Carlos Federico Graef Fernández

Quick Info

25 February 1911
Guanaceví, Durango, Mexico
13 January 1988
Mexico City, Mexico

Carlos Graef Fernández was an outstanding Mexican mathematician and physicist. As a lecturer, researcher and organiser his contributions were widely recognised.


Carlos Graef Fernández was the son of Carlos Francisco Graef Ziehl (1878-1945) and Gudelia Fernández Espinosa (1882-1953). Carlos Francisco Graef, born in Mexico City on 17 September 1878, became a mining engineer in Guanaceví, Mexico. He married Gudelia Fernández in Mexico City on 25 March 1910. She had been born on 23 February 1882 in Pachuca, Mexico. Carlos Federico Graef, the subject of this biography, was the eldest of his parents four children having as siblings Hermann Graef Fernández (born 1913 and became a distinguished doctor), Federico Graef Fernández (born 1914 and died at the age of two), and Laura Graef Fernández (born 1916).

Carlos spent his childhood and adolescence in Mexico City. Both his primary and secondary studies were at the Colegio Alemán de México, also known as the Deutsche Oberrealschule zu Mexico. This school, founded in 1894 on Canoa Street in Mexico City, was the first German school in Mexico. In 1918, the number of pupils having risen greatly, a new building was built. In November of that year, the German-type exam called the Abitur took place for the first time on the American continent. Graef completed his primary studies at the school in 1922 and continued with his secondary studies. In December 1928, he was awarded the "Abitur".

With German schooling, Graef decided that he was going to Germany for his university studies. He studied in the department of Civil Engineering of the Technische Hochschule in Darmstadt, Germany, for three terms from 1929 to 1930. It was unfortunate timing since the Wall Street Crash of October 1929 led to a collapse in the German economy. Conditions in the country rapidly deteriorated and, with his father struggling to fund his studies, Graef decided that he had to return to Mexico to continue his university studies. On 16 September 1930 he departed from Hamburg on the ship the Rio Panuco bound for the port of Veracruz in Mexico. Back in Mexico City, he enrolled in the National School of Engineering of the National Autonomous University of Mexico in 1931. He attended courses in the Departments of Oil Engineering, Mathematics and Physics from 1931 to 1933. One of his lecturers was Sotero Prieto, but sadly he died in 1935 while Graef was still a student.

In 1934 the rector of the National Autonomous University of Mexico, Manuel Gómez Morín, restructured the University and created the Faculty of Physical Sciences and Mathematics that included the School of Engineering, the School of Chemical Sciences and a Department of Physics and Mathematics. Graef enrolled in the Department of Physics and Mathematics which, he writes in [19]:-
... was located in the Palacio de Minería, home to the School of Engineers. The neighbourhood in the space of the two establishments was very fruitful. The professors in the Physics and Mathematics department were professors from the School of Engineering who had a special love for exact sciences. Those who were then dedicated to teaching these subjects had to give many hours of class every day to be able to collect a decent remuneration. There was no physics or mathematics institute, and the only income for them was from their teaching work.
An important event in Graef's mathematical development was attending lectures by Dirk Jan Struik. In the interview [29] Struik said:-
In the 1930s, I had a pupil by the name of Alphonso Napoles. He was very good in tensor calculus, and overall. Then he went to Mexico and soon became an outstanding member of the mathematics faculty at the Autonomous University in Mexico City. He became head of the department. Then he invited me to come to Mexico, which I did in 1934, at the time that I had a sabbatical. Included in the invitation was also an invitation by Manuel Sandoval Vallarta, who had returned to Mexico and became a very big shot in the scientific side of the government. I was in Mexico for six weeks in 1934, which was one of the great experiences of my life. The absolute differences between the Spanish and Indian cultures of Mexico and the Northern European culture of the United States was for me overwhelming. So I never forgot the deep impression I got of Mexico in those days. I was received very friendly. I was one of the very first outside lecturers in Mexico, and my pupils and colleagues showed me around. They showed me the pyramids and all kinds of things, and the colonial monuments and the nature. It was a very grand time. I'll never forget it.
Although Graef was still a student, he began teaching in 1933 when he was appointed professor of Analytic Geometry and Calculus in the Escuela Superior de Construccion of the Secretaria de Educación Pública of the Mexican Government. In 1934 he was appointed Professor of Higher Geometry in the Department of Mathematics and Physics of the Universidad Nacional de México. When teaching in the Preparatory School of the National University, Manuela Garín was one of his pupils. She said [34]:-
Graef and Nápoles Gándara were my mathematics teachers. With such teachers my vocation was further defined. Graef was a nice man, a very good teacher and a man of great human qualities. His laughter in the first yard could be heard as far as the third yard of the Preparatory School. ... Graef was our geometry teacher ...
In June 1937 Graef was awarded a John Simon Guggenheim Memorial Foundation fellowship to fund his research in mathematics at the Massachusetts Institute of Technology for a Ph.D. The Foundation's Report of 1937-38 states that Graef was a Latin American Exchange Fellow [6]:-
Appointed for studies of the theory of probability and mathematical theory of statistics; tenure, twelve months from September 17, 1938.
On 18 May 1938, Graef married Alicia Sánchez Castell in Mexico City. Alicia had been born on 2 March 1916 in Mexico City. Carlos and Alicia Graef had three children: Carlos Graef Sánchez, Alicia Graef Sánchez and Carolina Graef Sánchez. Alicia became a doctor of nuclear medicine, a branch of medical sciences that consists of the application of radioisotopes for the diagnosis and treatment of various diseases; Carolina became a prominent lawyer; and Carlos became an engineer.

Graef and his wife arrived in Laredo, Texas, USA, on 26 July 1938 travelling to Boston, Suffolk, Massachusetts. In Boston they were lodgers with Estehr Durman at 707 Commonwealth Avenue. In [11] Graef explains that the John Simon Guggenheim Memorial Foundation fellowship:-
... was in effect until February, 1940, and in March of the same year the President of Mexico, General Lázaro Cárdenas, awarded me a stipend to finish my doctoral studies. I received a stipend from the Universidad Nacional de México during all my residence in the United States, as well as scholarships from Massachusetts Institute of Technology for three terms.
He undertook research in the Department of Physics, advised by Manuel Sandoval Vallarta and was awarded a Ph.D. in 1940 for his thesis An analysis of periodic orbits of particles of primary cosmic radiation. Graef writes in the Abstract [17]:-
This thesis deals with the orbits traced by a charged particle moving in the magnetic field of a dipole, on a rotating meridian plane following the particle.
We use the geometrical method of attack to find properties of the orbits. The transformation of the dynamical problem into a geometrical one is done by the introduction of the characteristic surface, i.e. the surface whose geodesics have the same differential equation as the trajectories. The passing from the dynamical equations of motion to the geometrical equation of the geodesics is essentially an elimination of the time. The meridian plane is a conformal image of the characteristic surface, so that properties of the geodesics on the latter can be translated into properties of orbits on the former.

We found that the characteristic surface has singular points which correspond to the boundary between allowed and forbidden regions in the meridian plane. The curvature of the surface is proved to be everywhere positive.
Alberto Barajas writes [2]:-
This work, in the opinion of the famous topologist Solomon Lefschetz, was notable for the very original way of attacking a problem of differential equations in three dimensions, to the best of his knowledge, for the first time. One main result, that all periodic trajectories intersect the magnetic equator, was verified years later when the Van Allen bands, the cosmic ray death belts that surround the Earth, were discovered, so important to astronauts.
The thesis was approved in May 1940. By the time he submitted his thesis, he had three papers published: Representación de un Tensor por medio de Seis Vectores (1937); On Periodic Orbits in the Equatorial Plane of a Magnetic Dipole (1938); and Galactic Rotation and the intensity of cosmic radiation (1939). The 1938 and 1939 papers were joint publications with Shuichi Kusaka (1915-1947). Let us note in passing that Kusaka tragically died while swimming at Beach Haven, New Jersey on 31 August 1947 when he became separated from his friends.

While in Cambridge, Massachusetts, Graef attended courses in astronomy and astrophysics at Harvard University. After the award of his doctorate he remained at Harvard for a while and completed a World War II Draft Card on 16 October 1940. On the Draft Card, he gives his employer as Harvard University and gives: height, 5' 8"; weight, 180 lbs; complexion, dark; hair, brown; and eyes, grey. While he was at Harvard he met Luis Enrique Erro (1897-1955) a Mexican astronomer who became a member of the administration in 1940 tasked with building an observatory in Tonantzintla, Puebla. This site has particularly favourable atmospheric conditions for an observatory. Erro asked Graef if he would be the founding deputy director of the Observatory project. Graef agreed and returned to Mexico in 1941 where he was appointed Professor of Physics at the National Autonomous University of Mexico. The National Astrophysical Observatory of Tonantzintla was dedicated in February 1942 in a ceremony attended by the President of Mexico, Manuel Ávila Camacho. Erro was director and Graef deputy director of the Observatory until 1944.

On 30 June 1943 the Mexican Mathematical Society was founded. The First National Congress of Mathematics in Mexico had been held in November 1942, organised by Alfonso Nápoles Gándara, Alberto Barajas and Francisco José Alvarez. At this Congress, a commission was set up consisting of the three organisers of the congress together with Carlos Graef, which was given the task of setting up the Mexican Mathematical Society. At the first meeting of the Society on 30 June 1943, Alfonso Nápoles Gándara was elected as the first president of the Society and Carlos Graef was elected the first vice-president from 1943 to 1945.

On 20 February 1942 George David Birkhoff gave a lecture at the Astrophysical Congress in Puebla, Mexico, at a congress held to inaugurate the Tonantzintla Astrophysical Observatory. He outlined his new theory of relativity at the Congress, giving his lecture in English [2]:-
... the people who had committed to the director Luis Enrique Erro to translate foreign scientific papers for some reason did not arrive on time and Erro had asked Graef to help him in the emergency by taking care of the translation. G D Birkhoff spoke in English as the first speaker and Graef listened attentively without interrupting him. At the end of the presentation, he made a very precise synthesis of the work, underlining the most interesting conclusions, inserting witty comments and expanding explanations on graphs and diagrams. After a few hours of this unexpected way of translating, Harlow Shapley, could not contain himself and interrupted the session to say: "We are amazed at the transformation that a work undergoes when Graef translates it into Spanish. It becomes brighter and more understandable; as if the translator knew the article better than the author."
Graef invited Birkhoff to work at the Institute of Mathematics in Mexico in 1943 and they worked with others developing the new theory. Birkhoff then invited Graef to Harvard in 1944 and, as visiting professor of relativity and gravitation, he delivered a series of lectures. Graef published three papers in 1944, which are reviewed as [20], [21] and [22]. Graef entered the United States on 30 October 1944 and, while a guest at the Princeton University Mathematics Department, he was invited by Albert Einstein to visit him in December 1944. You can read Graef's description of this visit at THIS LINK.

This description of Graef's visit to Einstein was written in 1955 and was one of many popular articles that he wrote. Before that were: Galton's apparatus (1935); Coriolis' acceleration (1935); The geodesics (1935); Four dimensional space (1935); Hamilton's principle (1936); Elastic deformations (1937); The foundation of the Mexican Mathematical Society (1943); Morphological affinities between mathematics and painting (1946), Sculpture and science (1950); and Mathematical space and physical space (1955).

Returning to Mexico in 1945, Graef was appointed as director of the Institute of Physics, National Autonomous University. He held this post until 1957 when he became Director of the Faculty of Sciences. He was the main person involved in the founding of the Mexican Physical Society in 1951. Elected as the first President of the Society, he delivered the first expository lecture at the first meeting of the Society in the city of Querétaro on 22 April 1952 [35]:-
On Tuesday evening, Dr Carlos Graef Fernández, President of the Society, delivered an invited popular expository lecture on "Space, Time, Gravitation", and on Wednesday evening Manuel Sandoval Vallarta gave another invited lecture on "The Relation Between Cosmic Rays and Radio Waves Emitted by the Sun". Both were attended by capacity audiences of about 200 which filled the 17th century lecture hall to overflowing.
It is worth quoting from Graef's article [19] about his experiences of the reorganisation of the National Autonomous University of Mexico which occurred in 1952:-
When the faculties and schools of the National Autonomous University of Mexico were moved from their old sites in the City to their magnificent building in the Ciudad Universitaria in 1952, many new values were gained, but some very dear and appreciable values were also lost. For reasons of historical development, the faculties and schools moved to Ciudad Universitaria as intellectual islands, as they had been in Mexico City. There had been very little academic contact between one campus and another due to their different geographical location in the City. In this way the university dependencies went to their new city. Each one moved as an island. In the scree the links that had not existed in the centre were not established. Some university students hoped that by being all together in the Ciudad Universitaria a greater spiritual contact would be fostered; that the courses of the same subjects that are taught in different schools could be a bond of intellectual union. For example, we dreamed that all university students who had to acquire knowledge of analytical geometry would take that course in a Mathematics Department and have students from very different faculties as companions, all listening to the professional mathematician who taught them. The inertia of tradition was an insurmountable obstacle for this system that would have achieved a spiritual rapprochement among all university students. In many university structures around the world, including in the United States and Europe, this happens on a daily basis. When moving to Ciudad Universitaria, the link between the Faculty of Sciences and the Faculty of Engineering was lost.
We note that Graef worked in the Faculty of Science throughout his career although his initial studies had been in the Faculty of Engineering. Luis Estrada describes the teaching skills of Graef in [14]:-
Dr Graef distinguished himself as a great teacher. His classes mainly revolved around differential calculus, analytical geometry, and classical mechanics. He also gave courses on relativity, both special and general, on classical field theory, and on gravitation. Although he was aware of the achievements of modern physics - that based on quantum theory - he was, so to speak, a classical physicist. In his classes, he poured out enthusiasm and emotion when referring to the achievements of the human intellect in his explanations about space, time and the movement of bodies, which caused his students to continue studying what he pointed out. His expositions were very clear because he had the gift of communicating complex ideas in a simple way, for which he used analogies and metaphors based on everyday knowledge. He took advantage of his vast culture to exemplify his statements and told stories and anecdotes to support his teachings. He had a great sense of humour and liked to chat with the students, never missing a joke with which he was the first to laugh. His permanent attitude was to preach the charm of scientific knowledge and to enthuse everyone to take a look at it.
Graef became involved in the nuclear industry and was a member of the Mexican Delegation to the International Atomic Energy Commission of the United Nations Organisation in 1946. He served as a member of the Advisory Council of the National Nuclear Energy Commission from 1956 to 1962 and represented Mexico on the Board of Governors of the International Atomic Energy Agency in Vienna, Austria from 1960 to 1961. He then served as Coordinating Scientific Adviser of the National Nuclear Energy Commission from 1962 to 1965. Graef was appointed as General Director of the Nuclear Centre of Mexico in 1965 and continued in this role until 1970. After this he served on the General Counsel of the National Institute of Nuclear Energy from 1971 to 1977. There were several newspaper reports on statements made by Graef, mostly in his nuclear energy roles. For some extracts, see THIS LINK.

He retired from his position at the National Autonomous University of Mexico and was made Professor Emeritus in 1979. He continued to work, however, being the founding Director of the Division of Basic Sciences and Engineering at the Metropolitan Autonomous University, Iztapalapa from March 1974 to February 1976. He continued to work for the nuclear industry, being General Coordinator of the National Institute of Nuclear Research (1977-1978), General Coordinator of Mexican Uranium (1979-1980), and Manager of the Mexican Uranium Fuel Cycle (1980-1983).

Throughout his career, Graef received many honours and awards: in 1945 he received the Manuel Ávila Camacho Prize, awarded by the Instituto del Libro; the state of Durango awarded him their Francisco Zarco Medal; he received the National Prize for Sciences and Arts, in the area of Physical Mathematical and Natural Sciences, from the Government of Mexico in 1970; he received the Academic Medal of the Mexican Physical Society in 1982; in the same year he received the Nabor Carrillo Flores Award for Nuclear Science and Technology; the National Autonomous University of Mexico gave him their National University Award for Teaching Exact Sciences in 1985.

Let us end with three quotations. First from Fernando Alba [2]:-
With Graef's death, Mexico loses one of its great teachers, researchers, and organisers; but there are many seeds that he sowed and that will continue fighting to magnify the University and our Country, which he loved so much.
Next from Ariel Tejera [2]:-
Galileo said that the world is an open book; only that it is written in mathematical characters. For Dr Graef it was easy to read them and, furthermore, when he described them they became luminous.
Finally from Miguel Jose Yacaman [38]:-
Above all Graef had a warm and enthusiastic personality, and he was always willing to help students with homework problems or to discuss physics with colleagues. Listening to his class made one feel that physics was a beautiful and enjoyable science, worth being part of. Probably his best obituary was an anonymous note that appeared on the wall outside his classroom the day his death became known, written on a student's piece of paper: "Graef we loved you, we will miss you."

References (show)

  1. R Ai Camp, Graef Fernández, Carlos (1911-1988),
  2. A Barajas, Semblanza de Carlos Graef, Centre de Ciencias, National Autonomous University of Mexico.
  3. A Barajas, J Manuel lozano and C Graef Fernández, Homenaje al Doctor Carlos Graef Fernandez, Revista Mexicana de Física 30 (4) (1984), 599-628.
  4. Books for Mexico, Billings Gazette (27 April 1943).
  5. Carlos Graef Fernández, in Jay Kinsbruner (ed.), Encyclopedia of Latin American History and Culture (2008).
  6. Carlos Graef Fernández, John Simon Guggenheim Memorial Foundation.
  7. Carlos Graef: the Mexican who challenged Einstein, Mexicanist (15 April 2022).
  8. Carlos Graef Fernández, Prabook
  9. Carlos Graef Fernandez, Matemáticos en México, National Autonomous University of Mexico.
  10. Carlos Graef Fernandez,
  11. Carlos Graef Fernández. Biography, in J L Fernández Chapou and A Mondragón Ballesteros (eds.), Carlos Graef Fernández: Obra Científica (Universidad Autónoma Metropolitana, Mexico, 1993), 244.
  12. CV, Carlos Graef Fernandez, in J L Fernández Chapou and A Mondragón Ballesteros (eds.), Carlos Graef Fernández: Obra Científica (Universidad Autónoma Metropolitana, Mexico, 1993), 737-742..
  13. Dr Carlos Graef Fernández, Director en el periodo 1974-1977, Histórico de Directores, Universidad Autónoma Metropolitana Iztapalapa.
  14. L Estrada, A propósito del Dr Carlos Graef Fernández,
  15. J L Fernández Chapou and A Mondragón Ballesteros, Dr Carlos Graef Fernández, Universidad Autónoma Metropolitana.
  16. J L Fernández Chapou and A Mondragón Ballesteros (eds.), Carlos Graef Fernández: Obra Científica (Universidad Autónoma Metropolitana, Mexico, 1993).
  17. C Graef, An analysis of periodic orbits of particles of primary cosmic radiation, Doctor of Philosophy Thesis (Massachusetts Institute of Technology, 1940), in J L Fernández Chapou and A Mondragón Ballesteros (eds.), Carlos Graef Fernández: Obra Científica (Universidad Autónoma Metropolitana, Mexico, 1993), 90-244.
  18. C Graef Fernández, My Tilt with Albert Einstein, American Scientist 44 (2) (1956), 204-211.
  19. C Graef Fernández, Palabras Pronunciadas por el Dr Carlos Graef Fernandez Durante el Homenaje que le Rindio, La Sociedad Mexicana De Fisica, in J L Fernández Chapou and A Mondragón Ballesteros (eds.), Carlos Graef Fernández: Obra Científica (Universidad Autónoma Metropolitana, Mexico, 1993), 680-693.
  20. W Kaplan, Review: Periodic orbits of the primary cosmic radiation (Spanish), by Carlos Graef Fernández, Mathematical Reviews MR0010826 (6,75h).
  21. G C McVittie, Review: On Birkhoff's new theory of gravitation, by A Barajas, G D Birkhoff, C Graef and M Sandoval Vallarta, Mathematical Reviews MR0010796 (6,72d).
  22. G C McVittie, Review: The motion of two bodies in Birkhoff's theory of gravitation (Spanish), by Carlos Graef Fernández, Mathematical Reviews MR0011952 (6,240h).
  23. Mexico Hopes To Head Latin A-Center, Stars and Stripes Newspaper (22 April 1957).
  24. Mexico, U.S. Eye Nuclear Project, Albuquerque Journal (11 September 1970).
  25. A Mondragón, Carlos Graef Fernández, Boletín de la Sociedad Mexicana de Física 9 (2) (1995), 83-87.
  26. Nuclear Tests Caused Mexico's Hear Wave, Kingsport News (25 May 1957).
  27. E Pina Garza, Carlos Graef Fernández, Elementos 16 (2) (1992), 41-44.
  28. Plant Planned, Nevada State Journal (10 September 1970).
  29. A B Powell and M Frankenstein, In His Prime: Dirk Jan Struik Reflects on 103 Years of Mathematical and Political Activities, Harvard Educational Review 69 (4) (1999), 1-28.
  30. Premio Carlos Graef Fernández, Sociedad Mexicana de Física.
  31. Research Center Plan Announced, Brownsville Herald (21 April 1957).
  32. T Robles, Carlos Graef: el mexicano que retó a Einstein, National Autonomous University of Mexico.
  33. H S Ruse, Review: Birkhoff's theory of gravitation (Spanish), by Carlos Graef Fernández, Mathematical Reviews MR0053672 (14,807h).
  34. P Saavedra and M Neumann, Una pionera de la Matemática en México, Centro de Ciencias Matemáticas, UNAM (February 1997).
  35. M Sandoval Vallarta, Mexican Physical Society, Physics Today 5 (6) (1952), 30-31.
  36. Tortilla Eaters Absorb Less Fallout, Berkshire Eagle (17 June 1959).
  37. Tortilla Eaters Are Safer From Fallout, Scientist Says, Corpus Christi Times (16 June 1959).
  38. M J Yacaman, Carlos Graef-Fernandez, Physics Today 41 (12) (1988), 103.
  39. Yuma and the Nuclear Reactor, Yuma Daily Sun (3 October 1963).

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Written by J J O'Connor and E F Robertson
Last Update June 2023