There are several important aspects of Schlömilch's contributions which we shall examine. There are his excellent contributions to mathematical research, his fame as a writer of textbooks, and his founding of the journal Zeitschrift für Mathematik und Physik which he edited for many years. First let us list a few highlights of his research contributions. In 1847 he gave a general remainder formula for the remainder in Taylor series. He discovered an important series expansion of an arbitrary function in terms of Bessel functions in 1857. It is interesting to note that this 1857 paper, Über die Bessel'sche Function Ⓣ, was the first to treat Bessel functions of the first kind with integer indices as the object of a special mathematical theory independent of applied mathematics problems. He also made major contributions to the theory of the convergence of continued fractions.
He began publishing books early in his career with textbooks such as Handbuch der mathematischen Analysis Ⓣ (1845), Handbuch der Differential- und Integralrechnung Ⓣ (1846-48), Theorie der Differenzen und Summen Ein Lehrbuch Ⓣ (1848), Analytische Studien Theorie der Gammafunktionen Ⓣ (1848), Die allgemeine Umkehrung gegebener Funktionen Ⓣ (1849), and Grundzüge einer wissenschaftlichen Geometrie des Masses Ⓣ (1849). His textbooks presented Cauchy's techniques in analysis and through them these important methods became well known in Germany. After moving to Dresden in 1849, amazingly his output of textbooks became even greater with books such as Mathematische Abhandlungen Ⓣ (1850), Die Reihenentwickelungen der Differenzial- und Integralrechnung Ⓣ (1851), Handbuch der algebraischen Analysis Ⓣ (1851), Der Attractionscalcül Ⓣ (1851), and the two volumes of Compendium der höheren Analysis Ⓣ (1853). It would not be useful to give details of the many editions these books went through but, as an example, let us note that the first volume of Geometrie des Masses Ⓣ, published in 1849, had reached its seventh edition by 1888. Also the first volume of Compendium der höheren Analysis Ⓣ, published in 1853, had reached its fifth edition by 1881.
It was around 1853 that Schlömilch began to correspond with the published B G Teubner. It was through the collaboration with Schlömilch that Teubner went on to become one of the leading publishers of mathematics books and well as Schlömilch's journal Zeitschrift für Mathematik und Physik. Earlier Schlömilch had given much support to Johann August Grunert and his journal Archiv der Mathematik und Physik. The story of how the breakdown of his relationship with Grunert led to Schlömilch founding his own journal is told by Peter Schreiber in :-
Except in a few cases, the many famous mathematicians of the nineteenth century did not publish in Grunert's Archiv. ... One of the few contributors of the journal who has remained well known today, was Oscar Xaver Schlömilch. In the first twenty volumes of Grunert's Archiv he published 83 notes and 16 problems, thus becoming one of the most industrious of Grunert's cooperators. At this time there must have been a good personal relationship between Grunert and Schlömilch. For example, Grunert arranged for the printing of Schlömilch's book, 'Handbuch der Differential- und Integralrechnung' (1846-48), at C A Koch in Greifswald. Then an unpleasant dispute between Schlömilch and F W Barfuss, the director of an insurance company in Weimar, arose, which ran through several volumes of the Archiv. In a militant manner, Barfuss defended an obsolete point of view on the question of symbolic calculation with divergent series against Schlömilch. Possibly Schlömilch became vexed because Grunert repeatedly gave Barfuss the opportunity of printing nonsensical answers in the Archiv. Possibly Schlömilch changed his opinion about Grunert's (sometimes old-fashioned) style of mathematics and indeed about his scientific ability. In all events, in 1856 Schlömilch founded his own journal, the Zeitschrift für Mathematik und Physik, which was very similar to Grunert's Archiv with regard to readership and contents. In general, thereafter, the better known and more gifted authors preferred to publish in Schlömilch's journal.Schlömilch wrote to Teubner in 1854 explaining why a new journal was necessary. He explained that:-
Crelle's 'Journal für Mathematik' is totally unsuitable for a large group of teachers. The same is true for Poggendorff's 'Annalen der Physik und Chemie'. It only has a large readership because it is the only physics journal ... Thirdly the fact that Grunert's 'Archiv der Mathematik und Physik' is still a respected journal only goes to show the pressing need for a new journal.Teubner accepted the proposal immediately and the first volume appeared in 1856. Sometimes in its early years the Zeitschrift für Mathematik und Physik was called 'Schlömilch's Zeitschrift'. In 1857, in the second volume of his journal, Schlömilch published a harsh, damaging review of Grunert's Analytische Geometrie der Ebene und der Raumes für polare Koordinaten Ⓣ. In it Schlömilch was highly critical of Grunert's old-fashioned style and his criticisms have a personal rather cruel tone. From 1856 to 1859, Schlömilch was assisted in editing his journal by Benjamin Witzschel, then in 1859 Moritz Cantor replaced Witzschel as an editor. In the following year E Kahl joined the team and, until Schlömilch retired as an editor in 1896, these three jointly edited the journal. We should note that after Moritz Cantor joined the editorial team, articles on the history of mathematics began to appear in the journal. From 1875 Historisch-literarische Abteilungen der Zeitschrift für Mathematik und Physik Ⓣ appeared as a supplement to Zeitschrift für Mathematik und Physik.
Schlömilch's first highly successful publication with B G Teubner was a joint work Lehrbuch der analytischen Geometrie Ⓣ written with O Fort and published in 1855. A review in The Mathematical Gazette explains that:-
... rather more attention than usual is paid to questions that are capable of graphical interpretation or application. ... Conics are treated at first individually and in detail so as to bring out particular geometrical properties; then, starting with the equation of the second degree, they are discussed and reduced to their simplest forms and incidentally, as it were, we are introduced to the general properties of the curves of the second and higher orders. The proofs are well arranged and simply put.Other highly successful works included the two-volume text Compendium der höheren Analysis Ⓣ (1862) and the two-volume text Übungsbuch zum Studium der höheren Analysis Ⓣ (1867, 1868). In 1879 he published the first of two volumes of Handbuch der mathematik Ⓣ, the second volume appearing in 1881. A second edition of this text appeared in three volumes in 1904. After Schlömilch left his professorship at the Technical University of Dresden in 1874 to take up an appointment as minister responsible for education in schools in Saxony, he continued to hold his position as an editor of Zeitschrift für Mathematik und Physik. He also continued to write new textbooks and to bring out new editions of his many highly respected works.
Among the honours he received, we mentioned above his election to the Royal Academy of Sciences of Leipzig but we should also mention his election to the Royal Swedish Academy of Sciences (Kungl. Vetenskapsakademin) in 1862.
Article by: J J O'Connor and E F Robertson