# Piers Bohl

### Quick Info

Born
23 October 1865
Walka, Livonia (now Valka, Latvia)
Died
25 December 1921
Riga, Latvia

Summary
Piers Bohl was a Latvian mathematician, who worked in differential equations, topology and quasi-periodic functions.

### Biography

Piers Bohl's father, Georg Bohl, was a merchant and his mother was Ottilie Ehmann. Piers' family background was not an academic one. He had a younger brother Edgar Bohl, born in 1867 and possibly other siblings. Piers received his first lessons from private tutors and studied at the municipal elementary school in Walka (also known by the German name of Walk). In 1878 he went to the German Livonian State Grammar School in Fellin, Livonia, a town now named Viljandi in Estonia. Perhaps at this point it is worth briefly looking at the history of the region where Bohl was brought up and educated. The region of the Riga Governorate of the Russian Empire existed through the 18th century and was renamed the Governorate of Livonia in 1796 after the partition of Poland. Although it was part of the Russian Empire during the time Bohl was growing up, it was an autonomous region ruled by German nobles and the language of the people was German. Later in his life, however, Russification of the region took place. His home town of Walka (or Walk) was actually split into two when the Latvian/Estonian border went through the middle of the town in 1920. The Latvian half is now known as Valka, the Estonian half as Valga. Fellin, now Viljandi in Estonia, is about 80 km north of Walka.

In [4] the record of both Piers and Edgar Bohl at the Livonian State Grammar School in Fellin is described:-
The documents from the State Grammar School in Fellin, available in the Latvian History Archive, state that both brothers were ambitious students. Piers was ill for half a year and had trouble making up for what he had missed in class. Both brothers lived in the "Aluminat", a boarding school with the strictest discipline. School fees were usually paid for the lessons, but Piers Bohl's name (more often than that of his brother) can also be found on the scholarship list. From the archive records it can be seen that Piers found mathematics easy from the start and that he was one of the best students in class in this subject. His brother Edgar was a poor student, but he also graduated from high school, then studied medicine at Dorpat University (now Tartu University, Estonia) and obtained a doctorate. Nothing further seems to be known about Edgar Bohl.
At the Fellin Grammar School, Bohl was taught mathematics by Edward Hugo Weidemann (1854-1887), known as Hugo. Weidemann, who had graduated from Dorpat University, was an inspirational teacher who did not follow the standard curriculum but had his own ideas about how mathematics should be taught. Tragically, he died when only 32 years old, just three years after giving Bohl his passion for mathematics.

In 1884 Bohl graduated from the Fellin Grammar School with the Maturity Certificate and remained in, what today is Estonia, entering the Mathematics and Physics Faculty of the University of Dorpat in August of that year. We note that Dorpat University had all its teaching in German from 1802 to 1893. Russification meant that Dorpat was renamed Yuryev in 1893, was changed back to Dorpat during the German occupation during World War I, then became Tartu in 1919, the name it is known by today. When Bohl studied there, the three year course was examined with oral examinations by the lecturers at the end of each year. The Professor of Pure Mathematics at this time was Peter Helmling (1817-1901) and the Professor of Applied Mathematics was Ferdinand Minding, but neither had much influence on Bohl. Following Minding's death in 1885, Anders Lindstedt (1854-1939) became Professor of Applied Mathematics, but he left in 1886 when he was appointed Professor at the Royal Institute of Technology in Stockholm.

At the end of the first year, on 9 December 1895 Bohl completed his examinations on the theory of equations and determinants, differential calculus, integral calculus, analytical geometry and the theory of curves and surfaces. In each subject his examination was graded "very good". For his second year, completed on 18 August 1886, he was examined on the courses on mathematical geography, differential equations, the method of least squares, the theory of analytical functions, and modern algebra and geometry. Each paper was either graded "good" or "very good", a slightly less brilliant performance than in year one. Bohl's schooling had been in German, and students with this language background were required to write an essay in Russian at the end of their second year. This was rather a challenge for Bohl who had not performed particularly well in his school studies in Russian. He wrote his essay on Ivan Turgenev's novel Virgin soil (Russian), the final novel by Turgenev which is also his longest and most ambitious. Bohl's essay was, not surprisingly, the poorest of his examined works, graded "satisfactory". His performance in mathematics, however, was outstanding and he won a Gold Medal for an essay he wrote in 1886 entitled Darstellung und Anwendung der Invarianten der linearen Differentialgleichungen . It is a tribute to the quality of this essay that, although it was not published at the time, it was translated from German to Russian and published in 1973.

In his final year he took the following courses. (i) 'Theoretical and Experimental Physics' given by Arthur von Oettingen (1836-1920). Von Oettingen had studied physics and astronomy at the University of Dorpat, then continued his studies in Paris and Berlin before becoming professor of physics at Dorpat in 1868. (ii) 'General Astronomy' given by Ludwig Schwarz (1822-1894). Schwarz had been appointed as Director of the Dorpat Observatory in 1872 to succeed Thomas Clausen. (iii) 'Inorganic Chemistry' given by Carl Schmidt (1822-1894). Schmidt was awarded a doctorate from the University of Giessen in 1844, and appointed as Professor of Chemistry in the Mathematics and Physics Faculty of the University of Dorpat in 1851. (iv) 'Mechanics' given by Otto Staude (1857-1928). Staude studied for his doctorate at the University of Leipzig advised by Felix Klein, being awarded the degree in 1876. Staude was Professor of Applied Mathematics at Dorpat from 1886 to 1888. (v) 'Number Theory' given by Theodor Molien. On 17 August 1887, Bohl made a request to the Mathematical and Physical Faculty of the University of Dorpat to be examined in these five topics. After the five oral examinations were carried out in the week after his request by those who had taught the courses, Bohl was awarded "very good" in each of the five subjects. On 26 August 1887 he was awarded his Candidate's Diploma in Mathematics.

After graduating with his Candidate's Diploma, a few weeks later Bohl passed the examinations which qualified him to teach in High Schools. As part of the examination he had to write an essay and wrote about the purpose of high school education in Über den Zweck der gymnasialen Bildung . He then became a teacher [4]:-
After completing his studies, Bohl first worked as a tutor at the Levi estate in Estonia, then at the Kurland teacher training college. His first publications are "Das Gesetz der molekularen Attraktion" , which appeared in 'Annals of Physics and Chemistry' in 1889, and "Verallgemeinerung des dritten Keplerschen Gesetzes" , published in 1890 in the 'Journal of Mathematics and Physics'. As the titles of the works show, Bohl turned to physical questions during this period, but from today's perspective, some of his results and conclusions are incorrect.
By 1889 Bohl was registered as a research student at Dorpat University, where he was advised by Adolf Kneser who had been appointed to the chair of Applied Mathematics at Dorpat in that year. In 1893 Bohl was awarded his Master's Degree (equivalent to a Ph.D. degree) for the thesis Über die Darstellung von Funktionen einer Variablen durch trigonometrische Reihen mit mehreren, einer Variablen proportionalen Argumenten . This was for an investigation involving quasi-periodic functions, motivated from mathematical methods used in celestial mechanics. The examining committee for the thesis was Gustav von Grofe (1848-1895), Arthur von Oettingen and Adolf Kneser. Although Bohl was the first to study these functions the name is not due to him but is due to Ernest Esclangon who studied them later. Esclangon's work was in fact completely independent of Bohl's and, recommended by Paul Painlevé, was published in Comptes Rendus in 1902 [4]:-
Bohl read this note and found that Esclangon's concept of "quasi-periodic function" corresponded to that of "periodic function in a generalised sense", which he had defined in his master's thesis. He communicated this to P Painlevé and probably sent him a copy of his dissertation, and in 1903 Esclangon's second note in the "Comptes Rendus" acknowledged Bohl's priority in discovering the new class of functions. In the same note, Esclangon said that he had reached his results independently of Bohl, discussed the differences between the two works, and cited several new results. Bohl was very pleased with this reaction; in the subsequent correspondence between him and Esclangon, he in turn shared various new research results and was very benevolent about Esclangon's scientific projects. Esclangon was awarded a doctoral thesis for "Quasi-Periodic Functions" from the University of Paris.
The notion of quasi-periodic functions was generalised still further by Harald Bohr when he introduced "almost periodic functions". It is worth noting that the examining committee for Bohl's thesis had not realised the importance of his introduction of quasi-periodic functions. Had they done so, his name may well today be associated with this class of functions.

Bohl taught at Riga Polytechnic Institute from 1895. In 1900 he received his doctorate (perhaps equivalent to a D.Sc. in standard) from the University of Dorpat and, in the same year, he was promoted to professor at Riga Polytechnic Institute. The doctorate was a high qualification being essentially that required for a professorship, the Master's Degree being the passport to a university post. Bohl's doctoral dissertation applied topological methods to systems of differential equations. In this area he was following earlier work by Henri Poincaré and Adolf Kneser. The award of his doctorate, however, had not been straightforward. He had submitted his thesis Über einige in der Mechanik anwendbare Differentialgleichungen allgemeinen Charakters to the Mathematics and Physics Faculty of the University of Dorpat in February 1900. Now politics was playing a major role in the staffing of the University of Dorpat by this time since there had been a deliberate policy of Russification for a few years. The last German professors of mathematics at Dorpat were Friedrich Schur (1856-1932), who left in 1892, and Adolf Kneser who left in 1900. Platon Grave (1867-1919) had been appointed as a professor in 1898 and although he satisfied the criterion of being Russian he was not a research mathematician. Grave was one of the examiners of Bohl's doctoral thesis and he claimed it was not worthy of the degree. It is unclear whether Grave was not familiar with research mathematics to make a correct judgement or whether he was against Bohl because of the Russification policy which had got him the professorship. Eventually, despite Grave's objections, Bohl was awarded his doctorate in September 1900. We can confidently say that Grave was wrong in his objections since Jacques Hadamard thought highly of this work which, ten years later, was translated into French and published in the Bulletin of the French Mathematical Society.

The Russification had another impact on Bohl. From 1896, one year after he was appointed to the Riga Polytechnic, lectures had to be delivered in Russian. Although Bohl was able to do this he had never been particularly happy with that language and this must have presented him with difficulties. From 1895 to 1898 he was a lecturer, then an adjunct professor, and finally, from 1901, professor of higher mathematics.

As well as being an outstanding mathematician, Bohl was a top quality chess player. His abilities were seen while he was still a student [10]:-
Student P Bohl from Valka is an ingenious chess player. His game, however, has not yet fully developed due to a lack of strong opponents.
When he began teaching in Riga he was able to develop his game playing stronger opponents. The President of the Riga Chess Club wrote [10]:-
Bohl has the potential to become a first-rate master, and is currently the strongest new generation chess player alongside Hans Seyboth.
He played for the Riga City team and had many notable victories. He introduced what today is called the "Riga Variation" of the Ruy Lopez which was subsequently analysed by Emanuel Lasker in Lasker's Chess magazine. Mikhail Botvinnik, World Chess Champion from 1948 to 1963, was an admirer of Bohl's chess playing. He wrote (see [10]):-
Even though his opponents were chess players of all styles, Piers Bohl always pursued an open game, tried to attack, eagerly searched for difficult combinations. When it came to protecting himself, Bohl did so reluctantly and without much thought. Piers Bohl had a thorough knowledge of variations - no doubt he was studying them specifically. Of course, these were the variations that were typical of those times - open versions of the Ruy Lopez, Four-Knights, and so on. In terms of Bohl's practical level of play, in modern terminology he would probably be on the border between Category 1 and Candidate Master. This is a very remarkable level for a scholar who could only devote his free time to chess.
Latvia had been under Russian imperial rule since the 18th century so, in 1914, World War I meant that the Institute at Riga was evacuated to Moscow. Bohl went to Moscow with his colleagues. Conditions in Moscow were difficult with sharply rising prices and food shortages which led to riots. After the Russian Revolution of 1917 and the end of World War I in 1918, however, Latvia regained its independence (although this was to be short-lived) and in 1919 Bohl was able to return to Riga to fill a chair at the University of Latvia which had just been established. Sadly he was only to hold the chair for two years before his death. His health had suffered due to the difficulties of the war years and he suffered a stroke. This led to memory loss and there were many sad stories relating to his teaching from this point on. In the spring of 1921 his health deteriorated further and by the time teaching was due to begin in the autumn for the 1921-22 session, he was too ill to teach. On Christmas day 1921, while on a walk, he suffered a second cerebral haemorrhage which proved fatal.

Among Bohl's achievements was, rather remarkably, his proof of Brouwer's fixed-point theorem for a continuous mapping of a sphere into itself, see [9]. Clearly the world was not ready for this result since it provoked little interest.

Bohl also studied questions regarding whether the fractional parts of certain functions give a uniform distribution. His work in this area was carried forward independently by Hermann Weyl and Wacław Sierpiński. There are many seemingly simple questions in this area which still seem to be open. For example it is still unknown whether the fractional parts of $(\large\frac{3}{2}\normalsize )^{n}$ form a uniform distribution on $(0,1)$ or even if there is some finite subinterval of $(0,1)$ which is avoided by the sequence. [We note that a 2018 paper computes $(\large\frac{3}{2}\normalsize )^{n}$ modulo 1 for $n$ up to $10^{8}$ and statistical analysis of the results "strongly agrees with the hypothesis that $(\large\frac{3}{2}\normalsize )^{n}$ modulo 1 is uniformly distributed."]

Let us quote from [13] regarding some of Bohl's most important results:-
In his paper 'Über die Bewegung eines mechanischen Systems in die Nähe einer Gleichgewichtslage' (1904), he studied the existence and smoothness problems of stable and unstable manifolds for a quasilinear system of differential equations. In the course of studying these problems, as auxiliary results he established that a sphere is not a retract of a ball and proved that a continuous mapping of a ball into itself has a fixed point, a famous result which is often called the .Brouwer fixed-point theorem., although Brouwer published the result seven years later, see L E J Brouwer, 'Über die Abbildung von Mannigfaltigkeiten' (1911). In his paper 'Über ein Dreikörperproblem' (1906) Bohl proved a famous theorem about quasiperiodic functions, as well as some important results about differential equations with quasiperiodic coefficients. Of course, here we have been able to mention only a few results of this insightful mathematician.
Daina Taimina writes [14]:-
Bohl had no family and no close friends. He lived only for science and he was indifferent to glory. When he found some new result in mathematics he said he could not believe that nobody had noticed it before.
Adolf Kneser and Alfred Meder write [4]:-
Almost the only distraction that he allowed himself was to travel regularly during the summer months. He got to know a large part of Europe.

### References (show)

1. A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).
2. A D Myskis and I M Rabinovic, Mathematician Piers Bohl from Riga: With a commentary by the grand master M M Botvinnik on the chess play of P Bohl (Russian), Izdat. Zinatne (Riga, 1965).
3. A D Myskis and I M Rabinovic (eds.), P G Bohl, Selected Works (Russian) (Riga, 1961).
4. I Bula, Der Rigaer Deutsch-Baltische Mathematiker Piers Bohl (1865-1921), Journal of Baltic Studies 24 (4) (1993), 319-326.
5. Yu M Gaiduk, Evaluation of the scientific work of Piers Bohl by his contemporaries (Russian), Tartu State University, History of development, training of personnel, and scientific research II (Tartu, 1982), 28-39.
6. A Kneser and A Meder, Piers Bohl zum Gedächtnis, Jahresberichte der Deutschen Mathematiker-Vereinigung 33 (1925), 25-32.
7. L L Kul'vetsas, P Bohl's fourth thesis and Hilbert's sixth problem (Russian), Studies in the history of physics and mechanics, 1986 (Moscow, 1986), 62-93.
8. A D Myshkis and L E Reizin', Piers Bohl, a creator of qualitative methods mathematical analysis (Russian), Proceedings of XIIIth International Congress of the History of Science (Nauka, Moscow, 1974), 96-99.
9. A D Myskis and I M Rabinovic, The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P G Bohl (Russian), Uspekhi matematicheskikh nauk (NS) 10 (3) (65) (1955), 188-192.
10. Piers Bohl (born Oct-23-1865, died Dec-25-1921, 56 years old) Latvia, chessgames.com.
11. I Rabinovics, Famous Scientist from Riga Piers Bohl (1865-1921) (Latvian), Astronomical calendar for 1957 (Riga, 1956), 95-105.
12. L. Reizins, I. Henina, Piers Bohl. Commentaries (Russian), in L Reizins (ed.), P Bohl Collected Works (Russian) (Zinâtne, Riga, 1974), 5-7; 502-510.
13. A Sostak, The Latvian Mathematical Society after 10 years, European Mathematical Society Newsletter 48 (June, 2003), 21-25.
14. D Taimina, Some notes on mathematics in Latvia through the centuries, Department of Mathematics, Cornell University.
http://pi.math.cornell.edu/~dtaimina/mathinLV/mathinlv.html