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."