The course aims to provide broad historical-epistemological knowledge of the main themes of mathematics and mathematical thought, combined with a good general overview of the evolution of both. KNOWLEDGE AND UNDERSTANDING At the end of the course the student must demonstrate his/her knowledge and understanding of the topics treated in the course. APPLYING KNOWLEDGE AND UNDERSTANDING At the end of the course the student must be capable of applying his/her knowledge of the topics treated in the course, demonstrating his/her ability to link them together. MAKING JUDGEMENTS At the end of the course the student must have developed a critical attitude by reading and analysing the texts treated in the course. COMMUNICATION SKILLS At the end of the course the student must be able to speak appropriately about the topics treated in the course, with properties of language and confident exposition. LEARNING SKILLS At the end of the course the student must be capable, on the basis of his/her knowledge and his/her ability to analyse and link the topics, of consulting the works of mathematicians (in the original version or via secondary sources).

Notions of algebra, geometry, and mathematical analysis acquired in the courses of the first years of mathematics or equivalent courses, as well as basic notions of the history of mathematics

1) What is the history of mathematics. 1a. The methodology of the study of history of mathematics. 1b. How to do research in history of mathematics. 2) The history of Non-Euclidean Geometries. 2a. What is geometry. 2b. Euclids' Elements. 2c. The Postulates of Euclidean Geometry and the Fifth Postulate. 2d. Playfair's Axiom. 2e. Proclus. 2f. Girolamo Saccheri and Saccheri Quadrilaterals. 2g. Saccheri's Absolute Geometry 2h. Lambert and Lambert's quadrilaterals. 2i. Legendre. Legendre's Error. 2j. Gauss. "Nicht-Euclidische Geometrie". Gaussian Curvature. 2k. Schweikart. Astral geometry. 2l. Taurinus. Logarithmic-Spherical Geometry. 2m. Lobachevsky. Pangeometry. Imaginary geometry. 2n. Bolyai. Tentamen. Parallelism. Absolute Geometry. 2o. Models of Non-Euclidean Geometries. Hyperbolic Geometry. 2p. Beltrami. Klein. Beltrami-Klein model. Metric of the Beltrami-Klein model (Cayley's metric). Curvature of the Beltrami-Klein model. 2q. Poincaré. Poincaré's disk. Metric of the Poincaré's disk. Curvature of the Poincaré's disk. 2r. The half-plane model. Metric and isometries. 2s. The hyperboloid model. 2t. Elliptic geometry.

Boyer, C. B. (1968). A history of mathematics. New York: John Wiles & Sons. Burton, D. M. (2011). The history of mathematics: An introduction. 7th Edition. New York: McGraw-Hill. Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.

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Lectures with teacher-student dialogue, in-depth studies, and group exercises (participatory planning also with the use of digital technologies).

Students with special needs (including those with disabilities, workers, athletes, adults, parents, and detainees) who are permanently or temporarily unable to attend classes in person due to particular circumstances, will be allowed to participate remotely upon request to the teacher. The request, for which the student takes full responsibility, should be sent via email to the teacher well before the beginning of classes. The classes are held in person and are recorded. For information about digital teaching at the university, please visit the following link: [https://www.units.it/studenti/didattica-digitale]. Course materials will be available on the MS Teams platform of the University of Trieste. These materials must be integrated with the study of the texts indicated by the teacher (with appropriate references to supplementary materials). Erasmus students and those who do not participate to the lessons are invited to write an email to the lecturer.

The exam includes an oral test and the presentation of a seminar. The oral test consists of assessing the knowledge of the topics covered during the course and the ability to re-elaborate them. The seminar test involves preparing and presenting a seminar on a topic in the history of mathematics, previously agreed upon with the teacher. The score for the oral test and the seminar presentation is assigned using a grade expressed in thirtieths. The final grade for the exam is calculated as the average of the scores from the oral test and the seminar. The exam is passed with a score of 18/30. The student must demonstrate that they have acquired sufficient knowledge of the topics covered in the course. To achieve the maximum score (30/30 with honors), the student must demonstrate excellent knowledge of all the topics covered during the course, correctly answer all questions, and show the ability to connect the topics together.

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