I've been reading Courant's Integral and Differential Calculus for a bit, at some sections, there are problems which do not seem to be answerable with previously given material in the book.

Several days ago, I made this question but although the answer is fair, it's a little bit vague. I don't know much about German educational system nor its history, so at the time of publication of Courant's book, I believe people studied it in university (I may be mistaken, it seems that in the past, calculus was taught at high-school and still is at some countries). What were the subjects people typically ought to know when facing Courant's book through formal education?


3 Answers 3


By the time Courant's text was written, the German mathematics curriculum underwent a reform spearheaded by Felix Klein around 1900. It was later adopted by most European countries (but not the US). Prior to the reform only algebra, elementary geometry and trigonometry were taught at secondary schools. Note that all three were far less "mechanical" than today and involved deriving non-trivial identities and making Euclidean demonstrations. Indeed in classical education the purpose of mathematics was seen as exercising "brain muscles".

Klein's reform added analytic geometry and calculus to the list, and made the notion of function central to the curriculum. Klein's original motivation was to satisfy the new demands that industrial revolution imposed on workers' mathematical backgrounds. In alliance with German industrialists he established in 1898 the Göttinger Vereinigung für angewandte Physik (Göttingen Association for Applied Physics), broadened in 1901 to "und Mathematik". This was followed by reorienting the curriculum of secondary schools away from "brain muscles" and towards science and engineering. The reform was finalized in the then Austrian town Meran in 1905, the name "Meraner Reform" is sometimes used, its main document is known as the Meran syllabus. Fenker's textbook 'Arithmetische Aufgaben' gives an idea of what the syllabus looked like and what it replaced, see Bierman-Jahnke's chapter in Transformation - A Fundamental Idea of Mathematics Education, p.20.

Here is from History of the International Comission on Mathematical Instruction:

"Klein deplored what he used to call the "double gap": the discontinuity between school mathematics and university mathematics and the double forgetting of the respective knowledge: first one had to forget school mathematics upon beginning one's university studies and later as a teacher one had to forget university mathematics and return to school mathematics.

Klein proposed a radical reform. Since the mathematical courses at the technical colleges consisted of a basic preparatory "general" part and an advanced or higher part, he recommended that the basic studies be transferred to the preparatory schools, i.e. the secondary schools, and that only the advanced studies should remain as part of the college curriculum. However, they would be reformed and taught not as independent branches of knowledge, but rather "in permanent touch with the requirements of the students of mechanical and construction engineering.... Mathematics at the colleges can thus become once again what it ought to be: a specific power pervading the whole".

Transferring the preparatory courses to secondary school would help create a firm mathematical basis for the pursuit of college studies, due, in particular, to the "benevolent coercion of school," as opposed to a university environment where academic freedom reigned. But which subjects were to be considered as part of the basic core of this new secondary mathematical curriculum? According to Klein, this core would include analytical geometry, but primarily differential and integral calculus! In private letters to close friends and co-workers, Klein in fact always stressed that the key to his reform plans was to solve the question of mathematical preparation for the technical colleges."


First of all, there was no such subject as "calculus" in Germany in 1920s. The original title of the book is Vorlesungen über Differential-und Integralrechnung (1927), and the subject is called Analysis. I have no first hand experience with mathematics education in Germany; I was educated in Soviet Union, but Soviet system was modeled on the German one, and all that I know about German system shows that they were similar, and the textbooks were similar. (A Russian translation of Courant was used sometimes in some universities).

Analysis was one of the three principal subject taught to 1-st year students, the other two were called Higher Algebra and Analytic Geometry. So the background was high school mathematics. (In Germany it is called Gimnasium). High school gave a solid background in elementary algebra and geometry (rigorous, everything with proofs, including stereometry and trigonometry).

In general, European and American education systems were (and still are) very different. The main difference is that in Germany (and in Soviet Union) there is no analog of American "undergraduate education". When you enroll to a university you have to decide from the very beginning what you are going to study. If you choose Mathematics, then you are taught real (completely rigorous) Mathematics (not "Calculus").

I hope some German-educated mathematician will give more qualified answer.

  • 5
    $\begingroup$ Rechnung derives from rechnen, reckoning, calculation. Integral calculus is the standard translation of Integralrechnung, analysis would be Analyse. I recall Kolmogorov opining that it is a shame that students are admitted to the treasures of analysis only after the torture chamber called its rigorous foundations. $\endgroup$
    – Conifold
    Feb 2, 2018 at 1:33
  • $\begingroup$ @Conifold: I was not discussing the etimology. The subject which is called "Calculus" in the US universities does not exist in Europe. (I am telling from my own experience: I taught mathematics all my life in Europe and in US). $\endgroup$ Feb 2, 2018 at 20:30
  • $\begingroup$ Here is an example of a basic calculus course being taught in Europe: see drps.ed.ac.uk/16-17/dpt/cxmath08058.htm from Univ. Edinburgh. $\endgroup$
    – KCd
    Feb 2, 2018 at 23:12
  • $\begingroup$ @KCd: Yes, I know US slowly convert the world to their system:-) I had to state more carefully: it did not exist until recently. $\endgroup$ Feb 2, 2018 at 23:48

One must not forget that the Courant book was an edition of his lectures, see the description of its creation in Reid's biograpy of Courant. The exercises stemmed from the university courses and were not intended to be solved without tuition from the assistants.

Here are two other German calculus books from the late 19th and early 20th century, by Kiepert and Kowalewski. The latter book attracted Stanley Ulam to mathematics.


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