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Tilly studied the principles of geometry, Euclid's 5th postulate and non-euclidean geometry. In 1860 he achieved results similar to those of Lobachevsky but at this stage he had not heard about Lobachevsky. He learnt about Lobachevsky's work in 1866, then in 1870 he published a work on Lobachevsky space.
Tilly was the first to study non-euclidean mechanics, a topic he invented. Tilly corresponded with Jules Hoüel, the only French mathematician interested in these topics at that time. Until this point Tilly had worked in isolation.
He also wrote on military science and the history of mathematics in Belgium.
There were complaints that Tilly had unduly emphasised the scientific education of future officers. An inspector at the military school declared that Tilly was not allowed to use differentials. Tilly must have carried on with his methods of teaching despite this and he was dismissed from his post and forced into early retirement in 1900.
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