**Julio Rey Pastor**entered a world of accelerating scientific activity, particularly in Spain. The establishment of La Institución Libre de Enseñanza in 1876 had an enormous impact on the reinstatement of science in Spain. Founded by a group of professors who refused to adjust their methods to the official religious, political or moral dogmas, the aim of the institution was to defend the freedom of education. This idealism obliged them to pursue their goal on the fringes of intellectual society and thus create a private, education establishment.

Additionally, the German defeat of France in the Franco-Prussian War of 1870, marked the end of French hegemony in continental Europe and a significant shift from a French-orientated Spanish culture to one of German orientation. This was of great importance since it would transcend to Latin America through mathematicians such as Rey Pastor, Esteban Terradas and many others.

When Santiago Ramon y Cajal became the first Spaniard to win the Nobel Prize, together with Camillo Golgi in 1901 for their work on the structure of the nervous system, it seemed that the Spanish scientific community had finally awoken from a period of prolonged stagnation. Following this in 1907 was the creation of La Junta Para Ampliación de Estudios, which arose as a result of Spain's cultural self-reflection after losing her final colonies. The key objective of the organisation was: Pedagogic renovation through the provision of grants in order to elevate Spanish culture and bring it to the level of the most advanced countries in Europe, such as Germany. This was to have a direct effect on Julio Rey Pastor.

Educated at home until the age of twelve, Rey Pastor began studying at his local secondary school, El Instituto Sagasta, in 1900. He had broad interests in his youth and before he became completely occupied with mathematics, he wrote poetry. Having failed the mathematics section of the entrance examination for the Military Academy in Zaragoza, he began studying science at Zaragoza University in 1904. It was here that his true vocation for mathematics was awoken and he published his first paper in 1905, entitled *Sobre los números consecutivos cuyo suma es a la vez cuadrado y cubo perfecto* Ⓣ. He graduated with a PhD in algebraic geometry from Madrid University in 1910.

Between 1908 and 1910, Rey Pastor founded the Real Sociedad Matemática Espanola with the support of José Echegaray and General Benitez. In 1911, Pastor was appointed secretary of the society and, in the same year, he became Professor of Mathematical Analysis at the University of Oviedo. The university, situated in Oviedo in northern Spain, was an ancient one founded in 1608. It was here that he wrote the controversial inaugural address for the academic year 1913-1914, in which he candidly discussed the lamentable state of Spanish science since the sixteenth century. As a consequence, Rey Pastor was accused of being unpatriotic and his reputation suffered significantly. An example of his frank discourse may be seen his 1915 Inaugural Discourse, where Rey Pastor discusses the development of science in Spain.

You can see a translation of part of this address.

Between 1911 and 1916, La Junta Para Ampliación de Estudios funded Rey Pastor to carry out a series of visits to Germany. This resulted in two major publications on geometry in 1912 and 1916. The 1916 monograph was on synthetic geometry in *n*-dimensions and introduced [1]:-

In 1915 Rey Pastor moved to a chair at Madrid University, where he published the highly acclaimed,... concepts of great generality(for example the definition of the curve)and developing them in all their consequences.

*Fundamentos de la geometría proyectiva superior*Ⓣ. However he was not one to remain fixed in one place for a long time and went to Barcelona in 1915 to give a series of lectures at the Institut d'Estudia. His lectures there on

*n*-dimensional geometry and conformal mappings, developing the work of Schwarz, was written up by Esteban Terrades who attended the lectures, and the course was published in Catalan.

Rey Pastor was invited by the Institución Cultura Española to lecture at the University of Buenos Aires in 1917. Although he was still a young man, only 29 years old, he was asked to help promote mathematics in Argentina and a way was found to enable him to do this. A contract was proposed which enabled him to spend six months each year in Argentina and six months in Spain. Rey Pastor was pleased to sign the contract:-

It has been asserted that Don Rey Pastor was responsible for the creation of a distinctive Argentinean School of mathematical research and the reconstruction of science in Argentina. When he took up a 6-year contract at Buenos Aires University in 1921, the Science Faculty had only one PhD programme, which had barely progressed since 1900. Between then and Rey Pastor's arrival, there had been some small improvement in the engineering courses, but all the other further mathematics courses were in a severe state of stagnation. Rey Pastor needed to persuade the engineering professors of the importance of mathematics beyond the elementary textbooks, which they had previously been using.... to direct the advanced study of the exact sciences in Argentina.

He had been invited by the Institución Cultura Espanola, to give a lecture course at the University of Buenos Aires in 1917. This first course, given as a visiting lecturer, was an introduction to Klein's *Erlangen Programme*. In this course, Rey Pastor presented his students with the concept of geometry based on group theory, using methods of establishing invariants of each group, with topological methods being the most general. His second course, given in 1921, was a specialised one for engineering students and included the following topics: functions of a complex variable, conformal mapping, advanced geometry (non-euclidean), mathematical analysis and mathematical methodology. Many of these topics, although commonplace in Europe, were entirely new and revolutionary to Argentinean mathematicians. As a consequence, Rey Pastor succeeded in gaining popularity amongst the students, whilst receiving stern criticism from his contemporary old school professors, who regarded him a foreign usurper.

Rey Pastor focused mainly on teaching engineering, but he recruited many pure mathematics students to his courses since he regarded the engineering courses on the techniques of calculation good preparation for pure mathematicians. He believed it was important to maintain an understanding of both areas of mathematics and it was this characteristic, panoramic vision of his, which allowed his students to appreciate the profundity of the new concepts he was teaching them. By separating mathematics from its purely technical aspect, he attracted a wide audience on both sides of the Atlantic Ocean.

Rey Pastor founded the Sociedad Matemática Argentino in 1924. In 1927, he was given a permanent appointment at the University of Buenos Aires and held two chairs: one of Mathematical Analysis and the other of Higher Geometry. This was to have a profound and transcending impact on Argentinean mathematics. In 1928, he founded an influential mathematical seminar El semenario Matemático Argentina (similar to the one he had set up in Madrid). The seminar published a bulletin, which contained the first modern research in Argentinean mathematics. He also brought important foreign mathematicians to the university to give short courses, including: Frederigo Enriques (1925), Francesco Severi (1930), Tullio Levi-Civita (1937), Émile Borel (1928) and Jacques Hadamard (1930). In the 1940's, Pastor's best students began gaining international recognition. Amongst them were: Alberto González Domínguez, who became an important quantum physicist; Alberto Calderón, who would become the chairman of the Department of Mathematics at Chicago University; and many more who considerably enhanced the collective teaching capacity of the mathematical community.

In 1931, Rey Pastor published one of his most elegant works on analysis in the *Rendiconti del Circolo Matematico di Palermo*, an Italian mathematical journal. It dealt with the study of the method of summation of series. This article by Rey Pastor is framed by a long series of works, begun at the beginning of the twentieth century, on problems of summing series, convergence algorithms, singular integrals and comparative studies of series and integrals. He first presented his work in this area in 1926 in his course on "Series and Integrals" which he gave at Buenos Aires University. The same course, slightly amplified, was repeated in Madrid in 1928. The same year Rey Pastor presented a summary of his ideas in his paper given to the International Congress of Mathematicians at Bologna, which he attended with a large group of his Argentinean students. He continued working on problems related to the theory of summation of divergent series throughout the 1930's and published much of his work in international journals. The themes which Rey Pastor dealt with in this period had considerable influence on the development of Argentinean mathematics.

In 1952 he was expelled from Argentina despite his efforts to remain apolitical. However, he retired with the satisfaction of having initiated the transformation of Argentinean mathematics. He contributed exceptionally to the development of investigation into pure mathematics and trained a new school of Argentinean engineers with a modern outlook. Furthermore, his teaching opened the doors to the study of the history of science and laid the foundations for generations of secondary school teachers with his influential textbooks. In 1954 he returned to Argentina and joined La Academía de Lengua.

The history of mathematics had always interested Rey Pastor and late in his career his interests in historical topics extended to cartography. Of course Spain has a reputation for remarkable cartography so his monograph (written jointly with E Garcia Camarero in 1960) on the history of Spanish cartography was a particularly useful addition to knowledge of the topic.

Examining his textbooks gives us insight into Rey Pastor's ideas concerning teaching of mathematics. In the introduction to *Algabraic Analysis*, Rey Pastor comments that rather than follow the general tendency of elevating elemental problems to the point of abstraction, it is his goal to simplify complicated questions whilst maintaining a rigorous approach. He adds that all abstract thought requires a pre-existing knowledge base, which many students lack when they arrive at university and, which they expect to gain over the course of their degree. According to Rey Pastor however, it is a didactic mistake and historically absurd to attempt to approach the concepts of analysis in this contrary manner.

He died at his home in Buenos Aires.

**Article by:** *J J O'Connor* and *E F Robertson* based on an honours project by *Jenny Kirkland* (University of St Andrews).

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