44 research outputs found
L'isola di Laputa può volare?
No full textL'articolo presenta una proposta di lavoro interdisciplinare pensato per studenti di Scuola Secondaria di Secondo Grado. Si richiede la lettura di un testo di letteratura inglese, una ricerca sulle fonti storiche dello studio dei fenomeni magnetici e infine la valutazione quantitativa delle grandezze coinvolti nei fenomeni, alla luce delle conoscenze attual
ASTRONOMICAL PROJECTS FOR HIGH SCHOOL STUDENTS
No full textAstronomy is strictly related to mathematics but in Italian high schools is not very studied so therefore we decided to promote the study with interdisciplinary projects done with the help of the INAF of Merate (Italy).
The first project was done in the 2008/09 school year because the General Assembly of the United Nations proclaimed 2009 the International Year of Astronomy (IYA2009) since it was the fourth centenary of the publication of Kepler's first two laws of planetary motion in the Astronomia Nova and the first astronomical observations with the telescope by Galileo in Padua. The purpose of the first project was to replicate the Ptolemy’s geocentric model, supposing that the astronomer was a scholar of other planet in the solar system: Marptolemaeus, an hypothetical Martian astronomer, Indeed, as the earthly Ptolemaeus decided to put his own planet, the Earth, as point of reference, the same would do an astronomer born on Mars.
The students have chosen Mars because it is the planet most similar to the Earth.
The second project was intended to calculate the mass of Jupiter because the calculation of the mass of Jupiter is therefore essential to study the trajectory of the probes. Estimates of the mass of Jupiter were also made (1973) with the use of Pioneer probes.
In order to calculate the mass, the students have chosen to observe the system constituted by Jupiter and the four Galilean satellites: Io, Europa, Ganimede, Callisto, in the annual time frame in which Jupiter is visible.
The students acquired and analyzed 792 digital images of the Jupiter-Galileian satellite system, of the Moon and the Pleiades.
The third project as the aim to complete the mathematical and geometrical planning as well as the construction of a fully working sundial, equipped with a solar calendar. It has also been necessary to choose the most suitable kind of sundial, taking account of its future location: finally, due to some technical needs (such as the difficulty in drawing all the necessary hour-lines with precision on the surface of a spherical sundial) a horizontal one has been decided. The position of the hour-lines and date-lines has been calculated and laid out through the application of some theorems about spherical trigonometry in order to sort out a spatial geometry problem. An important part of the project consists in planning a spreadsheet (with the application Microsoft Excel) which calculates the equations of hour-lines and date-lines for a sundial working in Central Europe
Maria Gaetana Agnesi: suggestions from the past to new way yo teach calculus
No full textIn 2018 we celebrated the three hundredth anniversary of the birth of Maria Gaetana Agnesi, mathematician and benefactress born in Milan (Italy). We have examined the Analytical Institutions,the main work of Maria Gaetana:, that she dedicated to students education. We think that pre-university students can acquire the fundamental mathematical ideas in Differential Calculus using methods and ideas proposed in the books that go back to the origins of the Analysis. From this point of view, we can use many suggestions and examples, contained in Agnesi’s Books. [1] We can propose these arguments or problems by means of laboratory instruments, flipped classroom techniques, or by didactical methods and we think that the media are different but the meaning is the same. According to some scholars, the abstract approach, nowadays we use, can be didactically not very effective for the beginner student. They think that an intuition of infinitesimals can be oriented to lead to mathematical concepts. From this point of view, we present the way in which Maria Gaetana Agnesi presented the cycloid, a traditional curve that nearly every mathematician used as example for demonstrating the power of the differential calculus techniques
Luca Pacioli: letters from Venice
No full textIntroduction
In 1508, Luca Pacioli was the most famous Italian mathematician and Venice was at the height of his power. Born in Borgo San Sepolcro, Luca always had a special relationship with Venice. Between 1464-1470, he was at the service of the merchant Antonio Rompiasi and completed his mathematical studies. After becoming Franciscan friar, he began a life of travelling, staying at the major Italian Courts, but he returned several times to Venice where he published his principal books: the Summa, the De Divina Proportione and the Euclid’s Fifteen Books.
In 1508 he made the inaugural lesson of the Accademia di Rialto. He focused on Elements V in which Euclid develops the theory of proportions, foundation of Pacioli’s mathematical and philosophical program. Among all applications, cited in De Divina Proportione, we are interested to the woodcuts in which Luca builds his Alphabeto dignissimo antiquo, and we analyse the characters’ constructive principles.
The Text
In 1508 Pacioli made the inaugural lesson of the Accademia di Rialto to a large gathering of theologians, philosophers, physicians, scholars, artists, architects and illustrious Venetians. He spoke about the value and applications of the proportion and proportionality. The text of this lecture was published as introduction to the Pacioli’s edition of the Euclid’s Elements that a few months later he edited in Venice. The theme of the lecture was fashionable for the period, being the Renaissance scholars interested in scientific and mathematical works and having the lowest cost of books extended learning to a broader community of people.
A summary of Euclid was appeared already in Summa, the most famous book of Pacioli, written in vernacular language and edited in 1494. The Summa is a review of the whole of known mathematics covering arithmetic, trigonometry, algebra, tables of moneys, weights and measures and a large number of merchants’ problems. It contains a summary of Euclid’s geometry, in particular excerpts from Elements V. Pacioli transformed the material in a way suitable for a public with practical but only modest theoretical interests; there is no clear separation between definitions and enunciations and proofs are mostly replaced by explanations with reference to diagrams. Probably the Summa is a summary of the lessons that Pacioli delivered at the Universities of Perugia, Naples, and Rome.
In the 1508 inaugural lecture Pacioli explained his program to understand the universe through mathematics in general and through the theory of proportions in particular.
A few months later, Pacioli published De Divina Proportione, where in addition to the arguments of the Summa he states that the universe is written in mathematical language. This sentence must be understood in a very real sense, since the five elements, which explain the nature and complexity of all matter, are made by the so-called "platonic" solids.Therefore, the reality is understandably and the world is ordered according to clear parameters. The order induces harmony and beauty; the beauty is the right and the right is generated by God. The Universe is organically organised; the law that organises it is the proportionality. The proportion is not only geometric figure or quantitative ratio: it is Divine Proportion, because the world was created by God. Man can understand the order of the Universe through mathematics and this is not only a scientific path, but above all ethical.
De Divina Proportione comprises three independent works. At the beginning, Pacioli places the Compendium de divina proportione, the book about the Golden Section. Pacioli added a small Tractato de l’architectura. He states that he wrote it at the request of some “respectable masons, most diligent friends of sculpture and architecture”, and calls them his disciples. He promises them “norms and methods of arriving at the desired effect in architecture” [5].
The role of the Tractato is very important and alone justifies the composition of the book. Pacioli’s connection with architecture dates from his permanence in Rome, as a guest of Leon Battista Alberti. Later in Urbino he meets Francesco di Giorgio Martini and Bramante and in Milan (1494-1499) he collaborates with Leonardo da Vinci. The text of De Divina Proportione clearly depends on the close collaboration of these Renaissance scholars. The interest of Leonardo in mathematical aspects and his artistic point of view had an important influence on the book. Leonardo himself draws the geometrical illustrations for the manuscript.
The Tractato is based on Vitruvius’ book and is probably a translation into vernacular of Piero della Francesca’s Libellus de quinque corporibus regularibus. The work begins with a discussion on the proportions of the human body, in which Pacioli inserts the side profile of the head. The human body serves as example of perfect proportions, but also as a concrete model. Pacioli understands the figure of the man in circle and square; a third geometrical form, the equilateral triangle, drawn in a profile of a head, with some gridlines lacking, introduces the illustrations at the end of the book [1].
In the Tractato, Luca also fits the tables with the construction of the capital letters of the alphabet with the compass and the straightedge. His construction is based on the same square and circle construction that had guided his predecessors, but he inserted some fundamental differences.
Pacioli is a little less dogmatic of his predecessors, he used the thickness of the strokes to somewhat ease the distortions involved in fitting letters into a perfectly square scaffold [9]. His letters are calligraphic rather than epigraphic: we notice, for example, the unusual choice to make shorter the middle bar of the E. Pacioli does not offer complete hints as to how the geometric construction was applied to artistic lettering. Probably Pacioli thinks that geometric construction concept is so clear to readers that he does not need further explanations and that the type-cutters can follow their eyes and their judgment according to their own taste. The last supposition is supported by the caption with which he ends the discussion on the perfection of the two O and concludes: “you can take which you like, and form from it, as you will find set out in its place” [3].
The stylistic choice of Luca of the proportion of the Corinthian column indicates the choice of the thickness of the letter I that is similar to a small column. The ratio 1:9 was established for the construction of all letters, instead of the classical 1:10. Pacioli does not consider necessary to justify it. Maybe he adopts such a proportional relationship according to the tradition of Byzantine artistic model, establishing the height dimension of the human body in nine faces [6], although he suggests to the architects the Vitruvian proportions. In the design scheme of the head’s profile, the presence of the equilateral triangle is fundamental to the construction of the figure and justifies the choice of the Ternary subdivision [7].
Simple fractions of the square’s side determine the linear dimensions of each letter. In particular, in B the ternary division is evident for the horizontal side of the square; the vertical side is divided into two parts, whose ratio is 5/4. The choice of the lowered position of the middle limb of A and the drawing of M, with inclined external strokes from the vertical, suggests a relationship with Leonardo’s study on the centre of gravity of the human body.
Pacioli presents two different drawings of O; in both he engages with the construction of ovals. In the Middle Age, masons use build oval form fitting any measure by trial-and-error adjustment, in Renaissance we find the first published oval layouts. Serlio [8] solves the problem of laying out a surbased arch and explains it by an affine transformation of the circumscribed and inscribed circumferences [4]. In Codex Atlanticus Leonardo drew ovals by stretching a circle.
In the first O, Pacioli probably draws two ovals with orthogonal axes, using the method now called “by polycentric curves”, in the second he draws one oval with ten circles, using the method of the circumscribed and inscribed circumferences.
Many scholars accused Pacioli of plagiarism against Leonardo about Alphabet. The typographer and editor Goudy attributed to Leonardo two tables (C.L. Ricketts collection) in his view very similar to those of Pacioli. Documents do not support the ascription, but, even if that were the case, according to some experts of Leonardo, the similarity does not exist [10].
Leonardo’s notebooks contain no drawings of letters; only one example is in the scrolled motto on the reverse of portrait of Ginevra de’ Benci (1474). It is hard to believe that, twenty-two years old, Leonardo could have developed a lettering’s theory and, in any case, the similarity is not convincing.
A relationship could be seen in the double portrait of Pacioli (1495), but only if you accept attribution and meaning of inscription, both questionable [2].
Conclusion
Despite the lack of originality, Pacioli’s contributions to mathematics are important, particularly because of the influence that his books had over a long period. He is the prototype of the modern scientist and modern populariser; more than any other he is aware of the potential that the press and vernacular language provide for the possibility of enhancing the dissemination of their works and ideas to a wider community. Since 1550 when Giorgio Vasari wrote a biography of Piero della Francesca, many scholars accused Pacioli of plagiarism and many others defended. In our opinion, this is an unfair accusation: Pacioli relied heavily on the work of others, but he never claimed the work as his own. In particular, we affirm that the Pacioli’s alphabetic tables reveal his philosophy of the language and his cultural skills, so, in our opinion, the hypothesis of plagiarism must be rejected for artistic and cultural reasons
The 18th International Conference on Geometry and Graphics
No full textAbstract. In 1496 Fra Pacioli’s fame led to an invitation to join the court of Ludovico Sforza in Milan. Here Luca met Leonardo da Vinci and taught Leo- nardo the intricacies of geometry and in the meantime Leonardo informed Pacioli of the application of geometry to art and Architecture. The text of De Divina Proportione clearly depended on the close collaboration of these two Renaissance scholars. Leonardo himself drew the geometrical illustrations for the manuscript. In 1509 Pacioli published De Divina Proportione, integrated with the Tractato del’architectura, that begins with a discussion on the pro- portions of the human body. In this edition, Luca fits the tables with the con- struction of the capital letters. Pacioli’s alphabet is based on the same square and circle construction that had guided Leon Battista Alberti. We felt that this beautiful alphabet needed to be restored and we set out to construct an accurate replica with GeoGebra, using the Pacioli’s instructions although it was incomplete
Il tiro alla fune: un progetto fra matematica e sport
No full textIl tiro alla fune è uno sport che ha origine dai rituali praticati da antiche popolazioni ed ora è uno sport a tutti gli effetti. In questo articolo presentiamo un progetto che è stato proposto a un gruppo di studenti di Scuola Secondaria Superiore di Secondo Grado. Gli studenti, con l’aiuto dei loro docenti di matematica e fisica, hanno analizzato la fisica che sta alla base del tiro alla fune e poi hanno prodotto un modello. Dopo alcune lezioni introduttive, gli studenti hanno lavorato in modo indipendente producendo risultati che poi sono stati discussi con noi. Hanno poi proposto una simulazione nel laboratorio scolastico usando dapprima Excel e poi il programma di grafica LÖVE. Gli esperimenti sono stati effettuati nella palestra scolastica. Gli studenti hanno chiesto anche la collaborazione di alcuni giocatori di tiro alla fun
