Studied in Napoli and he was mainly a self-taught person. During the years 1848-60, owing to his patriotic feelings he was not allowed to become a professor. After the unity of Italy he begun to teach “geometria superiore” in Napoli (1861-71) and this was the first chair with this name in Italy (with Bologna's one which was held by L. Cremona); after the liberation of Roma in 1871 he went to the University of that city, where he taught algebra (and traslated the famous book by E. Netto ) from 1871 to 1885. In his last years he returned to teach in Napoli . He was a member of the Accademia Nazionale dei Lincei. He was also Rector of the University of Roma (1873-4) and dean of the faculty of sciences.

His main contributions are about forms and line geometry. Among his students we may cite: E. D'Ovidio, R. De Paolis, E. Caporali , D. Montesano, (all algebraic gometers) and A. Capelli , G. Frattini (the most prominent Italian algebraists of the end of last century).

In 1863 he founded the Giornale di Matematica, which was one of the more staunch defender of the ideas of non-euclidean geometry.

Some of his correspondance have been published in M. Castellana-F. Palladino (editors), Giuseppe Battaglini , Levante editori, Bari , 1996, which contains also a rich biography.

Main commemorations and studies: E. D'Ovidio, Mem. della R. Acc. Nazionale dei Lincei , (6), 1, 1894, 556-610; A. Capelli, Giorn. di Mat. , 20, 1894, 205-208; F. Amodeo , Giorn. di Mat. , 33, 1907, 229-274; G. Ferrario -F. Palladino, in AA. VV., Scienza in Italia 1840-1880 , Quaderni Pristem, Milano, 1994, 123-156.