**Tomás Rodríguez Bachiller**was born in Hong Kong. His father, Tomás Rodríguez y Rodríguez de Medio, had been born in Cobreros, Zamore, and had trained as a lawyer before having a career as a consul. At the time when his son was born, he was the Spanish Vice-Consul in what was, at that time, the British colony of Hong Kong. His second wife was Julia Bachiller Fernández Ruiz, from Montemayor de Pililla, in Valladolid, who became a full-time housewife looking after the family. There were five children in the family, the two oldest being the girls Julia and Pilar, both children of Tomás Rodríguez's first marriage, and three boys from the second marriage, Tomás (the subject of this biography), Ángel and Jesús. Ángel was born in Montemayor and Jesús was born in Puerto Rico, where Tomás Rodríguez was serving as consul at the time. Their mother, Julia, died as a result of childbirth when Jesús was born and was buried in Puerto Rico. Tomás

**Bachiller and his two brothers began their first studies in Ayamonte, Huelva, Spain, and studied there until the family settled in Madrid. From that time the three brothers were put in charge of the older sisters while their father continued his consular career, making frequent and long stays abroad.**

At the age of fourteen Bachiller went to Central University of Madrid to attend mathematics lectures by José Echegaray. At this time he was studying at the Cardinal Cisneros Institute in Madrid, a leading high school in Madrid. There he showed himself to be an exceptionally talented student, and he graduated on 16 June 1916. Following graduation, Bachiller enrolled as a student at the Central University of Madrid to study mathematics. His father, however, believed that his son would never be able to make a good career for himself as a mathematician so, in 1918, he enrolled as a student in the Madrid School of Civil Engineering and began studying there in parallel to his studies in mathematics at the Central University. He attended courses on mathematical analysis given by Luis Octavio de Toledo (1857-1934), Julio Rey Pastor (who was in Argentina much of 1917-18 ), and José Ruiz Castizo (1857-1929), who was also taught a course on rational mechanics. Bachiller studied geometry with Cecilio Jiménez Rueda (1858-1950), Miguel Vegas (1856-1943), Faustino Archilla (1870-1939) and José G Álvarez Ude (1876-1958).

In January 1921, Tullio Levi-Civita made a visit to Madrid and gave a course on "Classical and relativistic mechanics". Bachiller translated the lectures into Spanish and wrote them up for publication, but despite his efforts, the notes were never published. In 1921-22 Bachiller studied "Celestial Mechanics" given by José María Plans (1878-1934). In 1923 Einstein visited Spain and before the visit the Mathematical Society held two sessions at 8 Santa Teresa Street to prepare for Einstein's lectures. On 7 March 1923 Einstein lectured to the Mathematical Society and, after a question by Bachiller, discussed the matter with him for half and hour. When Einstein gave a series of three public lectures on relativity on 5, 6 and 8 March, Bachiller was told to attend them, take notes and prepare summaries. When Bachiller's abstracts of the lectures were published, Bachiller provided copies to Einstein who told him that "in no other country in the world had they done so well".

In May 1923 Bachiller applied to the Junta para la Ampliación de Estudios (now the Consejo Superior de Investigaciones Científicas) for a one-year scholarship in order to study the theory of functions and differential equations in Germany and Switzerland. He stated that he had knowledge of French, English, German and Italian. We learn from this application that at this stage Bachiller intended to write a doctoral thesis on the algebraic geometry of the style studied by Francesco Severi's school in Italy. This scholarship was not awarded but the Faculty of Sciences in Madrid awarded Bachiller a scholarship to enable him to spend the academic year 1923-24 studying in France. He arrived in Paris in November 1923 and attended courses at the Collège de France and at the Sorbonne. In particular he attended Émile Borel's course on elasticity, Jules Drach's course on contact transformations, Émile Picard's course on algebraic curves and surfaces, Élie Cartan's course on fluid mechanics, Jacques Hadamard's course on differential equations, the course that Claude Guichard (1861-1924) delivered on differential geometry and Laplace transformations, Henri Lebesgue's course on topology and Ernest Vessiot's courses on partial differential equations and on group theory. When Bachiller returned to Madrid in 1924 he delivered the first topology course ever given in Spain.

Back at the Faculty of Sciences in Madrid, Bachiller was appointed as an assistant on 2 October 1925 to give practical classes on "Elements of Infinitesimal Calculus", and on 13 November of that year also on "Spherical Astronomy and Geodesy". He was very active in working for the Journal of the Spanish Mathematical Society. He reviewed work of mathematicians across the whole of Europe, in particular writings articles on the topology of Maurice Fréchet, the doctoral theses of Luigi Fantappié on algebraic geometry and of Louis Antoine on topology, and the monograph *L'Analysis situs et la Géométrie algébrique* Ⓣ written by Solomon Lefschetz for the Borel Collection. Another of his tasks was to keep up a correspondence with foreign mathematicians who published in the first issues of the Journal, such as Hermann Weyl, Tullio Levi-Civita, Albert Einstein and Edmund Landau, as well as translating some of their articles into Spanish. Among the articles he wrote at this time we mention: *El professor F Klein* Ⓣ (1925); *Answering Problem 24* (as set out in *Revista Mat., Hisp.-Amerika* **2** (1920), 278) (Spanish); *La correspondencia biunívoca de Cantor y el teorema de Netto* Ⓣ (1926); *Conjuntos cerrados no densos* Ⓣ (1926); *Conferencias del Prof Dr Terradas* Ⓣ (1927); and *Sobre el numero de dimensiones de un conjunto* Ⓣ (1927).

On 6 May 1926 Bachiller married Pilar Pradilla with his friend Fernando Lorente de No as a witness. Pilar was 21 years old and from Torreblanca, Castellón de la Plana. Tomás and Pilar Bachiller had three children: Tomás, Luis and Agustín.

There is a rather strange gap in Bachiller's activities between 1928 and 1933. He had been a very active contributor to the Spanish Mathematical Society and to its journal as we described above. However this all stopped during these five years when he also seemed to vanish from the Mathematical Laboratory. It is likely that this was due to a disagreement with Rey Pastor. As Glick writes in [2]:-

It is likely that some criticism of Bachiller's writings by Rey Pastor and the pointing out of two errors in an article on algebraic curves published by the Spanish Academy of Sciences caused him to give up much of his activities. The Spanish Mathematical Society reported (see [1]):-His relationship with Rey Pastor was ambivalent. They considered themselves good friends but constantly quarrelled. On one occasion, when Rey Pastor criticized Blas Cabrera, Bachiller defended the latter, saying that while the physicist had accepted a scholarship in Strasbourg, largely at his own expense, Rey Pastor had gone to Argentina "for three pesetas."

However, Bachiller did not present a further version of his paper and neither he nor Rey Pastor were present at the following meetings of the Society. In fact in the 1931 list of members of the Spanish Mathematical Society Bachiller's name does not appear. Of course, Bachiller had never submitted a thesis for a doctorate so he may well have decided that he needed to concentrate on his research rather than do work for the Spanish Mathematical Society.Rodríguez Bachiller presented a proof of Poincaré's last theorem. Rey Pastor then observed that the first part coincides exactly with the one set out by him in1919in the public session held in honour of Jacques Hadamard, and as regards the modification introduced by Bachiller in the second part, it seems to him that it proves nothing, for it confuses the various concepts of curve of Jordan, Cantor and Menger. It was proposed to Rodríguez Bachiller that in the next session he present in writing the proof, corrected, so it can be examined again, and the author has promised to do this.

In October 1932 Bachiller's became an acting professor when the chair of differential equations became vacant. In July 1934 a competition for the Theory of Functions chair was announced. Two candidates applied, Bachiller and R San Juan, and a committee was set up to carry out the competition. The committee summoned the two candidates to appear before them on 15 June 1935 but only Bachiller was present. He was told that he would take the tests set out by the committee on 25 June. After he completed the tests successfully, the committee unanimously appointed him professor on 29 June 1935. However, since he still did not have a doctorate, he was told that he had to submit a thesis and be successfully examined before taking up the chair. He submitted his thesis *Axiomática de la dimensión* Ⓣ and defended it on 27 August 1935. He passed with the grade "outstanding" and took up the chair two days later. The thesis was never published. He had, however, published *Sobre el numero de dimensiones de un conjunto* Ⓣ in 1927 which discussed the definitions of dimension given by Brouwer, Menger and Urysohn.

The Spanish Civil War took place from 1936 to 1939. It was a very difficult time for everyone and Bachiller, who almost certainly sympathised with the Republicans, tried to stay out of the conflict. Glick writes [2]:-

He again became active in the Journal of the Spanish Mathematical Society but much reorganisation took place as some colleagues left Madrid. At the end of the Civil War he was treated with suspicion by the Franco regime and "disqualified from holding positions of leadership and confidence." He was, however, allowed to return to his chair in December 1939. He applied for a grant to allow him to spend four months in Rome undertaking research but his application was refused. Nevertheless, he was able to go to Rome but he had to make the trip at his own expense. He publishedRodríguez Bachiller did not take an active part in politics, but ideologically he was a democrat at heart. He pointed out the fright he got when he was in Berlin in1936; he went to greet the rector of the University, the mathematician Ludwig Bieberbach, and the latter replied with the Nazi salute.

*Comentarios sobre algebra y topologia*Ⓣ in 1941. André Weil writes in a review:-

In 1942 he published a paper in Italian, namelyA short survey of some fundamental concepts in topology, leading up to the following result: the n th homotopy group of a topological product is the direct product of the n th homotopy groups of the factors.

*Sulle superficie del quarto ordine contenenti una conica*Ⓣ. This article investigates the existence of birational transformations based on Severi's methods. However, after this he published no further research papers. He kept his interest in topology, and taught courses on it many times. As a result of Bachiller's visit to Rome, Severi and Fantappié were invited to give lectures in Madrid and Barcelona in the year 1942.

Bachiller had quite a wide range of different teaching duties in a number of different institutions. He ran, in collaboration with two others, a preparatory academy for civil engineers. During 1943-44 and 1944-45 he gave a new course, this time on topological groups, at the Institute of Mathematics "Jorge Juan" in Madrid. In April 1946, Bachiller delivered the inaugural lecture of the Mathematics Section of the Congress of the Spanish Association for the Advancement of Science held in San Sebastián. His lecture, *Estado actual de la teoría de la dimensión en los espacios topológicos* Ⓣ, was not published. In 1946 the Cultural Relations Board of the Ministry of Foreign Affairs announced that they were offering scholarships for research and study abroad. Some were aimed at graduates, others at professors. Bachiller won one of the scholarships for professors and on 4 February 1947 he arrived in the United States to begin a five month visit. The first three months were spent at the University of Illinois at Urbana where he worked with Oscar Zariski and then he went to Princeton where he worked with Solomon Lefschetz, Emil Artin and Claude Chevalley. Despite working on interesting research projects, he refused to publish his results. He also refused to publish the lecture notes of the courses that he gave.

He delivered topology courses at the Institute of Mathematics "Jorge Juan" up to 1951 when to changed to giving courses on logic and the foundations of mathematics, topics that had always interested him. In early 1955 he went to the Institute for Advanced Study at Princeton, where he visited Einstein a few days before his death. In the summer of 1956 he attended a topology congress in Mexico. During this time Bachiller continued his work as a translator, and made visits to Puerto Rico whenever he could. His final major work as a translator was with the three volumes of Severi's *Lecciones de análisis* Ⓣ (1951, 1956, 1958). Meanwhile, his first stay at the University of Puerto Rico in Mayagüez was in 1954, and he returned several times over the next years to deliver courses on topology and abstract algebra. Puerto Rico, of course, was special to the Bachiller family since Jeaús Bachiller had been born there and his mother, who had died in childbirth, was buried there. In fact Bachiller spent so much time in Puerto Rico that he had to resign from many of his roles in Madrid. He even had problems in 1965 because he was in Puerto Rico when he should have been lecturing in Madrid. However, he managed to continue to keep both his positions in Puerto Rico and in Madrid until he retired in 1969.

In Puerto Rico he enjoyed the same enthusiasm that he had experienced in Spain before the Civil War. He was a member of the Philosophy Society there and was described in these terms by a member (see [1]):-

There was Rodríguez Bachiller, a mathematician with a humanist bias, with an intellectual curiosity open to all horizons of culture. There were abundant examples of philosophical and literary works, classical and modern, many of them in their original language, in the very large library of his house in El Viso, and he could read works in French, English, Italian and German, and did so habitually. Another of his great hobbies was music. ...

**Article by:** *J J O'Connor* and *E F Robertson*