# Panagiotis S Zervos

### Quick Info

Zervata, Kefalonia, Greece

Athens, Greece

**Panagiotis Zervos**was a Greek mathematician who worked in the theory of systems of differential equations.

### Biography

**Panagiotis Zervos**was born in the village of Zervata on the Greek Ionian island of Kefalonia. He belonged to the ancient family of the island, which is first mentioned in a document dated 1264. The long-standing family tradition was that at least one son from each generation joined the Church. Panagiotis was a son of Spiridon Zervos, a merchant, and his wife Anastasia Montesanda. He was one of a large family having two brothers, Ioannis and Christos, and six sisters, one of whom, Maria, deserves special mention.

Maria Zervos became one of the first Greek women to graduate from the Department of Mathematics at Athens University. She became a school teacher and published a schoolbook on theoretical arithmetic, written in Greek, as well as at least four papers written in French:

*Sur les expressions de certains théorèmes de l'arithmétique élémentaire*Ⓣ (1927),

*Sur une expression nouvelle de la définition des nombres premiers*Ⓣ (1930),

*Sur quelques définitions et théorèmes de l'arithmétique*Ⓣ (1932) and

*Sur une classe de nombres composés*Ⓣ (1934). The second of these four papers was the published version of the talk she gave in the 'Matematiche elementari, Questioni didattiche, Logica matematica' Section of the International Congress of Mathematicians in Bologna in September 1928. Let is return to our biography of her brother Panagiotis.

Before Panagiotis had reached the age to attend elementary school, the family moved to Corfu. Spiridon Zervos, together with the rich merchant Margaritis, founded a wax works producing candles. Panagiotis attended the Economou School (1884-1889) and then at the Gymnasium of Corfu (1889-1893). Panagiotis's older brother Ioannis had already distinguished himself at this Gymnasium but Panagiotis ran into problems in the mathematics class. The mathematics teacher was unhappy when Panagiotis produced solutions to problems that were different from the 'official solutions'. He also got into problems in a written examination for helping another member of his class.

The problems at the school were minor compared with other tragedies that the family suffered. Two of Panagiotis's younger sisters, Angeliki and Philile, died and his father's wax business went bankrupt. The family left Corfu and moved to Alexandria in Egypt and Panagiotis, who had not yet completed his secondary education, entered the French school, the Collège des Frères. A year after moving to Egypt, Spiridon Zervos died in 1894 and left his family without financial support. Panagiotis's mother decided that her son should take a job as a watchman at a medical warehouse. This did not please Panagiotis who had become passionate about mathematics and wanted, above all else, to be able to study the subject at university. Therefore, at the age of sixteen, he travelled alone to Athens and, with no financial support, entered the Faculty of Mathematics of the University of Athens.

In the first two university years of 1894-1896 he lived alone in the attic of a house and earned his living by tutoring. In his first year of study, Zervos won a competition for students who needed financial support and received a cash prize. When Zervos began his third year at university, his whole family (except his older brother Ioannis) moved from Alexandria to Athens and he was able to support the whole family from the money he earned from tutoring since he had already gained a high reputation as a tutor. His younger brother, Christos, joined the army, but soon after died of tuberculosis.

The professor of mathematics at the University of Athens at this time was Cyparissos Stephanos (1857-1917), who had studied for his Ph.D. at the University of Paris. As well as serving as professor of mathematics, Stefanos was rector of the university 1894-1895. He taught three students who became leading mathematicians. One was Zervos, the subject of this biography, and the other two were Georgios Remoundos and Nicholas I Hatzidakis. Remoundos began his studies at the University of Athens in 1895, one year after Zervos. Nicholas Hatzidakis graduated one year before Zervos entered the university but his father Ioannis Hazzidakis, who had also studied in Paris, had been appointed as a full professor at Athens in 1884 and was the greatest influence on the mathematical development of Zervos. There was, however, one further mathematician teaching at the University of Athens at this time who was important to Zervos and that was his professor, Cyparissos Stephanos.

Zervos graduated from the University of Athens in 1899 with a degree which qualified him as a school teacher. He was awarded the grade 'excellent'. His examining committee included Cyparissos Stephanos and Ioannis Hazzidakis, and they contacted the Ministry of Education recommending that Zervos should begin immediately working as a secondary school teacher. On 24 August 1899, he was appointed as a teacher in Lixouri, Kefalonia, where he taught during the school year 1899-1900. In the following year he taught at the Varvakis Lyceum in Athens, followed by a year at the high school in Keratea. At this time, in addition to his teaching duties, Zervos was undertaking research attempting to determine the exact number of positive roots of algebraic equations with real factors. He published two papers in 1901, namely

*Quelques remarques sur la recherche du nombre des racines positives d'un polynôme*Ⓣ and

*Sur le théorème de Descartes*Ⓣ both in the journal

*L'Enseignement Mathématique*. His introduction to the second of these papers is as follows:-

In the following proof of Descartes' theorem we use Rolle's theorem. Laguerre also gave a proof of the same theorem based on Rolle's theorem. But our proof differs essentially from his. It shows how, given a polynomial, we find an upper limit of the number of positive roots by means of the upper limit of the number of positive roots of its derivative of a certain order. If we express by $n$ the number of sign changes of coefficients of a polynomial with real coefficients, the number of its positive roots is $n - 2t$, where $t$ is a positive integer or zero.The second of these papers contained results which were in Zervos's Ph.D. thesis entitled

*On the series and the theorem of Descartes*(Greek) for which he was awarded his doctorate by the University of Athens in 1901.

For a list of some of Zervos's papers, see THIS LINK.

Stephanos had encouraged Zervos to enter the state competition for a scholarship for further studies which took place in 1901. He did not enter, however, telling Stephanos that his family obligations did not let him take part. In fact it was this competition that was won by Remoundos and it allowed him to undertake research in Paris. Although he planned to carry on teaching and supporting his family, in fact circumstances did not allow that to happen. On 12 September 1902, he was dismissed from public service, because one of his relations, who was a politician, realised Zervos had not voted for him. Now out of a job, there was no reason why he could not go to Paris so, at the beginning of 1903 this is what he did. It was a brave move since he had very little in the way of financial support. Once in Paris, Zervos attended the lectures of Paul Painlevé and Jacques Hadamard for four semesters at the Collège de France.

At the beginning of 2005 Hadamard suggested to Zervos that he look at the Monge Problem. [1]:-

The Monge problem in one independent variable, in the broad sense, consists of explicitly integrating a system of k $(k ≤ n - 1)$ Monge equations:Zervos began looking to see what had been already done and discovered that the problem had been investigated by Joseph Serret in 1848, by Gaston Darboux in 1873 and 1887, by Georg Frobenius in 1877, by Friedrich Engel in 1889, by Sophus Lie in 1898, by Édouard Goursat in 1898, by Heinrich Weber in 1900, by Élie Cartan in 1901, and by Hadamard himself in 1901. With such a collection of leading mathematicians having studied the problem before him, Zervos thought Hadamard must be joking to ask him to examine it. After two months of unsuccessful attempts to prove the general case of the Monge Problem, Zervos found a counterexample showing it was not true in general. He published this result in$F_{i} (x_{1,} x_{2,}..., x_{n+1} ; dx_{1,} dx_{2,}..., dx_{n+1}) = 0 (i = 1, 2, ..., k)$,in which the $F_{i}$ are homogeneous functions of the $dx_{1}, dx_{2}, ..., dx_{n+1}$.

By the term "explicitly integrating," we mean expressing the x variables as well- defined functions of one parameter, $n - k$ arbitrary functions of that parameter and their derivatives up to a certain order, and that those functions can also contain a finite number of arbitrary constants. Monge solved that problem for the case $n = 2, k = 1$.

*Sur le problème de Monge*Ⓣ (1905). Zervos continued to undertake research on this problem for the rest of his life.

For a list of some of Zervos's publications, see THIS LINK.

After his return from France to Greece at the end of 1905, Zervos again faced family obligations and his income was through giving private lessons. He attended the International Congress of Mathematicians held in Rome in April 1908 and gave the talk

*Sur la correspondance entre les théories d'intégration des équations aux drivées partielles du premier ordre et d'intégration des systèmes de Monge*Ⓣ in Section I: Arithmetic, Algebra, and Analysis.

In September 1908, he began teaching at the 4th school of Piraeus and remained in that post until 1911. In the academic year of 1908-1909 he also taught Dynamics and Mechanics at the University of Athens. In 1911 he was moved to teach at the 4th school of Athens. He continued to work on the Monge Problem and in 1912, independently, he obtained results similar to those published by David Hilbert in

*Über den Begriff der Klasse von Differentialgleichungen*Ⓣ (1912) which appeared in the Heinrich Weber Festschrift. Zervos's results were presented to the International Congress of Mathematicians in Cambridge, England, in August 1912 as the lecture

*Sur les équations aux derivées partielles du premier ordre à trois variables indépendantes*Ⓣ. Zervos was able to generalise Hilbert's results and published these in

*Sur l'intégration de certains systèmes indéterminés d'équations*Ⓣ which appeared in Crelle's Journal.

In October 1913, he became a teacher at the Second Varvakis High School in Athens. Ioannis Hazzidakis retired from the chair of Differential and Integral Calculus in 1914 and Zervos was invited to accept a professorship in the Department of Higher Algebra at the University of Athens. But to everyone's surprise, Zervos, who was struggling to make ends meet, refused a professorship at the university and chose to remain a high school teacher. The professorship remained vacant, however, and Zervos was unanimously elected on 30 November 1917. This election was one of the last administrative acts of Cyparissos Stephanos, who died on 27 December 1917. Zervos visited Stephanos shortly before his death and he asked Zervos to take over the publication of his unpublished works. Zervos's mother died soon after this, however, and when some time later he managed to visit the house where Stephanos had lived, his sisters refused Zervos access to their brother's manuscripts. On 26 January 1918 Zervos gave his inaugural professorial address at the University of Athens entitled 'Relations of mathematics to other sciences and to philosophy'.

The Greek Mathematical Society was created in 1918, and Nikolaos Hatzidakis became the first president with Zervos as vice-president. The Society began publication of the

*Bulletin*of the Greek Mathematical Society in 1919 and Zervos published two papers in French in the first volume, namely

*Sur l'équivalence des systèmes d'équations différentielles*Ⓣ and

*Sur quelques remarques relatives aux théories de l'intégration des systèmes en involution du second ordre*Ⓣ. In 1920 he published four papers, two of them in the

*Proceedings*of the International Congress of Mathematicians which was held in Strasbourg in September of that year. At this International Congress of Mathematicians the International Mathematical Union was founded with eleven founding countries, one of which was Greece. One place on the board was for a Greek representative and Zervos held this place. He also represented Greece on the International Committee for the Teaching of Mathematics. He wrote papers on teaching mathematics such as

*On the teaching of Differential and Integral Calculus in Greece*(1920) and

*Modern trends in the teaching of Mathematics in different countries*(1936).

The political situation in Greece gave Zervos severe problems. Constantine I had been king of Greece from 1913 but internal pressure from Eleftherios Venizelos, the Greek Prime Minister, together with external pressure from Britain and France, led to him resigning in June 1917. Alexander, the second son of Constantine I, became king and it was Alexander who appointed Zervos to the chair at the University of Athens. Alexander died on 25 October 1920, after a freak accident, and Constantine returned as king on 19 December 1920. One of the first acts of Constantine was cancel all appointments made by Alexander so Zervos was dismissed. For the following two years he lived in extremely difficult financial circumstances. After the Greeks under Constantine were defeated by the Turks in Anatolia, Constantine abdicated the throne for the second time on 27 September 1922. His eldest son George became king as George II and Zervos was reinstated to his chair at the University of Athens on 1 December 1922. He continued in this position until he retired in August 1949.

In 1925 Zervos married the philologist Hariklia Papaioannou, a secondaary school teacher who published many books including

*Introduction to the study of ancient Greek poetry*(1974). They had one son, Spyros P Zervos (born 17 March 1930, died 23 January 2015), who also became an outstanding mathematician. Spyros studied at the École Normale Supérieure in Paris where he was awarded his doctorate in 1960 for his 115-page thesis

*Aspects modernes de la localisation des zéros des polynomes d'une variable*Ⓣ. He returned to Athens becoming a professor at the university where he continued except during 1968-1974 when he was dismissed by the Junta.

Since 1925 the Paris Academy of Sciences had been commissioning a series of monographs reviewing current mathematical problems, the series having the title

*Mémorial des sciences mathématiques*Ⓣ. Zervos, as a recognised world leader on the Monge Problem was asked to write

*Le problème de Monge*Ⓣ which appeared as No 53 in the series in 1932, see [2]. This was translated into English by D H Delphenich and is reference [1] below. The work is headed:-

Memorial to the Mathematical Sciences. Published under the Patronage of the Paris Academy of Sciences, of the Academies of Belgrade, Brussels, Bucharest, Coimbre, Kraków, Kiev, Madrid, Prague, Rome, Stockholm (Mittag-Leffler Foundation), of the French Mathematical Society, with the Collaboration of Numerous Scholars.The 1st Inter-Balkan Mathematical Conference was held in Athens in September 1934 organised by Zervos and Hatzidakis. Zervos was Chairman of the Organising Committee, while Hatzidakis was Chairman of the Executive Committee. The Honorary President was Constantin Carathéodory and the honorary vice-presidents were Gheorghe Titeica, Richard von Mises and Panagiotis Zervos. The conference proceedings were published and it contained two papers by Zervos, both written in French, namely

*Sur quelques équations différentielles indéterminées*Ⓣ and

*Sur l'intégration des systèmes différentiels indéterminés*Ⓣ.

The success of the conference led to the publication in 1936 of the research journal

*Revue Mathematique de l'Union Interbalcanique*with Zervos as its editor-in-chief. Zervos published an obituary of Gheorghe Titeica in the second volume of the

*Revue*in 1939.

We have seen from the title of Zervos's inaugural lecture that he was interested in philosophy. In fact, from 1930 to 1940, he taught a course for final year undergraduate students 'Chapters from Mathematical Philosophy'. He pressed for the founding of the Greek Philosophical Society and this came about in 1933. Zervos became its first president. He was elected to the Academy of Athens in 1944, having published the paper

*Sur l'intégration symbolique*Ⓣ in the Academy's journal in 1940 and after is election gave several talks to the Academy on Plato's Mathematics.

Zervos died in Athens at the age of seventy-four. His mathematical work was continued by his son Spyros P Zervos.

### References (show)

- P Zervos, Le problème de Monge,
*Mémorial des sciences mathématiques***53**(1932), 62 pages. - K Barakiti, Panagiotis Zervos - A Leading Greek Mathematician . https://www.palladio.edu.gr/ΕΠΙΛΟΓΕΣ/Άρθρα-Καθηγητών/ArticleID/29/Παναγιώτης-Ζερβός-Ένας-κορυφαίος-Μαθηματικός
- M P Janet, Zervos et le problème de Monge,
*Bull. Sci. Math.*(2)**95**(1971), 15-26.

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Written by J J O'Connor and E F Robertson

Last Update January 2019

Last Update January 2019