Veronese was fortunate, however, and the following year he was able to begin his studies again. He found himself a patron in Count Nicolò Papadopoli who supported him financially so that he could go to Zürich Polytechnic in 1873. There he set out on a course of study which involved both engineering and mathematics. However, he began to correspond with Cremona, who was at the University of Rome, on mathematical topics. He started work on a paper on Pascal's hexagram but, following Cremona's advice, he moved to Rome to complete his undergraduate degree.
In 1876 Veronese was appointed as assistant in analytical geometry on the strength of his paper on Pascal's hexagram which he had completed by this time. This is quite remarkable for one should remember that at this point Veronese was still studying the undergraduate course in Rome. He graduated in 1877 and continued to work for his doctorate in Rome. Veronese was in contact with Klein who was about to take up a chair of geometry at the University of Leipzig. It was arranged that Veronese would go to Leipzig in 1880 and to spend the year 1880-81 undertaking research under Klein.
Bellavitis died in November 1880 and his chair of algebraic geometry in Padua became vacant. The chair was filled by holding a competition which Veronese won and he was appointed to the chair in 1881. He held this chair throughout his life.
Freguglia, in , describes Veronese's study of geometry in higher dimensions. In 1880 Veronese described an n-dimensional projective geometry, showing that simplifications could be obtained in passing to higher dimensions. He illustrated the fact that difficulties arose when a simple surface in high dimension was projected onto 3-space. This was a very original approach to higher-dimensional projective geometry that Veronese developed. He is certainly considered to be one of the founders of that topic for with him what others had considered as linear algebra viewed geometrically became geometry.
This original approach was based on the supremacy of geometric intuitive techniques over the analytic and algebraic viewpoints. Veronese provided both logical and psychological motivations for his approach which greatly influenced the Italian school of geometry for many years.
Veronese invented non-Archimedean geometries which he proposed around 1890. However Peano strongly criticised the notion due to the lack of rigour of Veronese's description and also for the fact that he did not justify his use of infinitesimal and infinite segments. The resulting argument was extremely useful to mathematics since it helped to clarify the notion of the continuum. Any fears that non-Archimedean systems would not be consistent were shown to unnecessary soon after this when Hilbert proved that indeed they were consistent.
We should mention one or two further aspects of Veronese's life. He wrote a number of useful secondary school texts on mathematics and he also became involved in politics. He served as a member of the Parliament of Chioggia from 1897 to 1900, then later he served as a member of the Padua City Council, finally being a Senator from 1904 until his death.
He had two particularly famous pupils. Castelnuovo, one of the greatest algebraic geometers of the Italian school, was his pupil in the mid 1880s and Levi-Civita was one of his pupils about ten years later.
Article by: J J O'Connor and E F Robertson
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