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 François 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.

Pierre de Fermat 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. 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: it follows all the elements of which Fermat wrote in his outline; 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. (shrink)

This paper has the aim to provide a general view of the so called Jesuit Edition (hereafter JE) of Newton’s Philosophiae Naturalis Principia Mathematica (1739–1742). This edition was conceived to explain all Newton’s methods through an apparatus of notes and commentaries. Every Newton’s proposition is annotated. Because of this, the text – in four volumes – is one of the most important documents to understand Newton’s way of reasoning. This edition is well known, but systematic works on it are still (...) missing. We are going to fill this gap by means of a project exposed in the final remarks of this paper. In this paper we will: A) expound the way in which the notes and the additions to the JE were conceived by the commentators; B) provide some pieces of information about the commentators; C) summarize the most important of their notes; D) examine closely their notes as to a particularly important question: the so called "inverse problem of the central forces". (shrink)

In this research, we present the most important characteristics of the so called and so much explored Jesuit Edition of Newton’s Philosophi? Naturalis Principia Mathematica edited by Thomas Le Seur and Fran?ois Jacquier in the 1739-1742. The edition, densely annotated by the commentators (the notes and the comments are longer than Newton’s text itself) is a very treasure concerning Newton’s ideas and his heritage, e.g., Newton’s geometry and mathematical physics. Conspicuous pieces of information as to history of physics, history of (...) mathematics and epistemology can be drawn from it. This paper opens a series of study concerning Jesuit Edition, whose final scope is to put in evidence all the conceptual aspects of such edition and its role inside the spread of scientific ideas and inside the complex relation science, popularization & society. (shrink)

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 advanced mathematics in that period, focusing on the figure of Leonardo Pisano (c. 1170-c.1250), called Fibonacci, who played an influential role both in mathematics education and in advanced mathematics. (shrink)

This paper is divided into two parts, this being the first one. The second is entitled ‘Historical and Epistemological Reflections on the Culture of Machines around Renaissance: Machines, Machineries and Perpetual Motion’ and will be published in Acta Baltica Historiae et Philosophiae Scientiarum in 2015. Based on our recent studies, we provide here a historical and epistemological feature on the role played by machines and machineries. Ours is an epistemological thesis based on a series of historical examples to show that (...) the relations between theoretical science and the construction of machines cannot be taken for granted, a priori. Our analysis is mainly based on the culture of machines around 15th and 17th centuries, namely the epoch of Late Renaissance and Early Modern Age. For this is the period of scientific revolution and this age offers abundant interesting material for researches into the relations of theoretical science/construction of machines as well. However, to prove our epistemological thesis, we will also exploit examples of machines built in other historical periods. Particularly, a discussion concerning the relationship between science theory and the development of science art crafts produced by non-recognized scientists in a certain historical time is presented. The main questions are: when and why did the tension between science give rise to a new scientific approach to applied discipline such as studies on machines and machineries? What kind of science was used for projecting machines and machineries? Was science at the time a necessary precondition to build a machine? In the first part we will focus on the difference between Aristotelian-Euclidean and Archimedean approaches and we will outline the heritage of these two different approaches in late medieval and Renaissance science. In the second part, we will apply our reconstructions to some historical and epistemological problems concerning the relations of science/technology/constructions of machines. The problem of perpetual motion will play an important role in this context. (shrink)

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. (shrink)

This paper is the second part of our recent paper ‘Historical and Epistemological Reflections on the Culture of Machines around the Renaissance: How Science and Technique Work’. In the first paper—which discussed some aspects of the relations between science and technology from Antiquity to the Renaissance—we highlighted the differences between the Aristotelian/Euclidean tradition and the Archimedean tradition. We also pointed out the way in which the two traditions were perceived around the Renaissance. The Archimedean tradition is connected with machines: its (...) relationship with science and construction of machines should be made clear. It is enough to think that Archimedes mainly dealt with three machines: lever, pulley and screw. As underlined in the first part, our thesis is that many machines were constructed by people who ignored theory, even though, in other cases, the knowledge of the Archimedean tradition was a precious help in order to build machines. Hence, an a priori idea as to the relations between the Archimedean tradition and construction of machines cannot exist. In this second part we offer some examples of functioning machines constructed by people who ignored any physical theory, whereas, in other cases, the ignorance of some principles—such as the impossibility of a perpetuum mobile—induced the attempt to construct impossible machines. What is very interesting is that these machines did not function, of course, as a perpetuum mobile, but anyway had their functioning and were useful for certain aims, although they were constructed on an idea which is completely wrong from a theoretical point of view. We mainly focus on the Renaissance and early modern period, but we also provide examples of machines built before and after this period. We have followed a chronological order in both parts, starting from the analysis of the situation in ancient Greece. Therefore, in the first part, we have examined the relations between the Aristotelian/Euclidean and Archimedean traditions from ancient Greece to the early modern age. In this second part, we analyse the relations of Archimedean tradition/ construction of machines from ancient Greece to the 19th century, focusing on the mentioned period. We remind the reader that our aim is to prove an epistemological thesis, not to provide a complete historical endeavour. As a correlated article, the reader will find three previous paragraphs in the first above-mentioned article. (shrink)

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. (shrink)

Gottfried Wilhelm von Leibniz (1646-1716) has a prominent worldwide place in the history of scientific thought, from mathematics, logic, and physics to astronomy and engineering. In 2016, both his birth and death have been commemorated. Given the influence by Leibniz on Western sciences and philosophies and his polyhedric scientific activities, this special book chooses to focus on Leibniz's scientific works. In particular, we explore Leibniz's intellectual matrix and heritage within interdisciplinary fields, and present contributions from leading experts on the subject. (...) The book offers much-needed insights into the subject from scientific, historical, philosophical and nature of science perspectives. It also provides authoritative introductions to scholarly contributions, which are often dispersed in journals and books not easily accessible to every reader. Therefore, this volume also contains excellent chapters on topics which, generally speaking, have their place in any rounded science, history or philosophy topic. It provides an absorbing and significant read for historians, philosophers and scientists alike. Editors Raffaele Pisano is full professor at the Lille University, France. Michel Fichant is Emeritus professor at the Sorbonne University, France. Paolo Bussotti is senior lecturer-researcher at the Udine University, Italy. Agamenon R. E. Oliveira is full professor at the Federal University of Rio de Janeiro, Brazil. Foreword Eberhard Knobloch is Emeritus professor at the Berlin University of Technology, Germany. (shrink)

The interpretation of Parmenides’ Περί Φύσεως is a fascinating topic to which philosophers, historians of philosophy and scientists have dedicated many studies along the history of Western thought. The aim of this paper is to present the reading of Parmenides’s work offered by Federigo Enriques. It is based on several original theses: (1) Parmenides was the discoverer of abstract geometry; (2) his critics was addressed against the Pythagoreans rather than against Heraclitus; (3) Parmenides discovered and applied the contradiction and the (...) third excluded principles in the context of his research on foundation of geometry; (4) Parmenides’s metaphysical and physical conceptions have their bases in his speculation on geometry; (5) Parmenides used the principle of sufficient reason. Enriques’s reading is worth being expounded and discussed within the historical, philosophical and scientific context in which it is inserted. Since Enriques’s ideas are not widely known and discussed, my research has the purpose to fill this gap. The article also aims to provide elements to illustrate the discussion on Parmenides in the first half of the last century. (shrink)

Georg Cantor, the founder of set theory, cared much about a philosophical foundation for his theory of infinite numbers. To that end, he studied intensively the works of Baruch de Spinoza. In the paper, we survey the influence of Spinozean thoughts onto Cantor’s; we discuss Spinoza’s philosophy of infinity, as it is contained in his Ethics; and we attempt to draw a parallel between Spinoza’s and Cantor’s ontologies. Our conclusion is that the study of Spinoza provides deepening insights into Cantor’s (...) philosophical theory, whilst Cantor can not be called a ‘Spinozist’ in any stricter sense of that word.Keywords: Georg Cantor; Baruch de Spinoza; Infinity in philosophy and mathematics; Transfinite numbers; Set theory; Infinity and pantheism. (shrink)

This section provides the first full edited transcription—from Latin and Italian vernacular language—of the Opera geometrica. Further, as above said, facsimile texts are added, including our critical comments as footnotes to this chapter.

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. (shrink)

In this paper, we deal with Torricelli’s principle in mechanics according to which two heavy bodies linked together cannot move by themselves unless their common centre of gravity descends. At the beginning of the De motu gravium naturaliter descendentium, in his Opera geometrica (1644), Evangelista Torricelli (1608–1647) expressed this principle in a very brief section entitled Praemittamus. Our aim is to sustain the idea that Torricelli was constructing a general view of a kind of physics based on general principles, of (...) which the previous one is the most important and explicit statement. In our discussion, the following themes are addressed: (1) a comparison between the Torricelli’s principle and the virtual work principle; (2) the role of Torricelli principle within the history of the virtual work principle; (3) the logic behind the demonstrations offered in physics relying on general principles such as that of Torricelli. (shrink)

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...

Two topics, Science and History, have been studied for centuries, if not millennia, often in disjointed ways. The realization that Science is nourished by its History is quite obvious nowadays and is a standpoint shared by all the contributors in this book. Given this fact, the following chapters were written by interdisciplinary scholars who are experts in this field. The admirably written contributions concern History of Science, Epistemology, Philosophy of Science, but are not restricted to these disciplines. They also discuss (...) theoretical perspectives and ingenious results achieved by beautiful and clever minds working together or inspired by one another. Torricelli’s history is a fascinating one, filled with struggles and insights that have been studied and examined in depth by internationally renowned specialists. The gripping stories reconstructed, as well as the epistemological and logical threads drawn from long and painstaking research, are presented in a clear and absorbing fashion, in each instance with a unique style and always with considerably talented analysis. The preceding celebratory historical and historiographical chapter on Opera geometrica (Torricelli in Opera geometrica. Florentiae. Masse & de Landis (1644); Torricelli in Opere di Evangelista Torricelli edite in occasione del III centenario della nascita col concorso del Comune di Faenza da Gino Loria e Giuseppe Vassura. Montanari (1919–1944)) is also a structured introductory essay to the contents of this extensive Springer research project. Below, the reader will find summaries of selected distinguished essays written in order to celebrate the magnificence of Torricelli’s Opera geometrica. The whole volume has to be considered as an homage both to Opera geometrica and to Evangelista, a genial scientist. (shrink)

This paper is the second part of our recent paper ‘Historical and Epistemological Reflections on the Culture of machines around the renaissance: How s cience and t echnique Work’ (Pisano & Bussotti 2014a). In the first paper—which discussed some aspects of the relations between science and technology from Antiquity to the Renaissance—we highlighted the differences between the Aristotelian/Euclidean tradition and the Archimedean tradition. We also pointed out the way in which the two traditions were perceived around the r enaissance. t (...) he Archimedean tradition is connected with machines: its relationship with science and construction of machines should be made clear. i t is enough to think that Archimedes mainly dealt with three machines: lever , pulley and screw (and a correlated principle of mechanical advantage ). As underlined in the first part, our thesis is that many machines were constructed by people who ignored theory, even though, in other cases, the knowledge of the Archimedean tradition was a precious help in order to build machines. Hence, an a priori idea as to the relations between the Archimedean tradition and construction of machines cannot exist. i n this second part we offer some examples of functioning machines constructed by people who ignored any physical theory, whereas, in other cases, the ignorance of some principles—such as the impossibility of a perpetuum mobile —induced the attempt to construct impossible machines. What is very interesting is that these machines did not function, of course, as a perpetuum mobile , but anyway had their functioning and were useful for certain aims, although they were constructed on an idea which is completely wrong from a theoretical point of view. We mainly focus on the r enaissance and early modern period, but we also provide examples of machines built before and after this period. We have followed a chronological order in both parts, starting from the analysis of the situation in ancient Greece. t herefore, in the first part, we have examined the relations between the Aristotelian/Euclidean and Archimedean traditions from ancient Greece to the early modern age. i n this second part, we analyse the relations of Archimedean tradition/ construction of machines from ancient Greece to the 19 th century, focusing on the mentioned period. We remind the reader that our aim is to prove an epistemological thesis, not to provide a complete historical endeavour. As a correlated article, the reader will find three previous paragraphs in the first above-mentioned article (Pisano & Bussotti, 2014a). (shrink)

InCelebratingthisDIUMessayHistoriographywe describeIEMNTorricelli’s lifeLille Universityand worksUdine University in their scientific context, including the ArchimedeanArchimedean heritage in Torricelli’s works. We analyse the changes of Torricelli’s works and explain the novelties of the edition we are offering. Then, we provide a picture of the most significant results obtained by Torricelli (1608–1647), particularly in mechanicsMechanicsand geometryGeometry. Furthermore, we also focus on the Torricelli’s methodology, specifying how he provedProved two of his achievements, given their novelty and mathematical meaning: (a) the volumeVolume of the “solido acutissimo”; (...) (b) the geometryGeometryof theLogarithmic Spirallogarithmic spiralGeometric Spiral. AnContent ad hoc ad hocAd hocContentContent of the Opera geometricaOpera geometrica is proposed as well. It is based on the original chronological sequences of the contentsContent wrote by Torricelli. This is important to evaluate the birth of the scientific matters presented by Torricelli and the editorial process of the publicationPublication. (shrink)

Evangelista Torricelli exemplifies the use the moderns made of the ancients' mathematical methods. Celebrating Evangelista Torricelli's monumental Opera geometrica, this book marks 380 years since its publication (1644-2024). This homage to Torricelli introduces the magnificent major work in Mechanics and Mathematics of a brilliant Archimedean–and–Galilean scientist to modern readers. Opera geometrica deals with Motion & Mechanics and Geometry & Infinitesimals. In quibus Archimedis doctrina Torricelli also presents his mechanical principle of equilibrium – the foundation of the modern Principle of Virtual (...) Work/Static. This outstanding source and research book spotlights the relevance and originality of Torricelli’s Mechanics, and is the first and most profound analysis of the Opera geometrica to date. The historical study is achieved in extensive Introduction, 5 Essays and an accurate Transcription of Opera geometrica with parallel side–by–side text, including substantive explicative notes. The book is an accessible avenue to understanding this work by leading authorities who offer much-needed insights into the relationship Physics–Mathematics, Mechanics and Fundamentals. It appeals to historians, epistemologist and scientists. (shrink)