**Andreas Speiser**'s parents were Paul Speiser (1846-1935) and Elizabeth (Lilly) Sarasin (1861-1938), the daughter of Karl Sarasin and Elizabeth Sauvain. Paul had studied law at Basel, Göttingen, Berlin and Bonn before becoming a lecturer in commercial and tax law at the University of Basel. He was also politically active and served on the National Council. Paul Speiser married Salome (Saly) Sarasin (1853-1882) in 1873. They had five children: Paul (1875-1954), Esther (1876-1957), Salome (1877-1956), Anna Helena (born 1879, died a baby) and Felix (1880-1949). After the death of Salome Sarasin in 1882, Paul Speiser married Salome's younger sister Elizabeth (Lilly) Sarasin (1861-1938) on 18 March 1884. Their first child was Andreas (born 1885, the subject of this biography) followed by seven others Theophil (1886-1940), Clara Elizabeth (died a baby), Ernst (1889-1962), Marguerite (1891-1982), Paula (1892-1954), Ruth (1893-1976), and Marie (1901-1986).

Speiser's school education was in his home city of Basel and he graduated with his baccalaureate certificate from the Gymnasium at Easter 1904. He sought advice from Karl von der Mühll (1841-1912) about the best university at which to study mathematics. Von der Mühll had been born in Basel into a long established Basel family and had studied at the University of Basel, the University of Göttingen and the University of Königsberg before being appointed to the University of Leipzig. In 1890 he had returned to his home town when appointed as Professor of Mathematical Physics in Basel. Von der Mühll advised Speiser to study at the University of Göttingen and, having entered in 1904, he was taught by several leading mathematicians including Felix Klein, David Hilbert and Hermann Minkowski. While he was studying there he became friends with Constantin Carathéodory who had been a student of Minkowski's and was a docent at Göttingen for most of the time Speiser was studying there. Speiser undertook research on quadratic forms advised mainly by Minkowski but also helped by Hilbert. He had completed the work for his thesis *Theorie der binären quadratischen Formen mit Koeffizienten und Unbestimmten in einem beliebigen Zahlkörper* Ⓣ when, in January 1909, Minkowski died suddenly from a ruptured appendix. Speiser's oral examination, which took place on 3 March 1909, was conducted by David Hilbert.

After being examined on his thesis, Speiser went to Britain where he spent time in London but also visited Edinburgh in August 1909. Speiser later explained that he had a thought which led to a new direction in his studies while he was in Edinburgh:-

After leaving Britain, Speiser spent time in Paris before going to Strasbourg where he habilitated in 1911. He taught at Strasbourg for several years and during this time his work on the composition of quadratic forms led him to study group theory. His paperWhen I walked around Edinburgh in August1909, I suddenly thought: Maybe it's really true that mathematics is the source of art.

*Über die Komposition der binären quadratischen Formen*Ⓣ on the composition of quadratic forms was published in 1912. He published papers on group theory such as

*Zur Theorie der Substitutionsgruppen*Ⓣ (1914) and

*Gruppendeterminante und Körperdiskriminante*Ⓣ (1916). Notice that when he submitted the second of these to

*Mathematische Annalen*in July 1915 he was in Karlsruhe. In 1917 he published a paper in French,

*Équations du cinquième degré*Ⓣ.

Speiser married Emmy La Roche (1891-1980), the daughter of the banker and councillor Ludwig La Roche, in 1916. They had four children: Marie-Thérèse (1917-1982), Thomas (1919-1982), Renata (1921-1961) and Andreas (1925-1999).

In the summer of 1917 Speiser was appointed as an extraordinary professor at the University of Zürich and, in 1919, he became an ordinary professor at Zürich. In fact two chairs of mathematics at Zürich became vacant almost simultaneously. The other chair had been filled by Rudolf Fueter who was appointed after holding the chair of mathematics at the Technical University of Karlsruhe for only three years. Fueter took on the task of rebuilding the study of mathematics at the university and began the process of making it a cultural centre, even during a very difficult period of history. Speiser took over the course *Introduction to Analysis* which was the main course taken by mathematicians and physicists. He published three papers in 1919, namely: *Sur les lignes géodésiques des surfaces convexes* Ⓣ; *Die Zerlegungsgruppe* Ⓣ; and *Zahlentheoretische Sätze aus der Gruppentheorie* Ⓣ.

Several appointments strengthened the Mathematics Department at Zürich. Martin Disteli (1862-1923) had studied at Zürich where he was awarded his doctorate in 1888. After positions in Karlsruhe, Strasbourg and Dresden, he was appointed as an Ordinary Professor of Geometry at Zürich in 1920. He only held this chair for three years since he died in 1923. Eugenio Giuseppe Togliatti (1890-1977) studied at the University of Turin taught by Corrado Segre and Gino Fano and was appointed to the University of Zürich in 1924. He left in 1927 when he returned to Italy, taking up a position at the University of Genoa. Paul Finsler was appointed to the University of Zürich in 1927 where he was Speiser's colleague for seventeen years.

Certainly one of Speiser's greatest mathematical achievements was his book on group theory, *Die Theorie der Gruppen von endlicher Ordnung, mit Anwendungen auf algebraische Zahlen und Gleichungen sowie auf die Kristallographie* Ⓣ, which was published in 1923. The book ran to four editions, the last in 1956. Reviewing the second edition published in 1927, Philip Hall writes [20]:-

Philip Hall also reviewed the third edition published in 1937 and wrote [21]:-The author has succeeded in the astonishing feat of condensing all the principal propositions of the subject into the space of250pages. The style is extremely elegant and concise, with a minimum of comment and illustration. ... Altogether the author has written a most valuable book which should stimulate interest in this fascinating theory, so rich in contacts not only with other branches of mathematics but also with the latest speculations of natural philosophy.

For extracts from reviews of all four editions, see THIS LINK.The present edition preserves to the full the high stylistic qualities of the earlier versions. There are many mathematical books of which one may confidently feel that every word has been weighed. But there can be few in which they have been weighed with such skill as here.

In fact Speiser's work on group theory had already led him to think more generally about symmetry and, in particular, to think about how mathematics related to many cultural activities throughout history. Donald W Crowe writes [11]:-

Edith Müller's thesis, mentioned in the above quote, wasHow did these two disciplines, the cultural-historical and the mathematical, happen to meet and interact in this particular specialty: the study of patterned ornament? Many of us have answered that the interaction was especially fostered by Andreas Speiser's early group theory textbook, 'Theorie der Gruppen von endlicher Ordnung'. For, in all editions beginning with the second in1927, Speiser included a large section dealing with the symmetries of ornament, in which he introduced us to the plane crystallographic groups using the notation of his colleague, Niggli. This, combined with the thesis of his student Edith Müller on symmetries of ornaments in the Alhambra, as well as Speiser's other cultural writings, persuaded many of us that designers of ornament from ancient times had at least a subconscious understanding of basic ideas of group theory. Speiser was, in the eyes of many, the first to make this connection.

*Gruppentheorische und Strukturanalytische Untersuchungen der Maurischen Ornamente aus der Alhambra in Granada*Ⓣ (1944).

Speiser worked on several different mathematical topics. To illustrate this we give the titles of the papers he published between 1921 and 1930:* Über die geodätischen Linien auf einem konvexen Körper* Ⓣ (1921); *Die Zerlegung von Primzahlen in algebraischen Zahlkörpern* Ⓣ (1922); *Sur la décomposition des nombres premiers dans les corps algébriques* Ⓣ (1922); *Allgemeine Zahlentheorie* Ⓣ (1926); *Naturphilosophische Untersuchungen von Euler und Riemann* Ⓣ (1926); *Sur les groupes et groupoides* Ⓣ (1928); *Die Eulerausgabe* Ⓣ (1929); *Probleme aus dem Gebiet der ganzen transzendenten Funktionen* Ⓣ (1929); *Ein Satz über topologische Bäume* Ⓣ (1930); *Über Riemannsche Flächen* Ⓣ (1930); *Probleme der Gruppentheorie* Ⓣ (1930).

In addition to these papers, he published the book *Klassische Stücke der Mathematik* Ⓣ (1925), which contained translations into German of the writings of Plato, Aristotle, Euclid, Archimedes, Leonardo da Vinci, Kepler, Descartes, Pascal, Johann Bernoulli, Euler, Daniel Bernoulli, Sylvester, and Einstein. For extracts from two reviews of this book see THIS LINK.

From the above list of papers we see that he wrote about Euler and the work of the Euler Commission which had been set up to publish Euler's collected works. The first editor of the "Collected Works of Leonhard Euler" was Ferdinand Rudio who taught at the Eidgenössische Polytechnikum Zürich. He was forced to give up the editorship in 1928 due to ill health and at that time Speiser took over the editorship. In fact 35 volumes of the "Collected Works of Leonhard Euler" were published under Speiser's editorship. In 1965 he ended this role which was taken over by Walter Habicht (1915-1998). Habicht, who had studied for his doctorate at the Eidgenössische Polytechnikum Zürich, became a colleague of Speiser's in Basel in 1963. In 1937 Speiser wrote a report aimed at American mathematicians on the work of the Euler Commission [30]:-

Johann Jakob Burckhardt, who was a student of Speiser's at Zürich writing the dissertationThe "Collected Works of Leonhard Euler" will embrace approximately seventy volumes. From1911to1937twenty-six of these volumes have appeared, of which twenty volumes belong to Series I which contains the mathematical works. Series II contains the works on mechanics, technology and astronomy, while Series III is devoted to the works on physics and philosophy. In1911at the beginning of the undertaking the work could be regarded as well founded financially, thanks to the many subscribers and the large sum of money which was collected by the Euler Commission. Almost half of the subscribers were lost through the World War, furthermore the costs of printing rose to such an extent that we were compelled essentially to slow up the tempo of the edition.

*Die Algebren der Diedergruppen*Ⓣ, writes [11]:-

We have already mentioned some of his books that have followed this path. Here are some others.Speiser was of unusual erudition; his ambition was to pursue thoughts through the course of history and demonstrate their impact.

*Die mathematische Denkweise*Ⓣ (1932) was reviewed by Edward Switzer Allen [3]:-

Max Dehn, reviewing the second edition of 1945, writes [14]:-This book is not a treatise on how mathematicians think. It is a collection of essays on mathematical thought as it is revealed in art and music, in philosophy and astrology. It is the work of a man of broad culture - one whose contributions to group theory are well enough known, but who is also at home in yet more esthetic realms and is conversant with the history of serious human thought. ...[The reviewer is]urging that the book itself be read. It is not a necessity for one's library; it is a delight.

The bookIn this book we have the philosophy of a mathematician. It is written with the enthusiasm of a distinguished mathematician who penetrates the arts and the world in his peculiar way. It will transmit, I imagine, this enthusiasm to every mathematician who is not only a craftsman but possessed by sacred fire as the poet and philosopher ought to be.

*Elemente der Philosophie und der Mathematik*Ⓣ (1952) is dedicated to the memory of Rudolf Fueter:-

It was reviewed by Reuben Louis Goodstein who writes [18]:-... my best friend, with whom I was able to work during55semesters in Zürich and to whom I owe unlimited gratitude.

For a longer extract from this review and extracts from two other reviews, see THIS LINK.In the introduction Professor Speiser describes his book as finger exercises for beginners in philosophical and mathematical thought. Fundamental concepts in philosophy and mathematics are subjected to the Hegelian dialectic to bring out their inner content.

Speiser wrote articles on diverse topics, several of which he reprinted in the book *Die geistige Arbeit* Ⓣ (1955) which contains lectures and essays from various topics in art, science, philosophy and theology. His lectures also covered a wide range of topics as did the doctoral dissertations of his many students at Zürich. He was always concerned that his students would be well prepared and he took every opportunity to spend time talking to them. To understand something of his attitude to teaching, see THIS LINK.

Johann Jakob Burckhardt writes [11]:-

Although Switzerland remained neutral throughout World War II, nevertheless the country was on a war footing and it was a difficult time for all. During these war years mathematics teaching at the University of Zürich was almost entirely carried out by Speiser and Finsler. Speiser regularly delivered two four-hour lectures and ran the undergraduate seminar as well as the mathematics research seminar and the mathematical-philosophical seminar. Therefore, despite his love of Zürich, it is understandable that when he was asked to return to the University of Basel in 1944 he happily accepted. Back in his home town of Basel he found the tranquillity which enabled him to concentrate on the more and more stressful work associated with editing Euler's Collected Works. For example in 1947 he publishedSpeiser loved life in Zürich, especially the Old Town, where he lived for a long time and through which he liked to walk, often accompanied by his students. ... A highlight of his mathematical life in Zürich was the International Congress of Mathematicians in1932; Speiser was a Vice President. He was anxious to establish personal contacts between participants. Many will remember his hospitable home at Pelican Place, as well as at its headquarters in Melide, where they found a friendly reception from the Speisers.

*Einteilung der sämtlichen Werke Leonhard Eulers*Ⓣ which lists Euler's works giving their original places of publication and their location in the edition of Euler's Collected Works. In addition to editorial work on the volumes, he wrote many Introductions and Forewords to the volumes which were published over the twenty years 1944-64. In 1945 he was appointed permanent guest of honour by the University of Zürich. Speiser retired from his chair in Basel in 1955. In 1957 he was awarded an honorary doctorate by the University of Berne.

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

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