**Georg Scheffers**' father was a professor at the Academy of Art in Leipzig. Scheffers studied mathematics and physics at the University of Leipzig from 1884 to 1888. When Scheffers started his studies at Leipzig he was taught by Felix Klein, the professor of mathematics, but Klein left to take up a chair in Göttingen in 1886. Friedrich Engel, who had been a student at Leipzig, returned there as a lecturer in 1885 and taught Scheffers. Sophus Lie was appointed to Leipzig to succeed Klein and immediately had a strong influence on Scheffers. Lie acted as Scheffers' thesis advisor when he began to undertake research on plane contact transformations in 1888. He was awarded his doctorate in 1890 for his thesis

*Bestimmung einer Classe von Berührungstransformationsgruppen des dreifach ausgedehnten Raumes*Ⓣ. Lie continued to advise Scheffers as he continued to work on plane contact transformations and complex number systems for his Habilitation thesis which he submitted to Leipzig in 1891.

He lectured at the University of Leipzig from 1891 until 1896 when he was appointed to the Technische Hochschule in Darmstadt. Appointed initially as an extraordinary professor in 1896, he taught at Darmstadt for eleven years becoming a full professor there in 1900. In 1907 he was appointed to the Technische Hochschule in Charlottenburg, a town near Berlin named after Sophie Charlotte the wife of Frederick I, King of Prussia. Scheffers, who had succeeded Guido Hauck, held the chair of mathematics there until he retired in 1935. Before this, however, in 1920, Charlottenburg was incorporated into Berlin becoming an area of the city. Burau writes [1]:-

During the period that Scheffers worked with Lie in Leipzig he used his skill as a writer to assist in publishing Lie's important contributions. In 1891, he publishedScheffers' favourite field of study was geometry and, more specifically, the differential geometry of intuitive space. In this area he was a master at discovering many properties of particular curves and surfaces and their representations.; he also possessed a gift for giving an easily understandable account of them - although in a much wordier style than is now customary. His exceptional talent for vividly communicating material is also apparent in a later work on the grids used in topographic maps and stellar charts.

*Vorlesungenuber Differentialgleichungen mit bekannten infinitesimalen Transformationen*Ⓣ and, two years later

*Vorlesungenuber continuierliche Gruppen mit geometrischen und anderen Anwendungen*Ⓣ. These were substantial accounts of Lie's basic ideas in which Scheffers decided that it was more important to explain the ideas in a meaningful way even if this meant that the account lacked absolute rigour as a result. The first volume of an intended two volume collaboration between Lie and Scheffers was published in 1896 entitled

*Geometrie der Beruhrungstransformationen*Ⓣ. However, in the year this volume was published, Scheffers moved to Darmstadt and, although the two authors had done some preparatory work on the second volume, they made no further progress and it was never published. By this time Lie's health was also deteriorating and he left Leipzig for Christiania in 1898, dying early in the following year.

Even after Lie's death Scheffers produced work which had been clearly strongly influenced by Lie. For example Scheffers' most important work was a paper in 1903 on Abel's theorem which still showed Lie's influence. Lie had discovered and started an investigation of, investigated algebraic translation surfaces that can be generated in four different ways. He had published his results on their general properties and analytic representation. Scheffers gave a historical introduction to this topic in his 1903 paper, where he also continued development of the theory. Before this paper appeared, however, Scheffers had published one of his most important textbooks, namely the two volume work *Anwendung der Differential- und Integralrechnung auf Geometrie* Ⓣ. The first volume, subtitled *Einführung in die Theorie der Kurven in der Ebene und im Raume* Ⓣ, appeared in print in 1901, while the second volume, subtitled *Einführung in die Theorie der Flachen* Ⓣ, appeared in the following year. In 1908 and 1909, Eduard Study, the Professor of Mathematics at Bonn, published two papers in which he criticised the standard treatment of differential geometry in general, and Scheffers' treatment in particular. Scheffers reacted very positively to this criticism and incorporated many improvements into his treatment which he published as a second edition in 1910 and 1913. Dirk Struik, reviewing the third edition, published in 1922-1923, writes [2]:-

One of Scheffers most important textbooks was his revision of Serret's famous bookThis third edition has not many points of departure from the second. Scheffers' book remains one of the best textbooks on the subject; it deals with the material in a very pedagogic way and illustrates it with many interesting examples. Complex values of the coordinates are treated carefully. The point of view of invariants is emphasized ... Scheffers' book contains many historical footnotes. These are sometimes very interesting.

*Cours de calcul différentiel et intégral*Ⓣ published in the 1870s. It was not this original edition which Scheffers revised, rather it was the second edition of 1897-99 of the German translation which was first published in 1884. Scheffers' edition, the first volume of which was published in 1907, was almost a new book:-

To illustrate the changes Scheffers introduced was quote a few examples given in the review:-This third edition, by Georg Scheffers, is not only revised, but entirely rewritten and in part rearranged. All the figures have been drawn anew and many new figures added. The theorems have been stated more clearly and more use has been made of italics to make them stand out from the body of the text. Many more references to previous paragraphs have been inserted.

As the reviewer notes:-The articles on number have been remodeled according to Dedekind's theory and the proofs of the theorems on continuous functions and on the convergence of series have thus been given a real backbone. A new chapter on implicit functions, with a thorough discussion of the functional determinant and of the independence of functions and of equations has been inserted. ... The chapter on the complex variable has been entirely rewritten and shortened ...

To give another example of one of Scheffers' excellent textbooks, he publishedThis calculus is a geometer's calculus. Over three hundred of the twelve hundred pages are devoted to applications to geometry. With the exception of the paragraphs on center of gravity, there are practically no references to mechanics or physics. All the problems given are worked out in detail. In fact, detail is one of the features of the work. The reviser rejoices in saying that as far as possible he has eliminated from the text such phrases as "the reader will easily see", "the proof is left to the student."

*Lehrbuch der Mathematik: Eine Einfuhrung in die Differential-und Integralrechnung und in die Analytische Geometrie*Ⓣ in 1905. H Betz, reviewing the 6

^{th}edition of 1925 for

*The American Mathematical Monthly*writes:-

Scheffers wrote many other books. For exampleIt was written, as[Scheffers]says, for students of the natural sciences and engineering, who wish to acquire a sound working knowledge of higher mathematics by themselves. And it may be doubted whether any book could fulfil this avowed purpose more completely, especially if used by students as mature as those usually found in the first semesters of continental universities and technical schools.

*Lehrbuch der Mathematik für Studierende der Naturwissenschaften und der Technik*Ⓣ (1916) which was in its eighth edition by 1945.

*The Scientific Monthly*, reviewing this eighth edition said:-

We should also mention Scheffers' contributionThis book provides an unusually complete course in calculus for the sincere student. It is written with great care and clarity and contains many applications to physics and engineering.

*Besondere transzendenten Kurven*Ⓣ which appeared in

*Enzyklopädie der mathematischen Wissenschaften*Ⓣ in 1903, and another of his books which was

*Lehrbuch der Darstellenden Geometrie*Ⓣ (1919). There was also

*Wie Findet Und Zeichnet Man Gradnetze Von Land- und Sternkarten?*Ⓣ, his 98 page book on the grids used in topographic maps. A review in 1935 in

*The Geographical Journal*begins:-

Another review, this time by B H Brown inIn this little handbook the author sets out, as his title implies, to show how to draw the graticules of the various maps whose projections he considers. He is not therefore concerned with the equations of the projections, and though a certain amount of mathematics is necessary for his purpose, yet he concerns himself as far as possible with the geometrical constructions of the parallels and meridians, rather than with the trigonometrical calculations of their positions and sizes.

*The American Mathematical Monthly,*begins:-

Scheffers' excellent pamphlet contains an elementary account of the better known methods of map projection, with historical notes, and specific directions for the construction of some twenty types of map pictured therein. The author quotes with approval the dictum of Jacob Steiner to the effect that construction with the tongue is one thing, and construction with the pen quite another. This pamphlet is for those who use the pen.

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

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