**Karl Geiser**'s first name is often written as Carl. His father was Friedrich Geiser, a butcher who ran a local inn. Friedrich was influential in the district being a member of the Cantonal legislature at Berne. Karl's mother was Elisabeth Begert. There was a famous mathematician in Karl's family for his great uncle was Jakob Steiner; this certainly helped Karl in his career, particularly when he went to the University of Berlin. Before studying at Berlin he spent four semesters as a student at the Polytechnikum in Zürich (it was founded in 1855 as the Eidgenössische Polytechnische Schule and renamed the Eidgenössische Technische Hochschule in 1911). Then he spent a further four semesters at the University of Berlin where he was taught by Karl Weierstrass and Leopold Kronecker. Coming from a family with little money to support the education of their children, Geiser had to support himself while studying in Berlin. He did this by taking private pupils and Weierstrass and Kronecker both helped by finding pupils for him. He graduated in 1863 and was appointed as a dozent at Zürich Polytechnikum. He undertook research for his doctorate at the University of Bern advised by Ludwig Schläfli. He was awarded his doctorate in July 1866 for his dissertation *Beiträge zur synthetischen Geometrie* Ⓣ. In Zürich he took on the duties of a professor after the death of Joseph Wolfgang von Deschwanden (1819-1866), the professor of descriptive geometry, until the chair could be filled. Note that Theodor Reye also assisted with Deschwanden's teaching. In May 1867 Wilhelm Fiedler was appointed to fill Deschwanden's chair.

At Zürich Polytechnikum, Geiser was appointed as an extraordinary professor in 1869, and then in 1873 he was appointed to a full professorship of higher mathematics and synthetic geometry, with special responsibility for teaching mathematics to engineering students and to mathematics students. Geiser had a number of famous colleagues at Zürich Polytechnikum, first Georg Frobenius between 1875 and 1892, and then Adolf Hurwitz who filled Frobenius's chair. Hurwitz remained in Zürich until his death in 1917. Geiser taught algebraic geometry (his own research topic), differential geometry and invariant theory at Zürich. He was more interested in teaching mathematicians than engineers and he appears to have been a rather unpopular lecturer with some of the engineers [3]:-

He was rather exacting in his expectations from his students and, for this reason, was not always very popular, especially among the engineers.

His teaching did, however, make an important contribution to general relativity in an indirect fashion. Albert Einstein was a student at the Zürich Polytechnikum from October 1896 until he graduated in July 1900. He attended some of Geiser's lectures and towards the end of his life recalled his fascination with Geiser's *Infinitesimalgeometrie* course. In this course Geiser taught the Gaussian theory of surfaces and, in 1912, Einstein had what he described as "the decisive idea" of an analogy between general relativity and Gaussian surfaces.

Geiser published on algebraic geometry and minimal surfaces. In 1865 he published a paper *Über eine geometrische Verwandtschaft des zweiten Grades* Ⓣ on a quadratic transformation called 'quadric inversion' which was discovered in the same year by Thomas Hirst. Papers such as *Über die Normalen der Kegelschnitte* Ⓣ (1866) followed. One of his most important results explains how the 28 double tangents of the plane quadric are related to the 27 straight lines of the cubic surface. This appears in his 1869 paper *Über die Doppeltangenten einer ebenen Curve vierten Grades* Ⓣ published in *Mathematische Annalen*. He is also remembered by those working in algebraic geometry for his discovery of an involution, now named after him, which appears in his paper *Zwei Geometrische Probleme* Ⓣ (1867). Later papers include *Notiz über die algebraischen Minimumsflächen* Ⓣ (1871), *Zum Hauptaxenproblem der Flächen zweiten Grades* Ⓣ (1877), and *Über einen fundamentalen Satz aus der kinematischen Geometrie des Raumes* Ⓣ (1881).

However Geiser's most important contribution was not in his original research but rather in his political skills in organising the educational system in Switzerland [1]:-

Acquainted with many persons in the fields of politics and economics as well as with important mathematicians in the neighbouring countries, and a close adviser of the chairman of school supervisors, Geiser worked effectively within the professoriate to attract first-rate teachers. There devolved upon him, above all, the instruction of candidates for the teaching of algebraic geometry, differential geometry, and invariant theory.

Associated to his work in this area is his book *Einleitung in die synthetische Geometrie. Ein Leitfaden beim Unterrichte an höheren Realschulen und Gymnasien* Ⓣ (1869).

Arnold Emch, who was a student of Geiser's, explains how he [3]:-

... was very influential in bringing the Cantonal(or secondary)colleges of Switzerland to a higher level of teaching and of professional qualifications for teachers of those schools.

He also made a major contribution to making the Zürich Polytechnikum one of the leading institutions in the world. For ten years he was its director (1881-1887 and 1891-1895) and the mathematicians who taught there during this period indicate the high status it was achieving: Richard Dedekind, Heinrich Durege, Elwin Christoffel, Hermann Schwarz, Heinrich Weber, Theodor Reye, Wilhelm Fiedler, Georg Frobenius, Friedrich Schottky, Adolf Hurwitz, Hermann Minkowski and Ernst Zermelo. One of the important decisions which faced the Zürich Polytechnikum during the period he served as director was whether it was an institution devoted to practical subjects relating to industrial production or to technical and theoretical expertise in mathematics and similar subjects. It was in large part due to Geiser that when the laboratories were rebuilt in the 1880s and 1890s, theoretical knowledge in scientific experimentation was seen to be an important aspect of good teaching. He argued that the Polytechnikum should also teach students abstract ideas and show how they might be applied and adapted to assist industrial processes. Through his excellent leadership, an good balance was achieved making the Polytechnikum part university, part laboratory, and part industrial training centre.

Although Geiser was helped in his career by his relationship with Jakob Steiner, he repaid the debt by editing Steiner's unpublished lecture notes and treatises. For example he published *Die Theorie der Kegelschnitte in elementarer Darstellung. Auf Grund von Universitätsvorträgen und mit Benutzung hinterlassener Manuscripte Jacob Steiner's* Ⓣ (1867), *Construction der Fläche zweiten Grades durch neun Punkte: Nach den hinterlassenen Manuscripten Jacob Steiners dargestellt von Herrn C F Geiser in Zürich* Ⓣ (1868), and *Zur Erinnerung an Jakob Steiner* Ⓣ (1874).

Another important contribution which Geiser made, that was not in the area of research, was to organise the first International Congress of Mathematicians held in Zürich in 1897. He was president of the organising committee and in this role he gave an address at the opening ceremony on Monday 8 August in the Aula of the Polytechnikum [2]:-

In a long and florid address[Geiser]praised the Swiss mathematical glories: Jacob, Johann, and Daniel Bernoulli, Leonhard Euler, and Jakob Steiner.

In his speech, he also explained that Zürich had been chosen as the venue for the first International Congress because it was "at the crossroad of the large railways from Paris to Vienna and from Berlin to Rome". Following his speech, he was elected president of the Congress "by acclamation". We should note that, although 90 years old, he was able to be present at the next International Congress of Mathematicians held in Zürich in 1932.

Geiser was honoured with election as one of the first three honorary members of the Swiss Mathematical Society, elected in session 1911-12. He had been elected as a foreign member of the German Academy of Scientists Leopoldina on 9 December 1888.

Emch tells us something of Geiser's character in [3]:-

Although a good European in the best sense of the word, Geiser remained a rugged individualist of the old Swiss type. ... He hated all pretence and always worked in support of real merit. True friendship was guarded as a treasure by him. ... Although struck by loss of clear eyesight and temporary blindness which was later partly removed by a surgical operation, Geiser's mind remained active and alert. When89years of age, being especially fond of Italian culture, he insisted on being taken to Zürich from his Chalet on the lovely mountain-side of Lake Zürich to hear a lecture on Dante by the Swiss-Italian poet Chiesa. The writer last saw Geiser in the spring of1929, when he spent a friendly afternoon with him at his home on the lake.

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

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