Giuseppe Veronese’s love for geometry and mathematics started during his high school years. The lessons and writings on projective geometry and algebraic geometry of his teacher, Pietro Cassini, became the foundations of the young student’s thought. In 1873, Giuseppe Veronese enrolled in the Zurich Polytechnical School. In the beginning he attended the courses in the Section of Mechanics, but later on, in 1875, he asked to be moved to the Section of Pure Mathematics for which he felt he had a natural bent; his choice was supported by Professor Fiedler.

Nuovi teoremi sull'Hexagrammum mysticum, extrcat from Memorie Reale Accademia dei Lincei (1876-77) with a dedication of Giuseppe Veronese al Prof. Antonio Favaro

In 1876, on the occasion of a Matematisches Seminar, a Seminar for which each student had to report on a research project, Veronese presented a work on Hexagrammum Mysticum by Blaise Pascal (1640). While many mathematicians had already written on that subject, Veronese wanted to give his own contribution thanks to his love for art and his mathematical intuitions. His work ‘New theorems on the Hexagrammum Mysticum’ was presented to the Accademia dei Lincei and made him famous in the mathematical world at this time. The importance of that work allowed him to transfer to the University of Rome where, under the wing of his protector Professor Cremona, he was not only admitted to the fourth year of Mathematics undergraduate course, but also he became assistant to the Office of Projective and Descriptive Geometry (1876-1880). During the years 1880-1881 he perfected his studies under the guidance of Felix Klein. Klein had an important influence on Veronese’s scientific thought and helped to develop his ideas on Hyperspace Geometry, which later were synthetized in two articles (Die Anzahl der unabhängigen Gleichungen, die zwischen den allgemeinen Charakteren einer Curve im Raume von n Dimensionen stattfinden, Math. Ann., 18, 1881, 448 and Alcuni teoremi sulla geometria a n dimensioni, Transunti della R. Acc. Nazionale dei Lincei, (3), 5, 1880-81, 333-338) and a Memory (Behandlung der projectivischen Verhältnisse der Räume von verschiedenen Dimensionen durch das Prinzip des Projicirens und Schneidens, Math. Ann., 19, 1882, 161-234).

In 1881, Veronese was appointed to the Chair in Analytic Geometry at Padua University as a successor of Professor Giusto Bellavitis and later on, he became the Chair in Superior Geometry. His ideas resulted in a fundamental change in the approach to Geometry at Padua University.

The approach became essentially synthetic and intuitive instead of analytical. Veronese’s epistemological concepts developed during a climate of open debate between analytic geometry and synthetic geometry; euclidean and non-euclidean geometry. The debate was enhanced by the new, productive dialogue between Italy and the rest of Europe, a dialogue that had not existed until that time.

To develop his theories he used, for example, the studies of Cantor, Gauss, Poincarè, Peano and many mathematicians cited him in their works both confirming (Klein and Hilbert) and contrasting (Peano, Cantor, Viviani) his theories. His work focused on the study and demonstration of geometry in spaces higher than three dimensions and on non-Archimedean geometry. He studied a fourth order surface called. "Veronese Surfacee".

Frontispiece of Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare. Lezioni per la scuola di Magistero in Matematica, 1891

He published various Notes and Observations, which found full fulfillment in the work Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare. Lezioni per la scuola di Magistero in Matematica, 1891.
worked on the adaptation of his work writing Elementi di geometria, ad uso dei licei e degli istituti (primo biennio), trattati con la collaborazione di P. Gazzaniga and Appendice agli Elementi di geometria to adapt his Fondamenti di geometria to be used by teachers and students of middle and high school.
In these works his thought based on intuition and experimental method is clarified: "that every proposition and every reasoning are preceded and continuously enlivened by spatial intuition through the observation of figures traced on the blackboard or models that help the development of the imaginative geometry and reasoning" (from Elementi di Geometria).

He was a member of various institutes and academies including the Accademia dei Lincei and the Veneto Institute of Sciences, Letters and Arts of Venice of which he was president from 26 November 1908 to 7 January 1911.