**Joseph Raabe** is mostly remembered for *Raabe's Test* for convergence of series.

- Since his parents were poor, Raabe did not have the educational opportunities as a young boy so he had to learn mathematics largely working on his own.
- In Vienna, Raabe was inspired by Johann von Littrow (1781-1840), who encouraged him to submit his work for publication in August Crelle's Journal für die reine und angewandte Mathematik which began publication in 1826.
- Von Littrow had studied at the Charles University of Prague, spent a period in Russia, and had been appointed professor of astronomy at the University of Vienna in 1819, shortly before Raabe arrived in that city.
- He published a paper in Part 3 of Volume 1 (1826) of Crelle's Journal and Raabe's first paper in that journal, entitled Allgemeine Theorie der Epicykeln Ⓣ(The general theory of epicycles), appeared in Part 4 of Volume 1.
- It is worth noting that his name appears both in the index and on the paper as "L Rabe of Vienna" but there is no doubt that this paper is by Joseph Raabe.
- In the second volume of the journal, published in 1827, Raabe has five papers: Über den Stillstand der Planeten Ⓣ(On the stoppage of planets); Sphärische Polygonometrie Ⓣ(Spherical polygonometry); Eigenschaften der Curven, die sich auf bestimmten Oberflächen befinden; letztere zu beweisen Ⓣ(Characteristics of curves that are located on certain surfaces and to prove the latter); Untersuchung über die Directrixen der Curven Ⓣ(Investigation into the directrixes of curves); and Gleichungen der zweiten Ordnung in der Geometrie Ⓣ(Equations of the second order in geometry).
- For example, for the paper Sphärische Polygonometrie Ⓣ(Spherical polygonometry) his name appears in the index as "L Raabe", but on the paper itself as "Joseph L Rabe".
- For the paper Eigenschaften der Curven Ⓣ(Properties of curves) his name appears in the index as "J Raabe", but on the paper itself as "L Raabe", while for the paper Über den Stillstand der Planeten Ⓣ(About the standstill of the planets) his name appears in the index as "L Raabe", but on the paper itself as "Joseph L Raabe".
- By the autumn of 1831 it had reached Vienna and Raabe had no option but to stop giving his private mathematics lessons.
- He was studying under Johann von Littrow in Vienna when he became friends with Raabe.
- Eschmann and Raabe went to Zürich and Raabe found a home there and a position teaching mathematics at the Gymnasium.
- They made two inspiring appointments as privatdocent in mathematics, Gräffe and Raabe.
- Johannes Eschmann, Raabe's friend, was appointed to a similar position in astronomy.
- Today Raabe's name is mostly remembered for Raabe's Test for convergence of series.
- This test appears in Raabe's paper Note zur Theorie der Convergenz und Divergenz der Reihen Ⓣ(Note on the theory of convergence and divergence of series) which was published in Volume 11 (1834) of Crelle's Journal.
- Raabe did not publish any papers in Crelle's Journal after those we mentioned above in the first two volumes until this 1834 paper.
- In 1836 the University of Zürich appointed a professor of mathematics but the chair was not given to Raabe but rather to Anton Müller, a somewhat undistinguished mathematician.
- Not only was Raabe passed over, but so was his fellow docent, the talented Karl Heinrich Gräffe.
- Over the following years, Müller published no research of any merit while both Raabe and Gräffe continued to produce highly significant contributions.
- Raabe's work on the Bernoulli polynomials deserves special mention.
- The term "Bernoulli polynomials" was introduced by Raabe in this paper of 1851.
- Raabe was appointed as a professor at the Eidgenössische Polytechnische Schule in the year of its founding, but he did not hold this chair for long since he left his chair in 1858 and died in January of the following year at the age of 57 years.
- Richard Dedekind applied for Raabe's chair and was appointed in the spring of 1858, taking up the chair in the autumn of that year.

Born 15 May 1801, Brody, Galicia (now Ukraine). Died 22 January 1859, Zürich, Switzerland.

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Origin Ukraine

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive