The lecture course 'Introduction to Higher Mathematics', delivered regularly by Georg Feigl each semester during his teaching at the University of Berlin from 1920 to 1934, served a dual purpose. It was intended to encourage students to transition from school to the quite different type of teaching through lectures, and it was intended at the same time assist the teachers of beginners courses by providing a more satisfactory approach to basis material. In analytical geometry one might use the basic concepts of vector algebra and matrix calculus as a tool, without having to skip what comes on top of that, and in calculus you can, on a secure basis, build real numbers and give justification within the lecture where usually there is not enough time.
These two objectives have determined the nature of Feigl's introductory lectures and the selection of the material to be treated was an individual matter but Erhard Schmidt was also a relevant consultant. The beginners lecture have at the same time, given preliminarily material, while in the presentation following on from the teaching methods of the school, while the "Introduction" has shown many generations of mathematics students in Berlin both joy and benefits. It is hoped that it is also suitable in the present book form, to help overcome the initial difficulties of studying mathematics and transport all who want to study this science as a hobby or for professional reasons beyond those insights into higher mathematics. Provided is some background from school mathematics as well as the fundamental theorem of algebra.
I am happy to comply with the invitation from Springer-Verlag to publish Feigl's "Introduction". Because my acquaintance with this course stretches in a wide arc from the winter semester 1921/22, in which I took the course, to the summer semester 1934 when I delivered it as Feigl's representative was necessitated by his leaving. When putting together the manuscript I have used my lecture notes, those made in the summer semester of 1925 by Miss Rose Gadebusch (now my wife) and in the summer semester of 1932 by Miss Hanna von Caemmerer (now Dr Hanna Neumann, University College, Hull). I thank them for providing me with their notes. The processing was done in agreement with Dr Maria Feigl who I have to thank for her valuable assistance in reading the manuscript and for much stimulation and support. Also, I am very indebted to Dr Achim Zulauf who has called my attention to some significant improvements in the presentation, and to Dr Bodo Volkmann for help in correcting. When editing several chapters had to be positioned where they were featured in the lecture. Because of this a section on the foundations of geometry, which unfortunately could not be included within the scope of the book, is missing.
Feigl's "Introduction" has emerged out of repeated and supplementary courses, which were delivered by Feigl and other assistants of the Berlin Mathematical Seminar immediately after the war in the intermediate semesters of 1919 and although it was founded in autumn 1919 by some students of Mathematics and Physical Arbeitsgemeinschaft (MAPHA) at the University of Berlin, on their suggestion the Feigl's particularly pronounced didactically complementary course became, in expanded form, a permanent course for 'Introduction to higher mathematics' in the curriculum. With MAPHA and their beneficial action, the name of its founder, Alfred Brauer (now Professor of Mathematics, University of North Carolina, Chapel Hill, N.C., USA), indelibly linked. To him, the initiator of this introductory course and my friend for eight years, should this book therefore be devoted.
In April 1945 Georg Feigl succumbed to a stomach ailment at Castle Wechselburg in Saxony. With it, an academic teacher passed away, who showed a special love and empathic understanding to the students and especially to the beginners among them. May this book help to keep the memory of Georg Feigl alive among students today!
Mainz, December 1952.