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International audience; The aim of this paper is twofold: (1) to show the principal aspects of the way in which Newton conceived his mathematical concepts and methods and applied them to rational mechanics in his Principia; (2) to explain how the editors of the Geneva Edition interpreted, clarified, and made accessible to a broader public Newton’s perfect but often elliptic proofs. Following this line of inquiry, we will explain the successes of Newton’s mechanics, but also the problematic aspects of his perfect geometrical methods, more elegant, but less malleable than analytical procedures, of which Newton himself was one of the inventors. Furthermore, we will also consider the way in which Newtonianism was spread before in England and afterwards on continental Europe. In this respect the Geneva Edition plays a fundamental role because of its complete apparatus of notes, and because it appeared only thirteen years after the publication of the third edition of the Principia (1726). Finally, we will also confront some problems connected to the metaphysics of calculus. Therefore, the case of Newton is one of those in which, starting from mathematics applied to physics, it is possible to connect an impressive series of fundamental arguments such as the role of mathematics in science; the comparison between Newton’s geometrical methods and analytical methods; the way in which Newtonianism was spread as well as the philosophical implications of Newton’s mathematical concepts.

Among the mathematical methods which are taught in the last years of almost every high school, the mathematical induction deserves particular attention. It can be used both to define mathematical entities and to prove theorems. The second use is more common at high school level and is easier. Thus, I will basically focus on it, though analysing in depth two definitions by induction. The aim of this contribution is to offer the basic elements for a didact unit which could be developed in six/seven hours of lesson.

“Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon” (p. Newton 1846, p. 83). This is the famous first axiom or law of motion stated by Newton in his masterpiece The Mathematical principles of natural philosophy (ivi). Everywhere, in the courses of physics at the high school level the inertia principle is the first to be taught. However, there are many doubts that most of learners fully grasp its numerous and fundamental nuances, which are necessary for a satisfying introduction to physics. Therefore, I propose an interdisciplinary approach for the explanation of this principle in which history of science and analysis of the daily experiences are joined to offer a complete comprehension of the concept of inertia.

Pierre de Fermat (1601/7–1665) is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent (Fermat 1891–1922, II, pp. 431–436). He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: (1) it follows all the elements of which Fermat wrote in his outline; (2) it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods.

In this paper, I propose the idea that the French mathematician Michel Chasles developed a foundational programme for geometry in the period 1827–1837. The basic concept behind the programme was to show that projective geometry is the foundation of the whole of geometry. In particular, the metric properties can be reduced to specific graphic properties. In the attempt to prove the validity of his conception, Chasles made fundamental contributions to the theory of polarity and also understood that a satisfactory development of projective geometry has to overcome the specificity of that theory itself. In this perspective, he developed his ideas on duality and homography, showing their dependence on the concept that he was going to pose at the basis of geometry: the anharmonic ratio, today called cross-ratio. The conceptual itinerary that starts from Chasles’ studies on polarity and ends with his results based on the cross-ratio represents the itinerary of his foundational programme. I will follow this complex and interesting line of thought, place it in the context of what was then called descriptive geometry, and outline a connection with the later great results of Von Staudt, Cayley, and Klein.

This research deals with a possible use of history of mathematics in mathematics education. In particular, history can be a fundamental element for the introduction of the concept of integral through a problem-centred and intuitive approach. Therefore, what follows is dedicated to the teaching of mathematics in the last years of secondary schools, where infinitesimal calculus is addressed. The thesis here proposed is that the resort to Archimedes’ use of exhaustion method and to Newton’s initial lemmas expounded in his Principia Mathematica are useful means to reach a genetic comprehension of the concept of integral. Hence, two demonstrations by Archimedes and two lemmas by Newton are used to prove such thesis. A further idea here proposed is that history of mathematics can be of help for an interdisciplinary education. Keywords: interdisciplinary education, mathematics education, science history, secondary schools

The literature concerning the various methods by means of which the teaching of mathematics can be developed is simply huge and is increasing more and more. Several aspects are dealt with: the use of new technologies, especially as far as new computer programs or web sources are concerned; new techniques to develop calculations; researches concerning the possible relations between the everyday life of the pupils/students and the mathematical concepts; the best way to frame a lesson (frontal lessons, interactive lessons, discussions), and so on. This literature covers the entire school-life of a young boy/girl: from the elementary school to the university.

The concept of form is one of the most intuitive within our experience. When we say that two objects of different dimensions have or do not have the same form there is not properly a reflexion behind this claim. Rather, it is, at all appearances, based on our visual faculties, which is perfectly in order in the context of our daily life. This intuitive and visual notion of form is suitable to the necessities of our practical, or also esthetical, experience. However, on second thought, things are not so easy: suppose that I look at an object and I find that it is circular. I claim, hence, that it is a circle and my statement is correct. Another person looks at this object from another point of view and sees that this object is an ellipsis or a hyperbola or a parabola. He is not wrong.

In this paper we present the relations between mathematics and mathematics education in Italy between the 12th and the 16th century. Since the subject is extremely wide, we will focus on two case-studies to point out some relevant aspects of this phenomenon: 1) Fibonacci’s studies (12th-13th century); 2) Abacus schools. More particularly, Fibonacci, probably the greatest European mathematician of the Middle Ages, made the calculations with Hindu-Arabic digits widely spread in Europe; Abacus schools were also based on the teaching of the calculation with Hindu-Arabic digits. These case-studies are significant for understanding the connections between science, science education and the development of science within Western civilization. We think that the knowledge of such significant relations can be useful for the scholars who are nowadays engaged in mathematics education and in the research field of science-society relations. Finally, we attempt to outline the interaction between mathematics education and...

What is the possible use of history of mathematics for mathematics education? History of mathematics can play an important role in a didactical context, but a general theory of its use cannot be constructed. Rather a series of cases, in which the resort to history is useful to clarify mathematical concepts and procedures, can be shown. A significant example concerns differential calculus: Newton’s Methodus fluxionum et serierum infinitarum is a possible access-key to differential calculus. For, many concepts introduced by Newton ought be useful for the pupils/students (last or last but one year at the high school and first year at the university) to reach a more intuitive, geometrical and problem-oriented approach to calculus. The motivation to consider history of mathematics as an important didactical support is that the pupils/students often learn mathematics in a too formal manner, without understanding the real reasons for the introduction of several mathematical concepts. The problem is that the potential of such support is not exploited. The educational proposal is hence to show a concrete case to highlight what the teaching of mathematics based on history means. The conclusion is that a general theory, as differential calculus, should be considered by the pupils/students as a necessity, deriving from a specification, improvement and extension of the techniques used to solve significant problems posed and developed in the course of history. In this manner, mathematics appears as a human activity comparable with other activities and not as a merely formal exercise. Key words: mathematics education, history of mathematics, Newton, fluxions, tangents, maxima and minima, problem solving approach to mathematics education.

During his stay in Padua ca. 1592–1610, Galileo Galilei (1564–1642) was a lecturer of mathematics at the University of Padua and a tutor to private students of military architecture and fortifications. He carried out these activities at the Academia degli Artisti. At the same time, and in relation to his teaching activities, he began to study the equilibrium of bodies and strength of materials, later better structured and completed in his Dialogues Concerning Two New Sciences of 1638. This paper examines important details of four works dating to the Paduan period: Breve instruzione dell’architettura militare; Trattato di Fortificazione; Le Mecaniche; Le operazioni del compasso geometrico et militare. The two works on military architecture and fortifications were compiled from notes taken by students, and are not by Galileo’s hand, but are still illustrative of his work and thinking at the time.

In this chapter, we present the concept of force in Kepler. We follow the development of this concept during Kepler’s scientific career, starting from his early considerations in the Mysterium Cosmographicum (1596) until his ripest conceptions expounded in the Epitome Astronomiae Copernicanae (1618–1621). Kepler tried to supply a dynamical explanation to the planetary movements. This is an important novelty because astronomy was traditionally a kinematical science. Based on the main accredited literature we present a historical account and theoretical/nature of science (NoS) developments: (1) Keplerian forces; (2) physical astronomy; and (3) orbits, force, and the relation between distances and forces. Koyre had a prominent role in the studies on astronomical revolution and on Kepler. He was also important in fully clarifying the pivotal function that physical astronomy—and hence the concept of force—had in Kepler’s system of the world. Indeed, Koyre is still nowadays an almost unavoidable reference point for Kepler’s Forschung. We refer to several interpretations of his. This is the reason why we think appropriate to present this work in homage to Alexandre Koyre.

1 The Iuvenilia–Early Galilean works When Galileo Galilei (1564-1642) published the Sidereus Nuncius in 1610 [Galilei 1890-1909, III, pt 1, 51-96], he was a famous enough scientist, who was not young: for, he was 46. Nevertheless, this little book represented the fundamental turning point in Galileo’s life and scientific production. The Sidereus Nuncius was very successful and gave rise to numerous discussions. Some scholars defended Galileo—the most important was Kepler—, many others, with a...

Mathematics has always been a difficult issue, especially in the African countries. Mozambique is not an exception. This country had been colonized by Portugal until 1975. When the independence was obtained, a socialist regime was adopted (1977). The learning of mathematics entered the struggle against colonial and imperialistic ideas. Its best ally was Paulus Gerdes, one of the most relevant ethnomatematicians of the world, who carried out an intense promotion of this approach to mathematics in Mozambican school system. Albeit the great international impact of Gerdes’ ideas, Mozambique never implemented his methodology. When, at the end of the 80s, the country changed from socialism to liberalism, voting a democratic Constitution in 1990, its school system was aligned to the measures of International Monetary Fund (IMF) and World Bank (WB). The most recent ones are represented by the Millennium Development Goals. Despite the various reforms of Mozambican school system, the results of Mozambican children in mathematics are among the worst in Africa. The reasons of such a failure are here explained, through a historical approach based on national documents. The most recent experiences of school reform carried out by international agencies together with national institutions are stressed. The negative results obtained by the Mozambican learners as to mathematics are due to several reasons: 1) a lack of consideration of the Mozambican cultural substrate; 2) an improper massification of the school system, where the quality of instruction has been neglected; 3) the specific choice to marginalize mathematics education.

In some previous contributions of mine written for Scientia Educologica’s journals (Bussotti 2012; Bussotti, 2013; Bussotti, 2014) I dealt with the possible use of history of mathematics and science inside mathematics and science education. There is an abundant literature on this subject and I only tried to offer some ideas on possible educative itineraries in which history of mathematics and science could play a role. I had no claim to supply elements for a general theory on the relations history of mathematics-mathematics education and history of science-science education. In this editorial, I would like to deal with a possible interdisciplinary link between philosophical education and mathematics. This link is given by the infinity. The following considerations are valid for all those countries in which some high schools exist where philosophy is taught and, in general, for every course at a philosophical faculty in which the problem of the infinity is faced. Furthermore, they can also be useful in the teaching of mathematics at the high school when the concepts of infinity and infinitesimal (typically while dealing with calculus) are introduced.

In the period 2012-2013 I got the qualification (abilitazione) to teach history and philosophy in the Italian high schools. The course I followed was called TFA (Tirocinio Formativo Attivo, Active Formative Training). The final examination was constituted by various proofs. Two of them were the written presentations of one educational itinerary in history and one in philosophy. Both of them had to be structured in a series of interconnected lessons. In this editorial I will expose, with some minor modifications, the translation of the educational itinerary I prepared for philosophy. It concerns the scientific revolution of the 17th century. The interest of this itinerary is not limited to the schools in which philosophy is taught, but it can also provide ideas useful in a course of physics at the high school or of history and philosophy of science at the university. What follows is divided into two parts: 1) a general presentation of the aims and methods followed in the lessons; 2) the lessons of the educational itinerary. In my training in philosophy – developed in September and October 2013 in an Italian scientific high school – I presented the following lessons concerning the scientific revolution.

Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathematics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of mathematical analysis was profound: there were two complex examinations in which the theory was as important as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and mathematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance humanists, and with respect to mathematics education nowadays is discussed. Particularly the ways through which mathematical knowledge was spread in Italy between late Middle ages and early Modern age is shown. At that time, the term “scientific education” corresponded to “teaching of mathematics, physics”; hence something different from what nowadays is called science education, NoS, etc. Moreover, the relationships between mathematics education and civilization in Italy between the 12th and the 16th century is also popularized within the Abacus schools and Niccolò Tartaglia. These are significant cases because the events connected to them are strictly interrelated. The knowledge of such significant relationships between society, mathematics education, advanced mathematics and scientific knowledge can be useful for the scholars who are nowadays engaged in mathematics education research. Key words: Abacus schools, mathematics education, science & society, scientific education, Tartaglia

My research fields are history of mathematics and science, mainly physics and astronomy. I have also published some works on mathematics and physics education (as to these works see Bussotti 2012a; Bussotti 2012b; Pisano-Bussotti, 2012; Bussotti 2013). I have often wondered which role history of science can have inside science education, basically referring to high school and university students. This subject dates back at least at the second half of the 19th century when an important debate took place in Europe as to the most appropriate manner to teach Euclidean geometry. There were various positions: scholars who thought Euclid (fl. 300 BC) had to be completely abandoned, others who believed that the Elements had to be almost literally taught and, between these two opposite extreme opinions, a series of intermediate ones existed (for this problems see Bussotti, 2012a, where a series of references is presented, too). The discussion on the role of history of science/mathematics inside science/mathematics teaching is hence a long period debate and I have no pretension to provide an answer, but only to point out some questions and to develop a reasoning around them.

This study deals with Vittorio Checcucci’s ideas and proposals as to mathematics education. The scopes of this work are twofold: 1) the first scope is historical: my aim is to reconstruct Checcucci’s thought. This is a novelty because almost no contribution dedicated to Checcucci exists. The few existing contributions are brief articles whose aim is not to provide a general picture of his ideas; 2) the second scope is connected to mathematics education in the 21st century. A series of argumentations will be proposed to prove that many Checcucci’s ideas could be fruitfully exploited nowadays. For the first time, the thought of this mathematician is exposed to non-Italian readers because his ideas are worthy to be known, rethought and discussed in an international context. Key words: mathematics education, relations between theoretical and practical mathematics in the teaching, experimentations in mathematics education.

The paper try to provide a contribution to the scientific—historiographic debate concerning the relations between experiments, metaphysics and mathematics in Descartes’ physics. The three works on which the analysis is focused are the Principia philosophiae and the two physical essays: La Dioptrique and Les Meteores. The authors will highlight the profound methodological and epistemological differences characterizing, from one side, the Principia and, from the other side, the physical essays. Three significant examples will be dealt with: 1) the collision rules in the Principia philosophiae; 2) the refraction law in La Dioptrique; 3) the rainbow in Les Meteores. In the final remarks these differences will be interpreted as depending upon the different role Descartes ascribed to the three books inside his whole work. The concepts of intensity and gradation of the physical quantities used by Descartes will provide an important interpretative means. In this paper, we compare the aprioristic approach to physics typical for Descartes’ Principia with the experimental and mathematical one characterizing Descartes’ Essays.

Generally speaking the exponential function has large applications and it is used by many non physicians and non mathematicians, too. Nevertheless some crucial and practical problems happen for its mathematical understanding. Mostly, this part of mathematical cognitive programmes introduce it from the mathematical strictly point of view. On the contrary, since both physics experiments make a vast use of it, in this paper the exponential function will be explained starting from physical experiments and only later a mathematical modelling of it will be organized. The relationship physics-mathematics-geometry is crucial and indispensable in this kind of integrated and history&science education. The history and epistemology of mathematics and physics can be a significant means to make the epistemological and didactical research more profound and clear. Key words: interdisciplinary, elementary functions, geometric transformations, epistemological teaching, thermology and calorimetry.

Based on our research regarding the relationship between physics and mathematics in HPS, and recently on Geneva Edition of Newton's Philosophiae Naturalis Principia Mathematica (1739–42) by Thomas Le Seur (1703–70) and Francois Jacquier (1711–88), in this paper we present some aspects of such Edition: a combination of editorial features and scientific aims. The proof of Proposition XLIII is presented and commented as a case study.

This history of mathematics is a specific and, at the same time, wide field of research with proper methods, journals, congresses and results. However, some questions about its status are without any doubt legitimate. In particular: is the public to whom the work and the publications of the historians of mathematics are addressed, limited to the specialists in this field or is it broader? It often happens that the mathematicians engaged in the active research consider history of mathematics as a sort of curiosity, but nothing really useful for their researches. They are not interested in an inquire on the historical bases of their researches because they are concentrated in discovering new theorems and solving new problems. It difficult to valuate whether and inside which limits this way of thinking is correct. The problem is complex and cannot be dealt with in this context.