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the 4 arithmetical operations with whole numbers and fractions
the 4 arithmetical operations with letters
number systems (in particular the decimal system)
negative numbers
polynomials
calculation with powers (in particular square and cubic roots)
proportions
equations of first and second degree
diophantine equations, continuous fractions
logarithms
irrational and imaginary quantities
cubic and higher equations
progressions (sequences)
combinatorics
infinite series and analytic operations with infinite series
binomial theorem. (Tellkampf, 1829, Table of contents, translated)
This list contains the subject matter of the so-called "scientific course" from
Quarta (third year) to Prima (ninth and last year). Typical for the combinato-
rial analysis of the time are two elements of this catalogue: (a) the appearance
of combinatorics before the theory of series, because the latter was consid-
ered to be an application of combinatorics, and (b) the binomial theorem,
which was seen as the culminating point of school mathematics, because this
formula rules the basic arithmetical operations with power series. In the
Prussian syllabus of 1901, the binomial theorem still had this role. The con-
tents prescribed as compulsory in the 1812 Abitur edict are printed in italics.
This compulsory canon was supplemented by combinatorics and the binomial
theorem in the 1834 revision of the Abitur edict, which means that the core
elements of the combinatorial view were made compulsory only then.
The intuitive analogy between finite and infinite series and hence the uni-
versality and simplicity of algebraic analysis are dependent on the unrestricted
operations with infinite series. The restriction to convergent series thus con-
tradicts the spirit of the theory. After publication of A. L. Cauchy's famous

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4.2 The Role of Applications
After stating these facts concerning the mathematical structure of the syllabus,
I shall now take a closer look at the role of applications in school mathemat-
ics. I shall go back to Süvern's syllabus of 1812. While the latter was never
made compulsory, only being made known to the Provinzialschulkollegien as
a guideline in 1816, it strongly influenced the activity of the Prussian admin-
istration of education until the 1820s (cf. Jahnke, 1990b, chap. V). In this
syllabus, we find essentially the same contents as in the list above, but with
one remarkable difference. For Prima (9th and last grade), it additionally pre-
scribed probability theory, and ". . . the disciplines of applied mathematics,
in particular of the mechanical sciences instead of geometry" (Mushacke,
1858, p. 243, translated). Conversely, it can be noted that the everyday
practical applications in the sense of ordinary arithmetic occur only in a
marginal observation, which states laconically: "In Quinta, continuation of
numerical instruction in irregular systems, which involve the exercise of ap-
plied calculations in all regards" (p. 242). That this merely served as a formal
acknowledgement to external demand had been stated by the full professor of
the Berlin university responsible for mathematics, Johann Georg Tralles
(1763-1821):
textbooks (1821/1897, 1823/1899), the view that only convergent series
were permitted came to prevail in Germany. Textbook authors, while adapt-
ing to this trend, maintained the concept of algebraic analysis, adhering to
Cauchy's standards only superficially by supplementing some convergence
proofs. However, most of them did not follow Cauchy's conceptually revo-
lutionary step in treating algebraic analysis as a theory of continuous func-
tions.
CULTURAL INFLUENCES: A HISTORICAL CASE
422
However
,
I believe that the school should not accept the role of teaching that
which is useful to each social class; this would lead to a useless diversification in
subject matter at the expense of thoroughness . . . . Given this, I deem it most
useful to set aside all the so-called practical subject matters and to engage in only
pure mathematical sciences or that which can be considered as such. (cited in
Jahnke, 1990b, p. 244, translated)
The contemporary debate thus distinguished sharply between two types of
application: On the one side, were the everyday practical demands in the
sense of ordinary arithmetic; on the other side, applications like mechanics,
which could be conceived, as Tralle said in his above remarks, "as of pure
mathematics." I should like to designate these applications as theoretical ap-