A sketchy story of mathematics at Charles-Ferdinand University
The university of Prague was created in 1348 after an imperial decree of Charles IV. It included four nationes: the Bavarians, the Saxonian, the Polish and the Bohemian, which turned out to be one with more political weight from the 15th century onwards. A second college, the Academia Ferdinandea, was created in 1556 and was run by the Jesuits. It is better known by the name Klementinum. In 1654, Charles University and the Jesuit Clementinum were unified in a single institution, then called “Charles-Ferdinand university”. From that time until well into 19th century the university of Prague was organized in a traditional way, around four disciplines: philosophy, theology, law and medicine. The teaching of the first two, philosophy and theology, was monopoly of the Jesuits at the Clementinum, while the superior faculties of law and medicine were lay faculties, controlled by the state.
Mathematics was taught at the faculty of philosophy, and since attending philosophy courses was compulsory for all students, the dissemination of mathematical sciences in Prague and Bohemia was monopolized by the Jesuit order for about two centuries, from the first half of the XVIIth to the disbanding of the order in 1773.
The place of mathematics in the Jesuit curriculum had been a hotly debated topic among Jesuit educators and reformers soon after the creation of the order. Eventually, Christophorus Clavius’ strenuous campaign to obtain a revered place for mathematics in the curriculum won: “owing almost entirely to his dogged and tireless leadership – A. Alexander recounts – only a few decades later, Jesuits were setting the standard for the study of mathematics in Europe.” And, within Europe, in Prague too.
But what did students learn when they studied maths under the jesuits at the Clementinum? One thing that should be remarked is that the organization of the teaching kept pretty much stable until the dissolution of the order, and included, for example, disciplines we would not treat as part of mathematics today: civil and military architecture, optics (catoptrics and dioptrics), hydrostatics.
Before the Jesuits order was suppressed in the Habsburg empire in 1773-74, the system of university education had already undergone a gradual process of modernization and specialization through several reforms, starting from 1741. If we focus on the teaching of mathematical sciences, a significant change brought about by the reorganization of the curricula from the half of 18th century was an emphasis on mathematics and physics. One of the main consequence of these reformes was that new chairs were introduced, and the teaching of disciplines which did not belong to the traditional Jesuit cursus was promoted. For instance, alongside with the traditional chair of elementary mathematics, a chair of advanced mathematics (mathesis sublimior) was created in 1762. Concomitantly, the traditional Aristotelian framework which dominated the teaching of natural sciences was abandoned and substituted by an approach oriented towards experimental sciences.
Such reforms in the scientific education began, not without difficulties, under the leadership of the Jesuit Joseph Stepling (1716-1777), director of the philosophical faculty at Prague university from 1763 to 1776, and continued with his successor and most brilliant student, Jan Tessánek (1728-1788). Another important change was the introduction of the German as the official language for teaching in 1784, which opened the door to the circulation of German textbooks. Meanwhile, the modernization of technical higher education was prompted by the creation of a special chair of practical mathematics (1784), whose first professor was Franz A. L. Herget (1741-1800), followed by Joseph Havle (1763-1840) and A. Bittnar (1777-1844). Finally, the creation of a polytechnical school upon the model of the French Ecole polytechnique, in 1803, crowned the ongoing process of modernization in the 18^{th} century Czech lands.
The table below summarizes the historical evolution in the teaching of mathematics between 1760 and the end of the 18^{th} century. We note that the disbanding of the society of Jesus, although in some cases caused the removal of Jesuit professors from their chair, did not seem to have changed the structure of mathematical education, which continued to be part of the faculty of philosophy, except for an independent programme in advanced mathematics.
Chairs |
Chair: elementary mathematics. Elementary mathematics course (3 years). Included a teaching in applied mathematics (mathesis mixta) from the second year (introduced in 1775). |
Chair: Advanced mathematics (from 1762). Teaching of higher mathematics for those who excelled in elementary mathematics. |
Chair: Practical mathematics (from 1784) Before 1784, a course in practical mathematics was lectured during the 3^{rd} year of the elementary mathematics course. |
General content |
Algebra, arithmetic, trigonometry, geometry. “Applied mathematics”: possibly with elements of differential and integral calculus. |
Analytic geometry, differential and integral calculus, mechanics, hydrodynamics and astronomy. |
Land surveying, and trigonometry, both theoretical and practical. Topics related to construction and engineering. |
Teachers |
Joseph Bergmann (1723-1786) 1761-1767; Franciscus Zeno (1734-1781) 1767-1772 (or 1774?); Stanislav Vydra (1741-1804) till 1772/1774-1804;^{1} Ladislav Jandera (1776-1857) from 1804. |
Joseph Bergmann 1762- 1766; Franciscus Zeno 1766- 1772 (or 1774?); Jan Tesanek from 1774 to 1786 (although sources indicate that Tessanek was professor of mathematics since 1763); Gerstner from 1787 (appointed regular professor in 1789) till 1823. |
Before 1784, presumably the teachers of elementary or advanced mathematics. From 1784: Herget 1784-1799 Havle 1800-1802 Bittnar 1802-1804. |
Texbooks used for lectures |
Franciscus Zeno, Elementa algebrae, geometriae ac trigonometriae. Prague, typis Josephi Emmanueli Diesbach. Veneunt apud Antonium Elsenwanger, 1769. Joseph Anton Nagel, Mathesis Wolfiana in usum juventutis scholasticae, Wien, Johann. Thomae de Trattner, 1776. S. Vydra, Elementa calculi differentialis et integrali, Pragae et Viennae, Schönfeld, 1783 (possibly used for applied mathematics). A. G. Kästner, Anfangsgrunde der Arithmetik, Geometrie, ebenen und sphaerischen Trigonometrie und Perspectiv, 1758 (Used in Prague after 1784). |
J. Bergmann, Lectiones mathematicae in usum auditorum, Prague, typis Joannae Pruscha, 1765. L. Euler, Introductio in analysin infinitorum, 1748 (used in German translations, [Michelsen, 1788]). Used in Prague after 1789. W. J. Karsten, Lehrbegriff der gesamte Mathematik. Used after 1789. |
Not known. (Bolzano’s notes of Herget lectures are extant). |
1 We do not have consistent information about the years 1772-1774, probably due to diverse ways in which the ban of the Jesuits from university teaching was implemented.
When Bernard Bolzano, a key figure in the history of mathematics, logic and philosophy, stepped into Prague university as a teenager, at the end of 18th century, the education system of the jesuits was still in place in its general lines, even though the order was then gone for more than 20 years.
The disestablishment of an organization that had ruled for centuries was a process that affected the practical and intellectual aspects of the lives of the people involved. A paradigmatic change of sort, which has too often been neglected when studying the emergence of modern mathematics, such as in the case of Bolzano.