**Theodor Reye**'s parents were Amalia Augustina Beckmann (1810-1857) and Georg Wilhelm Reye (1799-1850). Georg Reye had been born in Ritzebüttel and he lived there all his life, being a local businessman. Theodor was one of his parents' ten children who were all born in Ritzebüttel. He had four older brothers, Auguste (born 1831), Wilhelm (born 1833), Julius (born 1834) and Adolf (born 1936). He also had two younger brothers, Otto (born 1840) and Arnold (born 1841), and three sisters, Amilie (born 1845), Antoine (born 1845), and Emilie (born 1847). Notice that of Amalia and Georg's ten children, the first seven were boys and the last three girls. Theodor was the only one of his parents' children to follow an academic career, the other boys becoming businessmen like their father or, in the case Adolf, a brewer.

Theodor attended school in Hamburg, being a pupil at the famous Johanneum. He then studied mechanical engineering at the Polytechnikum in Hannover before continuing his studies at the Polytechnikum in Zürich (this institution was later called the Eidgenössische Technische Hochschule or the Swiss Federal Institute of Technology). In Zürich, Reye was taught by Kurt Culmann (1821-1881), who had held the Chair of Engineering Sciences since the Polytechnikum was founded in 1855. The Chair of Mathematical Physics at the Polytechnikum in Zürich had been held by Rudolf Clausius since the founding of the Polytechnikum and, despite the influence of Culmann which we mention below, it was Clausius who influenced Reye to move from mechanical engineering to mathematical physics. Reye, following Clausius's advice, then went to the University of Göttingen to undertake research for his doctorate in Mathematical Physics. He received a doctorate from Göttingen in 1861 for his thesis on gas dynamics, *Die mechanische Wärme-Theorie und das Spannungsgesetz der Gase* Ⓣ. The background to Reye's thesis is the following. Victor Regnault (1810-1878) had made extremely accurate experiments concerning Boyle's law, first deduced on purely experimental grounds, which claimed that the product of the absolute pressure and volume of a gas at constant temperature is constant. Regnault had shown experimentally that the law is only exact within certain limits. Clausius had made marked contributions to the kinetic theory of gases in 1857 which provided a theoretical justification of Boyle's law but there was much dispute over this theory since it assumed an atomic theory which was not widely accepted. Reye's thesis sought to show that Regnault's experimental results were consistent with the kinetic theory of gases and provide further support for Clausius's theories. A couple of years later, Reye wrote a paper *Die Ausdehnung der atmosphärischen Luft bei der Wolkenbildung* Ⓣ (1865) applying the kinetic theory of gases to the expansion of air during the formation of clouds. It is one of the first applications of the kinetic theory of gases to meteorology.

While undertaking research at Göttingen, he had attended inspiring lectures on partial differential equations by Bernhard Riemann. These lectures had the effect of changing Reye's interests more towards mathematics, although he continued his interest in mathematical physics. After the award of his doctorate, Reye returned to the Polytechnikum in Hannover where he taught for a while before presenting his habilitation thesis to the Polytechnikum in Zürich and becoming a lecturer there in 1863. He was promoted to professor in 1867 and lectured in Zürich on mathematical physics until 1870. In Zürich he became a colleague of Kurt Culmann, who had taught him as an undergraduate, and it was now Culmann who had a major influence on the direction of Reye's research. The work of Jean-Victor Poncelet and Karl von Staudt had inspired Culmann to apply graphical methods to engineering problems. Under this influence, Reye's early interest in mathematical physics and meteorology turned to an interest in geometry while he continued to hold the lectureship in mathematical physics at Zürich. Led towards geometry by his interest in mechanics he began to study graphical statics and avidly read von Staudt's work on geometry. Culmann's fundamental monograph *Die graphische Statik* (Graphical Statics) was published in 1865 but Reye was by this time deeply involved in studying geometrical methods similar to the ideas of projective geometry contained in von Staudt's *Geometrie der Lage* Ⓣ (1847). The first volume of Reye's own major work *Geometrie der Lage* Ⓣ was published in 1866 [1]:-

Staudt's book was considered very difficult to read; Reye's 'Geometrie der Lage' ... was easily comprehended.

The second volume of Reye's *Geometrie der Lage* Ⓣ appeared in 1868. A second edition of the two volume work was published in 1877 and 1880. When Reye published a third edition between 1886 and 1892, it was as a three volume work and two further three-volume editions were published - the fourth between 1899-1910, and the fifth between 1909-1923. A French translation with title *Leçons sur la géométrie de position* Ⓣ appeared in 1881-1882, an Italian edition with title *La geometria di posizione* Ⓣ was published in 1884 and an English translation *Lectures on the Geometry of Position* in 1898.

While in Zürich, Reye married Marianne Sattler (1847-1933) in 1868. They had five children: Katharina (born 1869), Franciska (born 1871), Wilhelm (born 1873), Charlotte (born 1875) and Otto (born 1880). Only the first of these children was born in Zürich for, in 1870, the family left Zürich and moved to Aachen in Germany when Reye was appointed to the Chair of Geometry and Graphical Statics at the newly founded Polytechnikum in Aachen (now the Rheinisch-Westfälische Technische Hochschule). The reason for this move was due to an unhappy situation which had arisen in Zürich. Reye had been an assistant to Joseph Wolfgang von Deschwanden (1819-1866), the professor of descriptive geometry, and taught descriptive geometry at Zürich after Deschwanden became ill. Reye also taught projective geometry believing that the two courses should be linked. However, after Deschwanden's death a new appointment was made to the chair of descriptive geometry, namely Otto Wilhelm Fiedler and Reye wrote (in the Preface to the 2^{nd} edition of *Geometrie der Lage* Ⓣ):-

Unfortunately, I am now denied contributing to the propagation of my favourite science as a teacher in the way I did before; because recently I was ruthlessly deprived of my course of lectures on projective geometry, so that it could be assigned to the newly appointed professor of descriptive geometry at his request.

This unhappy situation for Reye was solved by his move to Aachen. However, he only worked in Aachen for two years before he was appointed to the Chair of Geometry and Mechanics at the Kaiser-Wilhelm University of Strasbourg. This move to Strasbourg was, in many ways, made for patriotic reasons. Germany had captured Strasbourg after a 50-day siege during the Franco-Prussian war of 1870-71 and annexed the city. After taking control of Alsace-Lorraine, the Germans had reorganised the University of Strasbourg and reopened it as the German Kaiser-Wilhelm University of Strasbourg in 1872. Two mathematics chairs were founded at this time, the first in Mathematics which was filled by Elwin Christoffel and the second in Geometry and Mechanics which was filled by Reye. Both Christoffel and Reye considered it their patriotic duty to assist in making the University of Strasbourg a German university. At Strasbourg, Reye and Christoffel [2]:-

... built a new mathematics department with a solid reputation ... The new institute included one, and later two associate professorships, occupied by George Roth(1845-1904), Eugen Netto(1846-1927), Karl Schering(1854-1925), Adolf Krazer(1858-1926)and Emil Georg Cohn(1854-1944). ... About two dozen candidates received their doctoral degrees in mathematics and astronomy at Strasbourg between1872and1895. ... Reye, who was a geometer in the tradition of Steiner and von Staudt, produced about a dozen doctorates during this time. These included B Klein(1846-1891; in1876), R Krause(1879), G Kilbinger(1880), St Jolles(1857-1942; in1883), Th Meyer(1884), A Pampusch(1886), C Arnoldt(1887), C Fink(1887), F Schumacher(1889)and H E Timerding(1837-1945; in1894). Of these, only Klein, Jolles and Timerding became university professors, namely at Marburg, Berlin(Technological University)and Braunschweig, respectively. The remaining half-dozen were in the field of astronomy.

Reye's contributions to the Mathematical Seminar at the University of Strasbourg are described in detail in [7]. He served as rector of the University during the academic year 1886-87. This required him to give an inaugural address and he chose as his subject *Die synthetische Geometrie im Altertum und in der Neuzeit* (Synthetic geometry in antiquity and in modern times). The lecture was published as a pamphlet in 1886 and reprinted in the *Jahresbericht der Deutschen Mathematikervereinigung* in 1902.

In 1872 Reye published the book *Die Wirbelstürme, Tornados und Wettersäulen in der Erd-Atmosphäre mit Berücksichtigung der Stürme in der Sonnen-Atmosphäre* Ⓣ. This important work on meteorology, which in particular studied cyclones, led to him being offered the role of director of the Hamburg Oceanic Observatory. This, however, was not the type of job that appealed to him for his real love was as an academic teacher. He refused the offer of the post and it was offered instead to Georg Balthasar von Neumayer (1826-1909) who, after setting up a physical observatory in Melbourne, Australia, to study terrestrial magnetism, had returned to Germany when appointed hydrographer to the German Admiralty. After he retired in 1909 Reye remained in Strasbourg until the end of World War I. Karl Geiser writes [3]:-

... the war created grave sorrows in his family; a son who had been under arms for the entire duration of the war came home safely, but a grandson was killed in action.

His daughter Katharina had married Ludwig Jost who was appointed Professor of Botany at the University of Strasbourg in the year that Reye retired. After the defeat of Germany in 1918, Strasbourg was returned to France and many Germans were expelled. Reye and his wife moved to Würzburg where they lived with one of their married daughters. Katharina and her husband were also forced to leave Strasbourg and Ludwig Jost was appointed as Professor of Botany at the University of Heidelberg in 1919. Karl Geiser writes [3]:-

In the autumn of1918, Reye decided to return to the old Germany, hoping that he would be able to spend the last days of his life in peace and quiet. Unfortunately, the relocation created various difficulties, which, when an order was rescinded by a counter-order, turned into the most repugnant delays and eventually forced the eighty year old to carry out the complicated relocation without the help of his son for which he had hoped. At least he could settle down in the comfortable home in Würzburg, which a son-in-law had chosen for him. In May1919he was able to celebrate his golden wedding; but the shocks, disappointments and the agitation of the previous months had such a strong after effect that he fell victim to them a few weeks after the anniversary.

Reye's work in geometry included a study of conics, quadrics and projective geometry. His 73 papers include *Die algebraischen Flächen, ihre Durchdringungskurven, Schnittpunkte und projektivische Erzeugung* Ⓣ (1870), *Über algebraische Flächen, die zueinander apolar sind* Ⓣ (1874) and *Die Hexaeder- und die Oktaeder-Konfigurationen* Ⓣ (1883). Most material from his papers was eventually incorporated into editions of his books. A configuration of 12 points, 12 planes and 16 lines which he published in this last mentioned 1883 paper is named after him. This paper was published in the first volume of the new journal *Acta Mathematica*. Reye's standing as a mathematician is illustrated by the fact that Gösta Mittag-Leffler, the founder of the journal, interrupted his honeymoon so that he could personally speak to Reye and ask for his support for the new journal by submitting a paper for publication. Reye's book *Synthetische Geometrie der Kugeln, mit einer Einleitung in die analytische Geometrie der Kugelsysteme* Ⓣ was published in 1879 with an Italian translation being published two years later. This book on spherical geometry did not, however, have the same impact as his classic text *Geometrie der Lage* Ⓣ.

Werner Burau writes in [1]:-

Reye treated in detail the theory of conics and quadrics and of their linear systems, that of third degree surfaces and some of the fourth degree, as well as many quadratic congruences and aggregates taken from line geometry. He was one of the leading geometers of his time, and he published a great deal on synthetic geometry.

It is not unreasonable to ask why Reye's work is not better known today. Part of the reason must be that some of Reye's work was later interpreted in the geometry set up by Corrado Segre, in particular it was interpreted in the setting of Segre manifolds. Reye's work on linear manifolds of projective plane pencils and of bundles on spheres went into the Segre manifold setting. Finally we note that although David Hilbert had a high opinion of Reye's work, their approach to geometry was very different. Hilbert proposed a purely axiomatic method in geometry while Reye rejects everything that cannot be justified with geometric intuition.

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

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