**Arthur Schönflies**'s father, Moritz Isaaks Schönflies (1812-86), manufactured cigars in Landsberg, a city about 130 km east of Berlin. Arthur's mother was Johanna Hirschfeld (1817-1879), from Pyryce in Pomerania, the daughter of Samuel Hirschfeld and Rösel Boas. The wealthy Jewish Schönflies family had a son Samuel Martin Schönflies who was about ten years older than Arthur. Martin Schönflies also trained as a mathematician and obtained a doctorate. Being much older than Arthur and following the same educational path, he was a strong influence on his younger brother. Arthur entered the Gymnasium in Landsberg in 1862 and studied there until 1870. The Franco-Prussian War began in July 1870 but the French troops had surrendered in September 1870 before Schönflies became a student at the Friedrich-Wilhelms University of Berlin. He studied there until 1875 working under Eduard Kummer and Karl Weierstrass but he also attended lectures by Siegfried Heinrich Aronhold at the Berlin Industrial Institute, an institution which would later become part of the Technical University of Berlin. After the award of a degree which entitled him to teach mathematics in Gymnasiums, he spent the year 1876 as a probation teacher at the Friedrich-Wilhelms-Gymnasium in Berlin. During this time, as well as learning practical teaching skills he was undertaking research for his doctorate following suggestions made by Aronhold. Since the Berlin Industrial Institute could not award doctorates, Schönflies had to have formal advisors at the University of Berlin. He obtained a doctorate from the Philosophical Faculty of the University of Berlin on 2 March 1877 for his thesis *Synthetisch-geometrische Untersuchungen über Flächen zweiten Grades und eine aus ihnen abgeleitete Regelfläche*.

Following the award of his doctorate, Schönflies continued teaching at a Friedrich-Wilhelms-Gymnasium in Berlin. In 1880 he went to Colmar in Alsace to teach at the Gymnasium there but he had his sights set on an academic career and was working on his habilitation thesis. He presented this thesis to the University of Göttingen in 1884 and began teaching there as a dozent. Felix Klein worked to set up a chair of applied mathematics at Göttingen and in 1892 Schönflies was appointed to this chair. However, Klein had great difficult with making the appointment due to the fact that Schönflies was Jewish. Hermann Amandus Schwarz left Göttingen in 1892 to fill Weierstrass's chair in Berlin and Klein wanted Adolf Hurwitz to fill the vacancy. However, Hurwitz was Jewish and Klein realised that he would not be able to get approval for two Jewish appointments. He wrote to Hurwitz (see [11]):-

There is ... - I must touch on it, as repugnant as the matter is to me, and knowing full well your justified sensitivity to this - the Jewish question. Not that your call as such would present difficulties; these I would be able to overcome. The problem is that we already have Arthur Schönflies, for whom I would like to create a firm position as salaried Extraordinariuis. And having you and Schönflies together is something I will not get past either the faculty or the Minister.

Hurwitz, perhaps because of anti-Semitism, was not appointed and the Prussian Ministry of Education appointed Heinrich Weber to fill Schwarz's chair. Klein was furious and redoubled his efforts to have Schönflies made an extraordinary professor. He wrote to the Prussian Ministry of Education saying that the situation (see [11]):-

... can only be somewhat remedied by having Schönflies named Extraordinarius. On the one hand, it is known that I have been working on his appointment for years, on the other, that my efforts have only met with resistance, so that I only dispensed from doing so as Hurwitz's call stood in question. Should Schönflies now be passed over, this impression[i.e., of Klein's impotence]will become a virtual certainty. I would then be forced to advise young mathematicians not to turn to me, if they hope to make further advancements in Prussia.

David Rowe writes [11]:-

Shortly after this letter was written Arthur Schönflies was appointed ausserordentlicher Professor in Göttingen, where for the next seven years he attracted droves of students to his classes in descriptive geometry.

While Schönflies was on the faculty at Göttingen, he married Emma Levin (1868-1939), the daughter of Albert Louis Levin, in Berlin on 7 April 1896. Arthur and Emma Schönflies had one son Albert (born 1898) and four daughters Hanna (born 1897), Elizabeth (born 1900), Eva (born 1901) and Lotte (born 1905). Two of his five children were murdered by the Nazis in 1944: Albert who died in the Auschwitz concentration camp, and Eva. Lotte, Hanna and Elizabeth died in 1981, 1985 and 1991 respectively. Schönflies left Göttingen in 1899 to take up a chair at the University of Königsberg, then in 1911 he became professor at the Academy for Social and Commercial Sciences in Frankfurt. He participated in the founding of the University of Frankfurt in 1914, became the first Dean of the Science Faculty, and played a significant role in establishing its first Department of Mathematics. Schönflies ended his career at the University of Frankfurt where he served as professor from 1914 until 1922 being rector of the University in the session 1920-21.

Schönflies worked first on geometry and kinematics but became best known for his work on set theory and crystallography. He published the book *Geometrie der Bewegung in synthetischer Darstellung* in 1886. Frank Morley writes [10]:-

The main idea of the book is to consider a body in two or more positions relatively to another body, and thence as a limit case to discuss the instantaneous motion. Full advantage is taken of the duality arising from viewing things from the standpoint of the one body or the other.

Klein suggested the problem of finding the crystallographic space groups in the late 1880s. By 1891 Schönflies had found the complete list of 230 such groups. His presentation of crystallographic space groups published in *Krystallsysteme und Krystallstruktur *(1892) used the latest aspects of group theory and became a classic on the subject. The book also includes the Schönflies notation, one of the two conventions still used today to describe crystallographic point groups. In fact the classification of the crystallographic space groups was made independently by E S Fedorov. Schönflies corresponded with Fedorov and they corrected some minor errors in both classifications before publishing their classification. Rolf Schwarzenberger writes [12]:-

In1879Sohncke had listed66groups consisting entirely of "movements"(i.e. orientation-preserving symmetries)thus completing a very incomplete list drawn up by Jordan ten years earlier. Both Fedorov and Schönflies were stimulated by noticing a mistake in Sohncke's work(one symmetry group was listed twice, giving65)and went on to count all symmetry groups, including those which include some orientation-reversing symmetries. Between1889and1891they got different results:

Fedorov229groups(+2omitted-1duplicated =230)

Schönflies227groups(+4omitted-1duplicated =230)

but eliminated errors by mutual correspondence. Only in1892did Fedorov publish a complete list showing the match between the two lists and making the often quoted remark: "... an extremely surprising circumstance has come to light, viz a coincidence in the work of two researchers such as has, perhaps, never been observed in the history of science" which has been misinterpreted as a statement that Fedorov and Schönflies worked in complete isolation from each other(and in England further embroidered to include the fiction that Barlow independently arrived at the same result; in fact he knew of the Fedorov/Schönflies papers but tried to obtain the same results direct from Sohncke's work by a different method and obtained a false result229even though - as Fedorov pointed out - "... he had more than one complete list in front of him").

Schönflies republished his classification in 1923 in *Theorie der Kristallstruktur*. R W G Wyckoff writes [17]:-

This book is essentially a new edition of Professor Schoenflies' 'Krystallsysteme und Krystallstruktur' which appeared in1891. Like its predecessor its primary concern is the deduction of the230crystallographically significant space groups. Though remaining purely geometric in the details of its reasoning, this deduction has been so rewritten as to make it somewhat shorter and more concise. The insertion of numerous figures is a great help towards the picturing of individual groups. As before these groups are described through the statement of both their sub-groups and the coordinates of equivalent positions within them. This description has, however, been amplified and improved through a listing of the symmetry properties associated with points lying in elements of symmetry. Since the discovery of X-ray diffraction the theory of space groups has become of immediate and every day use in experimental physics. The realization of this changed importance has inevitably influenced Professor Schoenflies' treatment and has led him to lay greater stress upon items of applied crystallographic interest. A chapter has accordingly been inserted which outlines the practical usefulness of space groups in studies of the positions of atoms in crystals. Although this account will not meet the needs of the practicing crystal analyst, Professor Schoenflies' books remain the only suitable source of information for those interested in the derivation of space groups.

In around 1895 Schönflies turned his attention towards set theory and topology. He wrote many works which were important at the time they were published but they were rather superseded by Hausdorff's *Grundzüge der Mengenlehre* in 1914. Three important papers on plane topology proved the topological invariance of the dimension of the square. He introduced the topological notions of accessible point, closed curve and simple closed curve. However, his work contains gaps and errors which were investigated by Brouwer who made some deep discoveries from studying these errors. Brouwer presented counterexamples to some of Schönflies's theorems showing that the notion of closed curve was more complicated than Schönflies had realised.

We should note the important contributions that Schönflies made to set theory in publishing the two-hundred and fifty page report on set theory *Die Entwickelung der Lehre von den Punktmannigfaltigkeiten* (1899), a substantial part of which studies transfinite numbers. In this work he was the first to give the name 'Heine-Borel theorem' to the theorem which today is always known by this name. Schönflies tells the reader that the proof of the Heine-Borel theorem is one of the most significant applications of transfinite numbers. He goes on to say (see for example [9]):-

... recent works have conferred on[countable ordinals]an ever increasing importance. They have found application in many domains and have shown themselves to be natural symbols, in particular whenever indefinitely many sequences of limit processes occur.

He went on to write two further important papers on set theory: *Zur Erinnerung an Georg Cantor* (1922) and *Die Krisis in Cantor's Mathematischem Schaffen* (1927). The first of these contains charming personal comments from Schönflies who had witnessed the mathematical revolution at first hand. For example (see [9]), Schönflies writes:-

[

Hurwitz]praised Cantor as the one to whom the credit for the recent success in function-theory was due. I was able to witness the radiant satisfaction which Cantor felt, and I can still recall the shining brilliance of his large round eyes reflecting that satisfaction.

In his 1927 article, Schönflies gives the reader insight into Cantor's thoughts. For example, Schönflies puts Cantor's contributions into three categories: general set theory and point-set theory; the theory of transfinite numbers; and his philosophical arguments. He writes (see [9]):-

[

Cantor]attached equal importance to all these, since he gave the same devotion to all his ideas, including the non-mathematical. This was not the position taken by the scientific community. If he didn't find the reception he wanted for his philosophical ideas he probably got over this fairly quickly. This was not the case as far as the second number-class[countable ordinals]was concerned .... For this[i.e. countable ordinals]he found no approval among the powers that be: he was not mistaken in thinking that even Weierstrass was in this respect at best only lukewarm towards him.

Schönflies also wrote on kinematics and projective geometry. He wrote textbooks on descriptive geometry and analytic geometry, and a calculus textbook *Einführung in die mathematische Behandlung der Naturwissenschaft *(1895) written jointly with Walter Nernst. This work, which had reached its eleventh edition by 1931, was written primarily for students of physics and chemistry, but engineers also made considerable use of the text. In 1895 Schönflies edited Plücker's complete works. Otto Laporte, a theoretical physicist, spent a year in Frankfurt in 1920. In an interview late in his life he spoke about Schönflies:-

When I went to Frankfurt, right away I was under the influence of some very great men. The oldest one there was a mathematician named Arthur Schönflies, whom you know, of course, as the man who first formalised the theory of space groups of lattices, but who has many, many other great achievements. He was then an old man, but he gave beautiful courses.

Among the many honours that Schönflies received we mention his election to membership of the German Academy of Scientists Leopoldina in 1896, to corresponding member of the Royal Society of Sciences in Liege in 1904, and corresponding member of the Prussian Academy of Scientists in 1918 who specifically cited his contributions to promoting Cantor's ideas on set theory. In 1910 he was awarded the Red Eagle Order IV Class by the German Emperor and appointed as a Privy Councillor in 1916. We also note that he was a founder member of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) in 1890.

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

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