He entered the University of Turin in 1879 and there studied for his laurea in mathematics. He has some outstanding teachers, Enrico D'Ovidio taught him geometry while Angelo Genocchi and Francesco Faà di Bruno taught him analysis. Faà di Bruno was particularly important as a teacher for he had studied in Paris under Cauchy and so brought a wider European breadth to his teaching. Gino Loria, who was to write famous texts on the history of mathematics, was a fellow student of Segre's and they remained friends throughout their lives. In 1881-82 D'Ovidio proposed as the topic of his course on higher geometry, the geometry of ruled spaces and Segre, just eighteen, was inspired to make a personal elaboration of Battaglini's theory of line complexes, adding new properties which he presented in a lecture at the school of Education :-
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.In his fourth and final year of study (1882-83), in addition to the compulsory courses on Higher Mechanics, Astronomy and Mathematical Physics, Segre again followed the course of higher geometry given by D'Ovidio and the analysis course by Faà di Bruno. Segre fully understood the importance of mastering both the geometrical methods as well as those of analysis. However this fourth and final year proved to be a very difficult one for him as his brother Arturo related (see  and also ):-
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.The degree was awarded for his thesis Studio sulle quadriche in uno spazio lineare ad n dimensioni ed applicazioni alla geometria della retta e specialmente delle sue serie quadratiche Ⓣ which he wrote while being advised by D'Ovidio. It was in two parts, one on quadrics in higher dimensional spaces with the other part on the geometry of the right line and of its quadratic series. The thesis received high praise and he published it as two papers in Volume 36 of the Memoirs of the Turin Academy of Sciences. Guido Castelnuovo writes in , forty years after Segre wrote his thesis:-
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.Even before he had completed his thesis, Segre had sent the paper Sur les différentes espèces de complexes du 2 degré des droites qui coupent hamoniquement deux surfaces du second ordre Ⓣ to Felix Klein for publication in Mathematische Annalen. It was one of fifteen papers he published in the two years 1883 and 1884. The paper appeared in 1883 and it started a correspondence between Segre and Klein which they actively pursued for many years. In fact in 1921, when both he and Klein were nearing the ends of their lives, he wrote, "[Klein] was my teacher, although we were at this distance!" For the academic year 1883-84, Segre was appointed as an assistant to D'Ovidio, the professor of algebra and analytic geometry at Turin. However, he was drafted to undertake military duties in November 1883 which did not please him. However, although occupied with his duties during the day, in the evenings he was still able to undertake research and read the latest papers published in the areas that interested him. He also continued to correspond with Klein telling him how impatient he was for the day when his military duties would end. The following academic year he was able to return to his duties as D'Ovidio's assistant and gain a teaching qualification in higher geometry. D'Ovidio was very pleased with his young assistant writing (see ):-
... 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.Another young mathematician at Turin, five years older than Segre, was Giuseppe Peano. He had been D'Ovidio's assistant and then became an assistant to Angelo Genocchi. The authors of  make an interesting comment about Peano and Segre at this time:-
... 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.In 1885 Segre was appointed as assistant to Giuseppe Bruno who held the chair of projective and descriptive geometry. During the two years 1886-88 he gave a course on the theory of plane algebraic curves. In 1887 he wrote to Klein (see ):-
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 Sn, by thinking about it always, that you become familiar with these arguments.By now his reputation was very high and he was offered a chair at the University of Naples. However, he preferred to remain in Turin and the university there was keen to keep him. As a consequence, the university asked the Minister of Education to split Giuseppe Bruno's chair of projective and descriptive geometry into two chairs. The request was refused so a new chair of higher geometry was created. Appointments to chairs were by competition and, in 1888, Segre entered the competition for the new chair of higher geometry in Turin. He was appointed and continued to hold this professorship for the next 36 years until his death. However, he didn't just teach university courses for he also taught at the teacher training college, the Scuola di Magistero, which was attached to the university. In fact, in 1916 he became the director of the teacher training college.
The interaction between Segre and Klein saw Segre become enthusiastic for Klein's 'Erlangen Programme'. Thomas Hawkins writes about this in :-
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'.In 1891 Segre went on an extended tour of Germany visiting the mathematical centres of Göttingen, Frankfurt, Nuremberg, Leipzig and Munich. He met many of the leading mathematicians, including Leopold Kronecker, Karl Weierstrass, Max Noether, Theodor Reye, Rudolf Sturm and Moritz Cantor. In Göttingen he met Felix Klein in person for the first time, although as we noted above they had corresponded since 1883. Segre wrote from Göttingen to Guido Castelnuovo in June 1891:-
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!Also in 1891, Segre wrote the article Su alcuni indirizzi nelle investigazioni geometriche Ⓣ which was translated into English and published by the American Mathematical Society as On some tendencies in geometrical investigations. It is an article advising young students how to go about research in geometry. It is a fascinating article, both for the advice it gives young people and also for the insight it gives us into Segre's thinking about geometrical research. Coolidge writes:-
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.In 1893 Segre married Olga Michelli; they had two daughters, Elena and Adriana.
Later in his career, Segre took on administrative positions in the university. For example he was director of the Biblioteca speciale di Matematica from 1907 until his death. Also he was dean of the Faculty of Science for two spells, the first being 1909-10 and the second 1915-16.
The papers that Segre published in the 1880s are studied in detail in  where the authors see this work as beginning a path that eventually led to the theory of vector bundles on an algebraic curve. They write:-
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.Segre worked on geometric properties invariant under linear transformations, algebraic curves and ruled surfaces studying transformations already considered by Alexander von Brill, Alfred Clebsch, Paul Gordan and Max Noether. In  Pierre Speziali says that through this work of Segre's:-
... 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.Using the methods which he had introduced, Segre was able to study Kummer's surface in a much simpler way. This surface, which had been discovered by Kummer in 1864, is a fourth order surface with sixteen double points. In a paper published in 1896, Segre found an invariant of surfaces under birational transformations which had appeared in a different form in a 1871 article by Zeuthen: this invariant is now called the Zeuthen-Segre invariant. In 1890 Segre looked at properties of the Riemann sphere and was led to a new area of representing complex points in geometry. He introduced bicomplex points into geometry. Motivated by the works of von Staudt, Segre considered a different type of complex geometry in 1912. Among other important work which Segre produced was an extension of ideas of Darboux on surfaces defined by certain differential equations.
Segre was one of only four invited plenary speakers at the International Congress of Mathematicians held in Heidelberg in August 1904. He gave the address La geometria d'oggidi e i suoi legami coll'analisi Ⓣ. In his talk he claimed that :-
... 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.For the remarkable number of papers published by Segre, see THIS LINK.
The reader might be surprised to learn that he had problems which limited the amount of work he could do later in his career. Julian Coolidge writes :-
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.In  his clarity of writing is mentioned and illustrated with these comments:-
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.This clarity is also emphasised by Henry Baker :-
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.Next we quote the summary of Segre's importance as described in the  article:-
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.Segre received many honours and we mention some of them. In 1889 he was elected to the Academy of Sciences in Turin, in 1898 he shared the Accademia dei Lincei's Royal Prize with Vito Volterra, and in 1901 he was elected to the Accademia dei Lincei. His achievements have been recognised recently with the conference 'Homage to Corrado Segre (1863-1924)' which was held in Turin in November 2013 to celebrate his 150th birthday. The conference announcement stated:-
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.Finally, we give several accounts of Segre as a teacher. Gino Fano writes in :-
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.Francesco Severi writes in :-
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.Beniamino Segre writes in  that Segre's lectures were:-
... 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.Here is Julian Coolidge's account of Segre as teacher and friend :-
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.
Article by: J J O'Connor and E F Robertson
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