History of Mathematics & Teaching of Mathematics

History of Mathematics and Teaching of Mathematics
Eger, 19-22 May 2016

László Buday


Domestic (OKM) and international surveys (PISA) increasingly more tasks point into the direction of the measurement of the space abilities. In the PISA and competence basis tasks have to use the students nearly 30 % of their spatial abilities. Beyond all these the Hungarian mathematics high school graduation part II. in all years contains one spatial-geometry computational task, to which one essential the interpretation of the plane figure belonging to him, and related mental operations using. The results of the mentioned measurement procedures, additional researches and educators' actual experiences show it, that the high-school students' space-perception not always suitable, requires a development. The GeoGebra DGS provides a distinguished opportunity to development. We report on the results of a research covering 5 school years, which one the development of the spatial abilities aiming at, and created GeoGebra tasks.


Edith Debrenti


Arithmetics is a work of the Scholarly Collection of the Reformed College of Sárospatak, which was released in Kolozsvár (today: Cluj- Napoca) in 1591. Thanks to modern technology this treasure is now available.  It has been digitalized by the Hungarian Institute for the History of Science.

Hungarian mathematics began to develop in the 16th – 18th century. The books published in this period are very important for the discipline, since their content and didactic considerations saved as a basis for elementary mathematics coursebooks later on. This period gave rise to six important Hungarian works.

Kolozsvár Arithmetics was used not only as a textbook. It was often used by merchants, who could study arithmetic operations in their native language. The book is also important from a historic point of view since the technical language of mathematics in Hungarian starts to emerge in this period.

The research method employed was document analysis. The following items have been examined under considerations of Dárdai Fischerné (2008):  topics in the book, questions and exercises, the learnability of the pedagogical text restricted to technical terms, and illustrations.

Keywords: history of mathematics, arithmetics, operations.


Roman Fric


Consider the following sequence: 1, 36, 27, 40, 55, 86, 57, 2. Guess what does it represent? The answer is unbelievable: professor Ján Jakubík was born in 1923 (passed away in 2015), his first paper appeared before 1954, and the other elements in the sequence are the numbers of his papers published in the consecutive decades, 1954–1963, 1964–1973, . . . , 2004–2013, 2014-2023. Let X be a mathematician and let ║X║80 be the number of papers published by X being an octogenarian. Let us call it the Jakubík number of X . Then for X = Jakubík we get ║X║80 = 57. 

How about that? 

The aim of our talk is to present some observations about a Slovak mathematician (most of his life is associated with Košice, also Cassovia, Kaschau, Kassa) related to research, education and culture. His story is a contribution to the history of mathematics in Central and Easter Europe. 

Keywords: Jakubík number, Slovak mathematician, history of mathematics


Ján Gunčaga 

Several Authors of Mathematical Textbooks in 19th Century Hungary 

The aim of the present paper is to give a brief account of mathematics education in the village schools in Hungary in the 19th century. It describes the book Didactics and Methodology by András Lesnyánszky (1795-1859) that was inspired by the work of another author of that time – Joseph Weinkopf (1787-1873). In 2014 we also commemorated the 200th anniversary of the author of a great number of mathematical textbooks published in Austro-Hungarian monarchy, Dr. Franc Močnik (1814-1892). The paper deals with some motivational approaches of these authors to primary mathematics education. 

Keywords: mathematics in primary and secondary education, András Lesnyánszky, Franc Močnik, Joseph Weinkopf 



Szilvia Homolya, Szilvia Szilágyi 

Zoltán Szarka’s Lecture Notes and Handbooks on Mathematics for Engineers 

The purpose of our paper is to present the authorship of Zoltán Szarka. He is one of the most famous author of mathematical lecture notes in Hungary. During his academic career he wrote more than 35 lecture notes and textbooks, partly with co-authors. These works incorporate several important areas of mathematics: analysis, linear algebra, probability theory and mathematical statistics. 

Keywords: teaching of mathematics, lecture notes, handbooks, biography, mathematics for engineers


Dirk Huylebrouck


In 2011, Dutchman Rinus Roelofs discovered an error in one of the geometric drawings Leonardo da Vinci made for Luca Pacioli’s book ‘De Divina Proportione’ (‘About the Divine Proportion’). It caused quite a stir, yet a little later other scientists such as Jos Janssen (The Netherlands), Tibor Tarnai and Andras Lengyel (Hungary) revealed more mathematical shortcomings in Leonardo’s work. I published a series of papers about these discoveries. Most of them were the results of joint efforts, but I included additional own remarks as well: ‘Lost in triangulation’, ‘Lost in enumeration’, ‘Lost in edition’, ‘Observations about Leonardo's drawings for Luca Pacioli’ and ‘About a Divine Error’. Some errors can be seen as inaccuracies or ‘slip-ups’ occurring during copying, but the error revealed in the last paper concerns a blunt statement claiming six summits of pyramids of the elevated icosidodecahedron lay in one plane, while this is simply not true. Admittedly, the erroneous theorem was formulated by author Luca Pacioli, not Leonardo, but he made the illustration for this ‘theorem’ and he did not give the slightest comment about this error in any of the seven thousand pages known to be of his hand. Moreover, in none of Leonardo’s notebook pages, there is any mathematically interesting statement, nor in geometry, nor in arithmetic, nor about squaring the circle, nor about the divine proportion. He even never ever wrote about the latter, the ‘divine or golden section’ (a terminology not even known in his time). A popular Australian exhibition, on a world tour for several years, is entitled ‘Leonardo the genius’, but at least from a mathematical point of view this title is unjustified.

Keywords: polyhedra, divine proportion, Luca Pacioli


Lost in Triangulation, Leonardo’s slip-up in geometry, EOS, 04 2011, reproduced in a shortened version in Scientific American, 04 2011, and in La Recherche (France).

Lost in Enumeration: Leonardo da Vinci’s Slip-Ups in Arithmetic and Mechanics, The Mathematical Intelligencer, Volume 34, Issue 4 (2012), Page 15-20.

Lost in Edition: did Leonardo da Vinci slip up?, ‘Leonardo - Art, Science and Technology’, vol. 48, issue 5, MIT Press, 12 2015, p. 458-464.

Observations about Leonardo's drawings for Luca Pacioli, ‘Journal of the British Society for the History of Mathematics’, Volume 30, Issue 2, 2015, pp 102-112.

A Divine Error, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015), Pages 93–98.


Tünde Kántor 


This factually substantiated overview offers, from the first time in the past century up to the present, a compilation of the names of distinguished university students who throughout their studies attained top grades in mathematics. A brief outline of the honour doctoral awards is followed by an overview of the varied history of distinguished promotion in Hungary and in Debrecen (promotio sub auspiciis Regis, promotio sub auspiciis Gubernatoris, promotio sub laurea Almae matris, promotio sub auspiciis Rei Publicae Popularis, promotio sub auspiciis Praesidentis Rei Publicae). Finally we are given an opportunity to get acquainted with the survey of the activity of distinguished honour doctors. 

Keywords: Honour Doctoral Awarding, history and forms of distinguished promotion in Hungary and in Debrecen, survey of the activity of distinguished honour doctors. 


Péter Körtesi


We will study the Chapter Famous curves of the MacTutor History of Mathematics archive, see: http://www-history.mcs.st-and.ac.uk/Curves/Curves.html. The GeoGebra softvare is suitable to represent both the set of functions, and the so called associated curves, like evolutes, or involutes, and to experience their relation. The curves are given either in explicit, implicit, parametric or polar coordinate form, and we will explore the power of the softvare to visualize them. The osculating circle, tangents, normals, convex boundary of family of curves will be mentionned as well. The above mentionned Famous curves chapter is suitable as well to offer practical examples for students in applying GeoGebra, and their results can be valourized on the GeoGebra Tube.

Keywords : Famous curves, MacTutor History of Mathematics, GeoGebra DGS, Evolute

Osculating circle


Michael Lambrou


Calculus is a major branch of Mathematics with great versatility. One of its characteristics is that unifies techniques used by earlier mathematicians to determine the tangent and the area under of curves, and related problems such as volumes, centres of gravity etc. Before the invention of Calculus mathematicians used ingenious ad hoc techniques for the determining tangents and areas but each case had to be studied separately. For example we find in ancient Greek mathematics the first exact definition of tangency (not identical to our modern) and the determination of tangent to a small number of curves such as the circle, the three conics and Archimedes’s spiral, each with it’s own method specific to the curve. General techniques appeared many centuries later, as for example a form of the Fundamental Theorem of Calculus by Barrow. In this talk we will study the methods of Euclid, Archimedes, Apollonius, Barrow and others for drawing tangents. Similarly we will discuss the ingenious methods used by ancient Greeks, the Arabs, Cavalieri, Kepler and Torricelli among others to determine curved areas and volumes such as the surface area and volume of a sphere, the area under a parabola and the volume of paraboloid of revolution, the corresponding one for hyperbola and others. Much of these techniques can be used in a school classroom to enhance teaching and to convey the message as to why one benefits from the study of Calculus.

Keywords: tangent, area, Calculus, Greek Mathematics, Archimedes, Apollonius, Cavalieri, Barrow, Kepler


Maria Mihailova


The article refers to the important developments in educational technology over the next five years, giving a definition to the key technologies which present these developments in K-12 and higher education worldwide. It is made a comparative analysis between two annual researches (2014/2015) of NMC Horizon Report depending on the developments’ temporal significance. For the aims of the research, seeking the important developments in education it is essential to define what an educational technology is and its role in adopting of learning geometry.

Keywords: education, development, technology, educational technology, flipped classroom, Internet of Things, Bring Your Own Device


Dénes Nagy 


János Bolyai’s discovery of the absolute and the hyperbolic non-Euclidean geometry was internationally recognized just a few years after his 1860 death. Indeed, the French and the Italian translations of his work were published in the 1860s. It was followed by the American George Bruce Halsted’s English translation with four editions in the 1890s. We will discuss some less known facts, including the activities related to Bolyai by the following mathematicians:

(1) 菊池 大麓 (Kikuchi, Dairoku, 1855-1917), the President of the University of Tokyo, then Minister of Education and “danshaku” (baron); he organized the publication in Japan.

(2) Henry Parker Manning (1859-1956), professor of Brown University; he made a much better English translation of Bolyai’s Appendix.

(3) Horatio Scott Carslaw (1870-1954), the Scottish-born Australian department chair of Sydney University; perhaps he made the greatest service for Bolyai’s international recognition.

(4) Вениамин Фёдорович Каган (Veniamin F. Kagan, 1869-1953) and Яков Викторович Успенский (James Victor Uspensky, 1883-1947), professors of Moscow Sate University and of Stanford University, respectively; Kagan – he was the grandfather of both Barenblatt (Berkeley) and Sinai (Princeton) – translated Bolyai’s Appendix into Russian, while Uspenskii wrote a book, where he gave credit to Bolyai (Petrograd, 1922). 

There are still a lot of tasks in connection with Bolyai, including the critical edition of many of his unpublished manuscripts.


It is interesting to observe the different ideas behind the basic terms for geometry in various cultures: the Greek γεωμετρία refers to “land-measuring”, the Sanskrit रेखागणित is associated with “line-culculating” (another term points to “figure-calculating”), and the Chinese幾何學 means “how-much-science”. In the Western culture “calculating” or “how much science” would be interpreted as arithmetic or algebra, not as geometry. Note that the Japanese adopted the Chinese characters in the given form, but they also developed a special field of geometry dealing with arrangement of figures. This activity was practiced even in small villages, and the results were posted in Shintō shrines and less frequently in Buddhist temples. Joseph Needham tried to link the roots of this activity to China, but we suggest rather native Japanese “ethnomathematical” motivations.

After comparing the ancient Greek and Roman mathematics in the West and the Japanese and the Chinese mathematics in the East, we formulate a thesis: The style of thinking in small countries is mostly geometric, while big empires prefer arithmetic-algebraic approach. Indeed, in the case of small countries the optimal usage of the available territories and the activity of land-measuring are important activities. On the other hand, in a large country or a big empire the recording of the data is essential. This fact is reflected even in the terms for geometry in India and China, where calculation is emphasized. The impact of these traditions can be observed even in modern time.

The usage of the dominant style of thinking – geometric or algebraic – is important for motivating students. However, at a later stage, we should develop stronger links between the “geometric brain” and the “algebraic brain” of each pupil.


Saeed Seyed Agha Banihashemi 


In recent decades the use of history of mathematics in mathematics education has been increasing around the world and its purpose is adjustment and challenging the notion of mathematics which it's dry and lifeless land, without question and mistake, show full of right answers.Hall 1 (2000) to explain scientific knowledge and insight of each person passes through history of science. Teaching mathematics with a historical perspective, in this respect major steps taken towards solving the problems of advanced and logical reasons and lead us to new ways of looking to the past. Today, in many countries the use of history of mathematics in mathematics education and special attention has been seeking to use this important tool in education. . The purpose of this article is the importance of the history of mathematics in mathematics education in the classroom. 

Keywords: history of mathematics, mathematics education, students.


Alena Šolcová

Mathematics and Physics in Prague of Slowly Fading Baroque. Joseph Stepling (1716 – 1778) – the Founder of the Prague Observatory

This contribution is concentrated on the change in the education of exact sciences in the Jesuit Academy in Prague in the 18th Century. The conceptual shift from the Aristotelian Science to the Newtonian Science was influenced by Joseph Stepling. It was he who played a central role in the change of the Jesuit educational curriculum. Joseph Stepling (1716, June, 29 – Prague 1778, July, 11) was born in Regensburg. His father from Westfallen was the secretary of the Emperor Embassy there. After death of his father Stepling as a child came to Prague with his mother (who was native of the Bohemia). Joseph Stepling was educated by private teachers and in the Jesuit College in Prague. The teachers led him to mathematical studies. The Jesuit teacher Sykora gave Euclid’s Elements to his hands. Stepling - 16 years old – firstly observed the eclipsis of Moon on 1733, May, 28th. He used the tables of French astronomer Philippe de la Hire (1640 – 1718).

 We will illuminate his scientific life and work and his impact on the education of exact sciences at the Prague University. Joseph Stepling made first steps to Newtonian Physics. He founded the seminar for students and colleagues. They solved problems of calculus (I. Newton and G. W. Leibniz, etc.) there. All his papers are in Latin language and later were translated to German by his pupil Anton Strnad, the director of observatory. In 1748 Stepling determined the geographical lenght of Prague for the new map of Germany. In 1751 the observatory in the Jesuit College in the Clementinum was founded. He studied the aberation of light of stars, and the electricity. In 1752 he started the regular meteorological observations. 
His correspondence with European mathematicians and astronomers: Maximilian Hell, Franz Huberti, Jean-Antoine Nollet, Ruder J. Boscowich, Nicolas-Louis de Lacaille, Christian Wolf, Leonhard Euler, etc., is important for the studium of mathematical knowledge of this time. 
He also influenced many his successors, among them Joannes Tesanek („Czech Newton“), Stanislaw Wydra (the professor of mathematics) , Anton Strnad (the professor of astronomy), Alois M. David (the director of observatory), Bernard Bolzano (the mathematician), Franz J. Gerstner (the professor of higher mathematics and the founder of Czech Technical University). 

From Stepling’s papers: 

Eclipsis Lunae totalis Pragae anno 1748 observatae, Pragae 1748. 

Exercitationes geometrico-analyticae de ungulis aliisque frustris cylindrorum, quorum bases sunt sectiones conicae infinitorum generu, Pragae 1751, 4°.; nova editio Dresdae 1760. 
Differentiarum minimarum quantitatum variantium calculus directus vulgo differentialis, Pragae 1764, 4°. 


Márta Takács


Since its introduction in the mid-sixties fuzzy set theory has gained recognition in a number of fields in the cases of uncertain, or qualitative or linguistically described system parameters or processes based on approximate reasoning, and has proven suitable and applicable with system describing rules of similar characteristics. It can be successfully applied with numerous reasoning-based systems while these also apply experiences stemming not only from the fields of engineering and control theory. Generally, the basis of the decision making in fuzzy based system models is the approximate reasoning, which is a rule-based system. Knowledge representation in a rule-based system is done by means of IF…THEN rules. Furthermore, approximate reasoning systems allow fuzzy inputs, fuzzy antecedents, fuzzy consequents. Fuzzy computing, as one of the components of soft computing methods differs from conventional (hard) computing in its tolerant approach. The model for soft computing is the human mind, and after the earlier influences of successful fuzzy applications, the inclusion of neural computing and genetic computing in soft computing came at a later point.

Keywords: fuzzy, soft computing, approximate reasoning.