D'Ovidio started from the ideas of Plücker, which had been taken up and developed by Felix Klein. According to these ideas, the geometry of ruled space is equivalent to the study of a quadratic variety of four dimensions imbedded in a linear space of five dimensions. In his lectures D'Ovidio examined the works of Veronese on the projective geometry of hyperspaces and those of Weierstrass on bilinear and quadratic forms.

The fourth year of university (1882-83) was especially difficult for him, a year that was exceedingly painful, during which my family went through economic collapse and the sad end of my poor father's life. He composed his thesis in that terrible predicament, and took his degree in July 1883 with honours.

Still today those who read ... the two closely connected works remain surprised by the confidence and breadth of vision and mathematical means with which this young man, Corrado Segre, deals with this broad topic. The dissertation appears to be the work, not of a beginner, but of an experienced and skilled mathematician.

... he has a keen intellect, is industrious, painstaking, and able to successfully treat the most difficult geometrical problems and solve them with clarity and elegance. As a teacher he has already proved his worth during the year as my assistant, being successful, clear, accurate and effective.

... in the mid 1880's, these two very young researchers, Segre and Peano, both of them only just past twenty and both working at the University of Turin, were developing very advanced points of view on fundamental geometrical issues. Although their positions were quite different from one another, they were in some way more complementary than opposed. So it must come as no surprise that Turin was the cradle of some of the most interesting studies on such issues.

What you tell me about the effect on you made by the synthetic reasoning about the geometry of $n$ dimensions does not surprise me; it is only by living in $S^{n}$, by thinking about it always, that you become familiar with these arguments.

Shortly after [Segre] assumed the chair of projective geometry in Turin in 1888, he decided it would be worthwhile to have an Italian translation of the 'Erlangen Programme' because he felt its contents were not well enough known among young Italian geometers. ... Segre convinced one of the students at Turin, Gino Fano, to make a translation which was published in 'Annali di Mathematische' in 1890. Fano's translation thus became the first of many translations of the 'Erlangen Programme'.

No one who has never been here can imagine what kind of man Klein is, and what kind of organisation he was able, with a skill that no one else could have, to impose on the mathematical studies in this university: it is something that has made an extraordinary impression on me. And I have already been most vividly impressed by the scientists that I have met on this journey!

The point which Segre enforces most strongly is that at all costs the beginner must learn to choose really significant subjects. To acquire the instinct to do this, he should study the mathematical classics. He must especially avoid trivial questions and perfectly obvious extensions or generalizations. The young geometer should also be familiar with the leading methods and results of modern analysis, and welcome help from every quarter, regardless of whether his own preference is for synthetic, algebraic, arithmetical or transcendental methods. With regard to rigour, Segre recommends the geometer to be as rigorous as he can, and to own up like a man when he makes use of methods of doubtful parentage.

To appreciate the innovative character of the ideas put forward by the very young Segre, it is convenient to recall that in those years it was harshly debated upon the usefulness of studying hyperspace geometry. Some authors maintained that addressing the geometry of the hyperspace was an unfruitful intellectual game not certainly helpful to understand the "real" geometry in two or three dimensions. On the other side, Veronese and Bertini at first, and then Corrado Segre were perfectly aware that not only the geometry of hyperspaces would shed new light on the geometry of curves and surfaces of ordinary space, but also that these latter could be viewed - and this is certainly innovative - as points (defined by a number of parameters) belonging to new algebraic varieties that could not be placed in ordinary space.

... it thus became possible to reduce the classification of surfaces to that of curves. The insufficiencies of the earlier theories proposed by A Möbius, Grassmann, Cayley and Cremona were thus soon revealed.

... geometry and analysis deal with what is essentially the same subject matter, but with different emphasis, terminology and aesthetic aim ... He lays special stress on various branches of algebraic geometry, particularly those properties of figures which are unaltered by birational transformation.

At some period during the early part of his career, he had a severe attack of brain fever brought on by overwork. He complained sadly that his capacity for labour was never the same thereafter. When one reflects that the number of his published titles is 128, it fatigues the imagination to speculate as to what he might have accomplished had he retained his early strength.

Segre wrote a long article on hyperspaces for the 'Encyklopädie der mathematischen Wissenschaften', containing all that was then known about such spaces. A model article, it is notable for its clarity and elegance.

As a man, one thinks of him as always mainly led by his devotion to his subject; as a writer, as concerned above all things to conduct his reader to an eminence from which may be seen spread out, each in fitting place, all the details of a theory hitherto believed to be incapable of oversight. This is achieved, repeatedly, by the use of higher space, and his power of communicating his own intense sense of the reality of this.

Through his teaching and publications, Segre played an important role in reviving an interest in geometry in Italy. His reputation and the new ideas he presented in his courses attracted many Italian and foreign students to Turin. Segre's contribution to the knowledge of space assures him a place after Cremona in the ranks of the most illustrious members of the new Italian school of geometry.

Corrado Segre is widely recognized as one of the founders of the Italian School of Algebraic Geometry, where he counted G Castelnuovo as an assistant, F Enriques as a postdoc, F Severi as a student, G Fano and A Terracini as students and colleagues. Moreover Corrado Segre is regarded as a crucial figure in the scientific evolution and in the history of Algebraic Geometry.

He considered it a true mission to direct his students towards the upper levels of mathematics, and especially of geometry, encouraging them whenever possible to produce original work ... He lavished infinite care and treasures of knowledge on his 36 courses of advanced geometry, the subjects of which he himself expounded in writing, in his clear, distinct hand, in little books which his old and recent students were very familiar with, always very precisely worded and with numerous bibliographical quotations, with complements which gradually occurred to him, often with original ideas and opinions, with indications of topics for further research, from which he drew the subjects he suggested for degree dissertations.

He was one of the most careful in the preparation of his Lectures, that I have ever known. In fact they were written in advance word for word and in definitive form in little booklets, which he took with him to his lectures, so that he could give bibliographical indications from them, always exact and exhaustive.

... of a highly skilful, elegant, stimulating interweaving of synthetic considerations and algebraic developments, the latter being kept to a minimum and carried out in such a way that they make entirely clear the geometric content of the results, at times even of the individual stages, and supply appropriate checks on the most delicate points.

Behind Segre the geometer and the expositor, there remained always Segre the teacher and friend. An interesting figure he was in the lecture room. Of medium height and frail, half seated on the end of the table, gesticulating rapidly with his left hand, while his right whirled his watch chain about with astonishing angular velocity, his whole being was thrown into the task of driving home the essentials of what he had in mind. The subject matter of his course was new every year. There was no limit to the amount of care and patience which he would bestow on one of his pupils. ... He had a sympathetic, sensitive nature, a spiritual quality, which impressed all who had the privilege of his friendship.