NMAK15016U History of Mathematics 2 (Hist2)

Volume 2020/2021

MSc Programme in Mathematics


History of geometry 1770-1920.

During the indicated period geometry underwent a development that had wide ranging influence on our understanding of

1. what mathematics is and

2. the nature of space.

It is often mentioned as one of the few examples of a revolution in mathematics. Arround 1830 the discussion of the parallel postulate led to the creation of non-Euclidean geometry. Whether this geometry was consistent remained an open question until Gauss' and Riemann's works on differential geometry made it possible to create a model of the non-Euclidean plane as a surface of constant negative Gauss curvature. This model was also interpreted in projective geometry that was also devellopped in the 19th century. The century ended with different new attempts to give axiomatic descriptions of geometry, among which Hilbert's is the most famous. The considerations concerning non-Euclidean geometry was not only an exercise in axiomatics. For all the actors it was also a question of understanding the nature of (physical) space. The discovery of non-Euclidean geometry led to a rejection of Kant's opinion that geometry (for Kant this meant Euclidean geometry) was an a priori but synthetic intuition. Instead various empirical, conventional or formalistic epistemologies were put forward. The mathematical and philosophical considerations of the 19th century created a background for the revolutionary ideas that Einstein put forward in his special and in particular general theory of relativity. In the course all these interacting subjects will be discussed.
Students are required to take an active part, write a project and give seminars.

During the course the student will learn to investigate the history of a piece of mathematics, to analyze a mathematical text from the past, and to use the history of mathematics as a background for reflections on philosophical and sociological questions regarding mathematics. Moreover the course will give the students a more mature view on the mathematical subject in question. The course will be particularly relevant for students who aim for a career in the gymnasium (high school) but all mathematics students can benefit from it.

Learning Outcome

After having completed the course, the student will have a rather deep knowledge of the history of geometry in the period 1770 to 1920 and about the historiographical questions related to this history
After having completed the course the student will be able to
1. Read a mathematical text on foundational issues concerning geometry from the period 1770 to 1920 (in translation if necessary).
2. Find primary and secondary literature on the subject of the course.
After having completed the course the student will be able to
1. Communicate orally as well as in written form about the selected topic from the history of mathematics (history of geometry).
2. Analyse a primary historical text (if necessary in translation) within the subject of the course.
3. Analyse, evaluate and discuss a secondary historical text on the subject of the course.
4. Use the historical topic of the course in connection with mathematics teaching and more generally reflect on the development of the selected topic.
5. Place a concrete piece of mathematics from the selected topic in its historical context. 
6. Independently formulate and analyze historical questions within a wide field of the history of mathematics.
7. Use the history of mathematics as a background for reflections about the philosophical and social status of mathematics.
8. Use modern historiographical methods to analyze problems in the history of mathematics.

Primary sources (mostly in English translations) and secondary papers.

Hist1 is usefull but not absolutely necessary. Moreover Geometry 1 or similar.

Academic qualifications equivalent to a BSc degree is recommended.
6 hours per weeks divided between lectures by the professor, seminars given by the participating students and discussion sessions.
  • Category
  • Hours
  • Lectures
  • 21
  • Preparation
  • 163
  • Theory exercises
  • 21
  • Exam
  • 1
  • Total
  • 206
Peer feedback (Students give each other feedback)

The professor and the other students will give oral comments on the seminars both concerning the presentation and the content.

The professor will give written feedback on the project.

During the discussion sessions the students will answer a series of questions. The answers will give rise to comments from and discussions among the participants and the professor



7,5 ECTS
Type of assessment
Oral examination, 30 minutes
30 minutes oral exam with 30 minutes preparation time. The student will start the exam by giving a 10 minutes presentation of the project.
Exam registration requirements

In order to qualify for the exam the student must write a project on a topic within the subject of the course and give an oral presentation of it in class. Moreover he/she must give a 1-1½ hour seminar presentation on another topic.

Only certain aids allowed

During the 30 minutes preparation time all aids are permitted. During the exam itself the student is allowed to consult a note with at most 20 words. Other aids are not permitted.

Marking scale
7-point grading scale
Censorship form
External censorship

Same as ordinary exam. If the student has not written the required project he or she must hand it in no later than three weeks before the beginning of the re-exam week.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.