34740 - History of Mathematics

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education

Academic Year 2021/2022

Learning outcomes

At the end of the course, the student has an in-depth historical and epistemological knowledge of the main key themes of mathematics and mathematical thought. He also possesses a good general overview of the evolution of mathematics and mathematical thought.

Course contents

From the origins to the sixth century

1. Introduction to the course. Methodology. The prehistory. Methodological problems in the approach to the history of Mathematics. The search for concrete signs of the use of Mathematics.

2. Mathematics in Mesopotamia. 2.1 The neolithic in Mesopotamia and Mathematics. The city. The "tokens" and their possible use in Mesopotamia after the advent of agriculture. First baked clay tables with notations denoting exchanges of goods: their mathematical meaning.

2.2 Sumerians and Chaldeans in the protohistory and history of Mesopotamia. Cities and writing among the Sumerians. Positional numeration of the Chaldean peoples. Advantages and disadvantages of positional notation. First tables of mathematical use for Sumerians.

2.3 Babylonian numeration and algebraic problems in Babylonia: their characteristics. The scientific and organizational figure of O. E. Neugebauer. Arithmetic operations among the Babylonians. Mathematical and historical recalls around the Pythagorean triples.

3. Mathematics in ancient Egypt. Chronological lines of the history of ancient Egypt. Arithmetic among the Egyptians: system of numeration, ancient Egyptian algorithms. Recent historiographic criticism of Egyptian mathematics.

The method of false position: mention of the use of this method in Egyptian Mathematics. Simple and double false position.

4. The rise of Mathematics in the archaic eastern Mediterranean. The appearance of the Achaean people. Recalls on the Minoan and Mycenaean civilizations. The rise to the throne of Cyrus the Great. First Greek colonies, Italic colonies. Presumed results of mathematical character of Thales.

5. From Pythagoras to Euclid. Reference to ancient literature on Pythagoras (Porphyry of Tyre and Jamblico) and presumable historical contextualization of his life. The school of Pythagoras. The "Pythagorean proposition" in ancient times and its alleged demonstrations. Notes on Pythagorean vision of universal harmony (according to Porphyry). Commensurability of magnitudes in the presumed Pythagorean vision. Hippocrates of Chios and the quadrature of the lunulae. Plato and his dialogues: Menon, Timaeus, Theetetus. Archytas of Tarentum [https://www.britannica.com/biography/Archytas-of-Tarentum] . Aristotle and the Mathematics.

6. Ruler and compasses constructions. Figures drawn in one step with ruler and compasses starting from a finite set of points of the plane. The notion of a constructible point (starting) from two given points A, B.

The classical construction problems of geometry with ruler and compasses. The cyclotomic problem. The Gauss - Wantzel Theorem. Descartes' theorem (about the "algebrization of the construction with ruler and compasses"). Every constructible number is algebraic. Non-squareability of the circle by ruler and compasses.

7. Euclid. The figure of Alexander the Great: its historical contextualization. The foundation of Alexandria: the Library of Alexandria, the Museum, the Serapeion and the Fund of the Ships; the Lighthouse of Alexandria. Euclid

7.1 The first books of the Elements.. Critical analysis of the first propositions of the Elements. The first and fundamental Euclidean geometric constructions.

7.2 On magnitudes and their ratios. Recall the notion of Archimedean and non-archimedean fields. Examples of magnitudes. The notion of area for polygons. Hilbert's third problem and Dehn's method. Magnitudes in the same ratio. Inequalities between ratios of magnitudes.

7.3 The Euclidean method of exhaustion. Propositions (X, 1), (XII, 1), (XII, 2). Notes on applications of (XII, 2), contained in the same Book XII.

7.4 Critical conclusions on the Elements and the problem of the foundations.

8. Archimedes. The scientific personality of Archimedes; references to his main books. The method of exhaustion in Archimedes. The "Measure of the Circle". The Sandreckoner. The heliocentric hypothesis of Eratosthenes. The spiral of Archimedes: rectification of the circumference, trisection of the angles by the method of the spirals.

9. From Apollonius to Heron. The treatise on conics. The problem of Apollonius. The birth of trigonometry: Hipparchus of Nicaea and Menelaus.

10. From Heron to Diophantus. Use of the marked ruler and effective trisection of the angles. The scientific and technological personality of Hero of Alexandria (Metrics, Mechanics, Diopter). The scientific personality of Claudius Ptolemy. The theorem of Ptolemy; tables of chords and contributions to trigonometry. Diophantus of Alexandria: his algebraic approach to Number Theory; some diophantine problems.

11. Pappus and the conclusion of an extraordinary scientific cultural season. Pappus: the "Collection". Pappo's classification of geometric problems. Pappo's theorem and its projective meaning. Theon of Alexandria and Hypatia. Proclus Lycius Diadochus: his importance in the History of Mathematics. Some further brief considerations on science and technique in Hellenistic and Alexandrian ages.


Euclide, Gli Elementi, a cura di Attilio Frajese e Lamberto Maccioni. 1996.
Thomas Heath, A History of Greek Mathematics, Clarendon Press, Oxford, 1921,
Morris Kline, Storia del pensiero matematico, Einuadi Editore, 1991.
Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, John Wiley and Sons, New York. Several other recent editions.

Teaching methods

Frontal lessons. Class attendance is not mandatory, but strongly recommended for the learning process

Assessment methods

Oral exam test.

Teaching tools

Research and reading on the web of historical and mathematical texts as instructed and guided by the teacher. Notes provided by the teachers before and after the lesson. Reading and commentary on some passages extracted from mathematical texts in the Italian version: For the first module, short deciphered texts of ancient tablets and papyri; translations of passages from Platonic dialogues, from the Elements, from texts of Hellenistic mathematics. For the second module, reading and commenting on some excerpts from mathematical texts in Italian translation: Liber abaci (Pisano), Ars Magna (Cardano), Algebra (Bombelli), Geometria Indivisibilibus (Cavalieri), Discoursde la méthodee Géométrie (Descartes), Nova Methodus (Leibniz).

Office hours

See the website of Salvatore Coen

See the website of Maria Giulia Lugaresi