The 1804 examination for the chair of
Elementary Mathematics at the
University of Prague
Davide Crippa Elías Fuentes Guillén
GAČR PROJECT (2019-2021)
History of mathematics in the Czech lands:
from the Jesuit teaching to Bernard Bolzano
Bolzano’s mathematical works
• Praise for his early (1804-10) “ideas and innovations in methodology” (Russ).
• “Epoch-making [1816-17 works] on the foundations of real analysis” (Ewald).
• Reine Zahlenlehre (1830s) is “a surprising achievement for its time” (Russ).
• He has an “imputed role as pathfinder for the later work on sets by Cantor
and Dedekind” (Simons).
The general impression is that we “can only speculate on the direction the
history of mathematics might have taken” if he had become professor of
mathematics and that he remained “isolated” (Rusnock and Šebestík).
“a man who had a great influence on the occupancy
of these chairs, and because he was fond of me,
sent for me especially to ask me which of the two I
would prefer[;] I replied that I could hope with some
confidence to be of more meaningful use by lecturing
on Religion than it would ever be possible [for me to
be] by lecturing on Mathematics.” (Bolzano, 1836)
Our research sheds light on the mathematical practices in the teaching and
learning of mathematics in Prague that were common at the turn of the 19th
century, as well as on the institutional context within which Bolzano lived.
The 1804 Concursprüfung and Bolzano’s early manuscripts constitute valuable
sources of information on his knowledge of mathematics at the time and his early
mathematical practices.
Bolzano’s education
1784: teaching in “all educational institutions above elementary level” should be
in German (Armour); at gymnasia and universities, mathematics, among other
subjects, was emphasised.
1791-1796: Roman Catholic Piarist gymnasium in Prague’s New Town.
• The Piarists cooperated in the school reform after the Pope’s 1773
suppression of the Society of Jesus, which until then had played a key
role in education throughout much of Catholic Europe.
1796: Faculty of Philosophy (three-year studies for a decade; plus 1 year).
1800-1804: Faculty of Theology.
Three chairs and their context
In 1804 the chair of Elementary Mathematics at the University of Prague
(Philosophy) fell vacant and two new chairs of Religious Doctrine were
established in Prague, one there and one at the Old Town gymnasium.
The chairs of Religious Doctrine were part of the Holy Roman Emperor Franz II’s
efforts “to educate obedient citizens of the State” and avoid replication of the
French Revolution: “I do not need scholars, but obedient citizens. It is your
obligation thus to form our youth” (Rusnock & Šebestík).
Theology Medicine Law
Philosophy
(propaedeutic studies; mathematics)
Mathematics in Prague
During the last decades of the 18th century the number of hours devoted to
natural sciences and mathematics increased and new chairs were created in
fields which were considered apt to play a key role in the modernisation of
Bohemia, such as physical geography, astronomy, higher mathematics and
practical and applied mathematical sciences (Tomek).
In 1804 there were 3 chairs in mathematics:
• Elementary Mathematics: Stanislav Vydra (since 1772).
• Practical Mathematics: Franz Anton Herget (since 1784).
• Higher Mathematics: Franz Joseph Gerstner (since 1788-89).
Teaching of mathematics c. 1800
Mathematical subjects
Math chairs Pure Applied
Elementary Arithmetic, geometry, trigonometry.
From Kästner’s Anfangsgründe.
Mechanics, optics, astronomy. Mainly
from Kästner’s Anfangsgründe.
Higher Introduction to the analysis of finite and
infinite quantities, and differential and
integral calculus (1st year). From Euler.
Higher mechanics and hydraulics with
application to mechanical engineering
(2nd year); optics and theoretical
astronomy, with application to
gnomonics, chronology, geography and
nautics (3rd year). From Karsten and de
la Lande.
Practical Various arts and crafts. Land-surveying, practical arithmetic, nautical arts, civil and
hydraulic engineering, etc.
Elementary Mathematics
He was one of the few professors who, despite being members of the Jesuit
Order, were not dismissed after its suppression.
• Among other things, he wrote Historia Matheseos In Bohemia Et Moravia
Cvltae (1778) and Počátkowé Arytmetyky (1806), the first published
textbook of elementary mathematics written in Czech.
He followed Kästner’s work (imperial order of 1784) but not in a strict manner, as
Bolzano’s notes on his lectures reveal.
• E.g., he began with a general introduction to mathematics similar to that
by Kästner, but included a section in Napier’s bones (not in Kästner’s).
Notes by Bolzano
“Anfangsgründe”
The learner should start from the foundations
and go upwards (elementary vs higher)
Mathematical sciences should be divided
according to their object (pure/applied).
“Kästner proved that which is generally passed
over because everybody already knows it.
That is, he attempted to make clear to the
reader the reasons upon which his judgment
rested – this I liked the best of all.” (Rusnock
and Šebestík; Bolzano, 1836)
The 1804 Concurs
Two candidates took part in a written and
oral examination for the chair of
elementary maths: Bernard Bolzano and
Ladislav Joseph Jandera (1776-1857).
Written exam 25th October 1804, 9am till
the afternoon.
Oral exam: 26th October 1804, half an hour
(“trial lesson”?)
Committee and questions
Examination board: F. A. Steinský (Diplomatics, Heraldry,
Antiquity Studies and Numismatics), F. Schmidt (Natural
Sciences), A. David (Practical Astronomy), A. G. Meißner
(Classical Ancient Authors and Aesthetics), A. von Zürchauer
(Economics), J. K. Mikan (General Natural History and
Technology), F. N. Titze (General World History) and F.
Něměcek (Theoretical and Practical Philosophy), J. F. Gerstner
(higher mathematics and director of the philosophical faculty for
the math. and phys. studies).
Nine questions submitted to a higher commission, 3 were
chosen and opened during the examination.
The 1804 Concurs: Questions
I.Calculate both the surface area and
the volume of a spherical cap from its
given height and the diameter of the
sphere.
II.What proofs are known for the
theorem on the equilibrium of the
lever? And what does in particular
speak for or against each of them?
III.At what speed does the water flow
from small outlets of large vessels?
And how can this speed be
established on a theoretical basis?
The 1804 Concurs: Oral question
(on 26th October) “Shall the 14 proposition from
Kästners Geometrie be chosen, that they should
explain orally: “parallelograms included between
the same parallels and having the same base have
equal area” (Gerstner, Protokoll)
Expectations from the committee
“The objects of these questions were chosen in
such as way that their answers would not only prove
a complete familiarity with the adopted textbook, but
also be the occasion [for the candidate] to
distinguish himself thanks to his superior knowledge
(durch höhere Kenntnisse aus zuzeichnen).”
Gerstner, Protokoll 38039/3376
Bolzano: Question 1
How can one calculate a curved surface, or a plane one
bounded by curved lines, as well as a solid confined within
such a surface?
Archimedes formulated postulates that should not be placed
among the “Grundsatzen”.
Infinitesimals remove the difficulty “for the eye, but not for the
understanding.”
Use of Lagrange’s method that has the advantage of brevity,
uniformity and generality: it applies algebraic methods to
geometric investigations.
Theorem I
“If the quantity 𝑀 should always lie within the two quantities 𝑁
and 𝑁 + 𝑖𝑃, however small 𝑖 is taken, then it must be [that] 𝑀 =
𝑁.”
The theorem is not Lagrange’s, but uses Lagrange’s symbolism.
Let 𝑀 − 𝑁 > 0, and 𝑖𝑃 − (𝑀 − 𝑁) ≥ 0.
Let 𝑀 − 𝑁 = 𝑛!
(a positive quantity)
Then 𝑖𝑃 − 𝑛!
≥ 0 (for every 𝑖)
Hence 𝑖 ≥
"!
#
, against the assumption.
Theorem II
“An arc of a curved line, for which the equation between the
orthogonal coordinates is 𝑦 = 𝑓𝑥, is a primitive function of the
abscissa x, whose first derivative function is expressed by the
formula
𝐹‘𝑥 = [1 + 𝑓‘(𝑥)!]
"
!
Bolzano gives general theorems about arc-length, area and
volumes (also in Lagrange), and applies them to the case
presented in the problem. For exemple:
Positive evaluation of Bolzano’s approach
“He showed the difficulties, that are inherent to the postulates
of Archimedes and the recent differential calculus, and thus
chose the newest, and most solid of all know methods (die
neunste und strengste unter allen bekanntne Methoden), that
is the theory of functions of Lagrange, and by it he solved
completely and correctly the given problems”
(Protokoll B - written exam)
Question 1: Jandera
Jandera’s solution reproduces Kästner’s method and order of
propositions (“Methoden und Ordnung wie Kästner in dem
vorgeschriebenen Lehrbuche behandelt”), consisting in dissecting an
area or surface into infinitesimal strips. Jandera makes a mistake (“den
Inhalt des ganzen Abschnitts unrichtig berechnet hat”)!
“The unanimous opinion of the Professors was as follows: that
although the first one, namely Bernard Bolzano, went deeper into
the subject matter, he deviated somewhat from Kästner in the
order of the assumptions by sending the theory of the similarity of
triangles on ahead, which in [the work of] Kästner is addressed later.
However, the other one, namely Mr. L. Jandera, stood out due to a
clearer analysis and a more comprehensible lecture for the
students, maintained the order of Kästner and not only
exhaustively proved that proposition in the same period of time,
but also drew from it a few subsequent propositions.”
Opinion about the oral examination
CG Publicum1796-1805 98/755, Protokoll A
Summary Report
“er [Bolzano] mehr durch
gründliche Behandlung
und Erudition, als durch
Vielheit der Gegenstände,
und ängstliche
Ordungsabmessung sein
Talent hat geltend
machen wollen”
Bolzano’s talent:
“Thorough reflection and
erudition”
Jandera’s talent:
multiplicity of topics
(probably referred to the
oral question),
respect of the order, a
stronger voice
Gerstner: it was “regrettable” that there was
no chair available in Higher Mathematics in
addition to that of Elementary Mathematics,
“the unanimous testimony of all the
professors” would be to appoint Bolzano
to the former and Jandera to the latter.
6/12/1804: Final decision of the commission: Bolzano was
recommended for the chair, but Jandera should not be completely ruled
out, as he had shown a “great deal of knowledge of mathematics” and a
familiarity with the chair due to the couple of years he had spent as
Vydra’s assistant.
Administrative motivations: Vienna urgently needed a teacher of
religion, and Bolzano, who had taken part in the examination for
the chair of religion was the best, and only available candidate.
The “official” story
Jenaische allgemeine literatur-zeitung: Intelligenzblatt, 1805