An outline of Egyptian, Babylonian, Greek, Indian and Arab mathematics. Leonardo Fibonacci and the Liber abaci. The mathematics of the abacus in Italy in the Middle Ages and Renaissance. The figure and work of Luca Pacioli. Algebra in Italy in the 16th century: Dal Ferro, Tartaglia, Cardano, Ferrari, Bombelli. The algebraic work of Viète. The Géométrie of René Descartes. Quadrature methods from antiquity to the seventeenth century. The birth of the calculus.
Boyer C. B., Storia della Matematica, Milano, Mondadori, 2009.
Franci R., Toti Rigatelli L., Storia della teoria delle equazioni algebriche, Milano, Mursia, 1979.
Freguglia P., La geometria fra tradizione e innovazione, Torino, Bollati Boringhieri, 1999.
Giacardi L., Roero C.S., La matematica delle Civiltà arcaiche: Egitto, Mesopotamia, Grecia, Torino, Università Popolare di Torino, Editore, 2010 (I ed. 1979).
Giusti E., Piccola storia del Calcolo infinitesimale dall’antichità al Novecento, Pisa-Roma, 2007.
Maracchia S., Storia dell’algebra, Napoli, Liguori Srl, 2005.
Itinera Mathematica. Studi in onore di G. Arrighi per il suo 90° compleanno. A cura di R. Franci, P. Pagli, L. Toti Rigatelli. Centro Studi sulla Matematica Medioevale. Università di Siena, 1996.
Un ponte sul Mediterraneo. Leonardo Pisano, la scienza araba e la rinascita della matematica in Occidente, a cura di E. Giusti e con la collaborazione di R. Petti, Firenze, 2002 .
Ulivi E. Dispense del Corso di Storia della Matematica.
Learning Objectives
Knowledge acquired:
Rudiments of the History of Mathematics from Antiquity to the first half of 18th century
Competence acquired:
General picture of developments in arithmetic, algebra and geometry, with particular reference to the 13th to 17th centuries, of broad use also in teaching
Skills acquired (at the end of the course):
Capacity for direct analysis of ancient mathematics texts, bibliographical research and coordination of the theoretical aspects of mathematics with its teaching and related historic developments
Prerequisites
Courses to be used as requirements (required and/or recommended): none
Courses required: none
Courses recommended: none
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 72
Hours for laboratory: 0
Hours for laboratory-field/practice: 0
Seminars (hours): 10
Stages (hours): 0
Intermediate examinations (hours): 0
Further information
Attendance of lectures, practice and lab:
Not mandatory
Teaching tools:
Computer and projector
Egyptians, Babylonians and Greeks:
Numbering systems and arithmetic operations. First-degree equations in Egyptian mathematics. First-, second- and third-degree equations and systems in Babylonian mathematics. The problems of “application of areas” in Greek mathematics and second-degree equations. Classical problems ascribable to third-degree equations. Notes on Diophantus’s Arithmetica.
Indians and Arabs:
Numbering systems and arithmetic operations. First-, second-, third- and fourth-degree equations and systems in Indian mathematics: Aryabhata, Brahmagupta and Bhaskara. Importance and influence of Arab culture in the West. First-, second- and higher-degree equations in Arab mathematics: al-Khwarizmi, Abu Kamil, al-Karaji, al-Khayyam, al-Kashi.
Arithmetic in the Middle Ages and Its Teaching:
Severino Boezio and the De institutione arithmetica, Aurelio Cassiodoro, Isidoro of Seville, Beda, Alcuino of York, Gerberto d’Aurillac.
Leonardo Fibonacci and the Mathematics of the Abacus:
Life and works of Fibonacci. Analysis of the fifteen chapters of the Liber abaci. Abacus teachers and schools in Italy in the Middle Ages and Renaissance. The mathematics of the abacus and treatises dealing with it. Operations with integers and fractions: various methods of multiplication and division, multiple fractions and unitary fractions, approximation of roots. The methods of simple and double false position. Problems of mercantile arithmetic: barter, coins, companies. Recreational mathematics. Algebra. Analysis of the algebraic problems in chapter XV of the Liber abaci. First-, second- and higher-degree equations in the mathematics of the abacus: Jacopo da Firenze, Paolo Gherardi, Dardi of Pisa, Antonio Mazzinghi, Piero della Francesca; Benedetto da Firenze and the great “encyclopedias” of the 15th century.
Luca Pacioli:
The figure and work of Pacioli: the Summa de arithmetica, geometria, proportioni et proportionalità and the mathematics of the abacus, the De viribus quantitatis and the Divina proportione.
Developments in Algebra in the 16th Century:
Invention of formulae for the solution of the third- and fourth-degree equations: Scipione Dal Ferro, Nicolò Tartaglia, Gerolamo Cardano and Ludovico Ferrari. Raffaele Bombelli’s Algebra: relations between algebra and geometry; the irreducible case of the third-degree equation and complex numbers.
The Algebraic Work of François Viète:
Life and works of Viète. The Isagoge and Viète’s method of analysis; the “logistica speciosa” or new algebra using letters. Summary of the contents of the Notae priores and the Zeteticorum libri quinque. The De aequationum recognitione et emendatione tractatus duo: zetetic analysis, plasma, sincrisis and the “remedia” for the reduction of an equation to canonical form. Algebraic solution of third- and fourth-degree equations. The Canonica recensio and the Supplementum geometriae in reference to the geometric solution of second- and third-degree equations.
René Descartes:
Life and works of Descartes. Analysis of the three books of Géométrie.
Quadrature methods:
The "exhaustion" method. Archimedes and the Method: sphere, cylinder and cone. The volume of the sphere according to Al-Haytham. The revival of the Archimedean studies in the Renaissance: Luca Valerio. Bonaventura Cavalieri and the
Theory of Indivisibles. The infinite hyperboloid of Evangelista Torricelli: curved indivisibles. The indivisibles after Cavalieri: observations.
The birth of the calculus:
The Nova methodus of Leibniz; Newton's method; the dispute on the calculus. I nod to the spread of calculus in Europe.