilles, in France, and become a French citizen in 1800. He began to undertake scientific work in Marseilles, joining the observatory there in 1803, and started to work out a theory of the universe and its origins. Then in 1810 he moved to Paris and, in the same year, he married Victoire Henriette Sarrazin de Mountferrier, whose brother was the mathematician Alexandre Mountferrier. It was at this time that he adopted the surname Wronski but he did not use it consistently, rather alternately used Wronski and Hoehne. When writing papers, however, he used the name Hoëné Wronski without the use of any first name. Bushaw writes in  that Hoëné Wronski:-
... was brilliant, erudite, industrious, versatile, and ambitious. His many projects ranged from the design of water works, navigational instruments, and railway car wheels, through mathematics, astronomy, and other sciences to the farthest reaches of metaphysics ... Wronski also had a difficult personality, and has been accused, not without plausibility, of arrogance, charlatanry, paranoia, and other blemishes of character.
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. He did, however, have strong financial support from a financier Pierre Arson after he arrived in Paris and this continued for many years. Eventually they fell out over the financial arrangement which had been put in place and this dispute became very public with high profile court actions in 1819. A piece of work which he had undertaken during this period resulted in a publication Résolution générale des équations de tous degrés in 1812 claiming to show that every equation had an algebraic solution. This contradicted Ruffini's results which were already published but of course Ruffini had failed to convince many of the truth. Wronski's work here, although of course wrong, nevertheless still has important applications and contains interesting novel ideas.
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. As always seemed to happen with Wronski's ventures, it degenerated into an argument which had little to do with the science. He then tried to get the Royal Society to show an interest in his work on hydrodynamics. This again quickly degenerated into an argument which had nothing to do with the science.
His main work involved applying philosophy to mathematics, the philosophy taking precedence over rigorous mathematical proofs. He wrote on the philosophy of mathematics. His book Introduction to a course in mathematics was published in London in 1821. He criticised Lagrange's use of infinite series and introduced his own ideas for series expansions of a function. Out of this came his "universal Hoëné-Wronski series" or "la série universelle de Wronski". This consisted of the development of a function as a series in terms of arbitrary functions. Also he gave his "highest law" or "La loi suprême" which was a law giving a general rule to calculate the coefficients. The coefficients in this series are determinants now known as Wronskians (so named by Muir in 1882).
In 1827 he published his Canons de logarithmes . Bushaw writes:-
There may be no more striking example of cleverness [in the history of tables] than Wronski's 'canons of logarithms'. This does not seem too strong a claim to make for a scheme that made it possible to put a seven-place table of common logarithms, in readable type, on a single page less that 17 cm by 22 cm in size. ...
The booklet Canons de logarithmes :-
... includes instructions, examples, and theory for the use of six one-page tables of common logarithms. For good measure, it contains a summary of the "general solution of the fifth degree equation". In the Prospectus, Wronski argued that conventional tables are expensive, are bulky, have too many pages to turn, and frighten people. He had therefore devised his extremely compact "canons" (the term goes back to Napier) with none of those disadvantages.
This was not Wronski's only work in calculating and aids for calculating :-
Wronski was apparently a tireless reckoner. One form this interest took was designing computational aids not only for scientists but also for non-scientists - school children, soldiers, business people, and others. Some of these devices should be called pre-electronic calculators, and in fact Wronski called one of his inventions the 'calculateur universel'. He left unfinished several projects of this kind ...
Among other things he did in the 1830s was design caterpillar vehicles to compete with the railways. However they were never manufactured. For many years all Wronski's work was dismissed as rubbish. This was largely due to his personality leading to :-
... grandiose exaggeration of the importance of his own research, violent reaction to the slightest criticism, and repeated recourse to non-scientific media as allies against a supposed conspiracy.
However a closer examination of the work in more recent times shows that, although some is wrong and he does have an incredibly high opinion of himself and his ideas, there is also some mathematical insights of great depth and brilliance hidden within the papers.
Although not actually his last words, nevertheless, during last stages of his final illness he said:-
God Almighty, there's still so much more I wanted to say!
Article by: J J O'Connor and E F Robertson