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Luca Pacioli and the Summa [other ideas for an article ...]

The figure of Luca Pacioli is certainly crucial in this century. His biography, by itself, allows us to understand many of the traits of Monaco Borgo Sansepolcro, born into a family of modest economic conditions around 1445.

's father was Bartolomeo Pacioli Luca Pacioli, but it seems that Luke did not live with his family, he was instead Sasepolcro, but with the family Befolci. As for Pacioli, the most important component of this market town was the study of Piero della Francesca, who spent much time at his studio in Sansepolcro, intent to follow the many orders received. The hypothesis that has received part of his education in the study of Piero della Francesca is very likely. One of the main reasons why you believe this is that Pacioli seemed to be very knowledgeable on the subject of the works of Piero della Francesca and Pacioli's works seem to be much influenced by his presence.

At the end of the studies conducted in the schools of abacus, Sansepolcro Pacioli left to move to Venice (1464) to be in the service of the wealthy merchant Antonio Rompiansi who lived nel quartiere malfamato della Giudecca. Bisogna supporre che dovesse certamente avere una buona educazione riguardo alla matematica di base grazie ai suoi studi a Sansepolcro e sicuramente doveva avere una buona educazione letteraria per essere stato assunto come tutore dei tre figli di Rompiansi. Ad ogni modo, durante il suo soggiorno a Venezia colse l'occasione per approfondire ulteriormente i suoi studi matematici avendo come maestro Domenico Bragadino. Durante questo periodo Pacioli prese confidenza sia con l'insegnamento, grazie al suo lavoro di tutore, sia con gli affari, grazie al suo ruolo nell'aiutare Rompiansi con i suoi commerci.

Fu proprio durante questo soggiorno a Venezia che Pacioli scrisse la sua prima opera: un libro di matematica arithmetic which he dedicated to his employer. This was completed in 1470, probably the year he died Rompiansi. After leaving Venice, to Rome as the guest of tarsferì Leone Battista Alberti, who was secretary in the Papal Court. Alberti was able to provide Pacioli many religious contacts. During this time Pacioli became a theologian and a few years later became a Franciscan friar in the order.

In 1447 Pacioli began a life of travel, spending time in universities to teach mathematics, especially arithmetic. I teach at the University of Perugia dall'1477 all'1480 and stayed there while writing his second book on arithmetic for the classes they teach. I teach in Zadar e durante il suo soggiorno scrisse la sua terza opera sull'aritmetica. Nessuno di questi libri fu pubblicato e solamente quello scritto per gli studenti a Perugia ci è arrivato. Dopo Zara, Pacioli insegnò alle università di Perugia, per una seconda volta, a Napoli e a Roma. Sicuramente Pacioli conobbe Federico di Montefeltro duca di Urbino durante questo periodo, a cui venne attribuito questo titolo dal papa Sisto IV nel 1474, e sembra che Pacioli abbia insegnato al figlio Guidobaldo di Montefeltro, l'ultimo regnante della famiglia Montefeltro dopo la morte del padre nel 1482. La corte di Urbino era un famoso centro culturale e Pacioli aveva contatti con gente importante per molti anni. Nel 1489, dopo due anni trascorsi a Roma, Pacioli tornò a Sansepolcro. However not all went for the best in his city. The Pope had allowed certain privileges and religious men of the small town were jealous of him. In fact, Pacioli was forbidden to teach there in 1491, but the jealousy seemed to be attached to respect for his wisdom and numeracy, and in 1493 he was invited to preach sermons in Lent.

During this period in Sansepolcro, Pacioli worked on one of his most famous: the Summa de Arithmetica, geometry, proportions proportional er which he dedicated to Guidobaldo then Duke of Urbino. Pacioli went to Venice in 1494 to publish the Summa. This book provides a summary of the mathematics known at that tempo anche se mostra poco per quanto riguarda idee originali. L'opera studia l'aritmetica, l'algebra, la geometria e la trigonometria e, nonostante mancasse di originalità, avrebbe fornito una base per il più grande progresso nella matematica che avvenne in Europa poco dopo la sua morte. Come detto nella Summa era:

… non indirizzato ad alcuna particolare parte della comunità. Un'opera enciclopedica scritta in italiano, contiene un trattato generale sull'aritmetica teorica e pratica; elementi di algebra; una tavola di monete, pesi, e misure usati nei vari stati italiani; un trattato sull'amministrazione delle entrate e delle uscite; e un riassunto della geometria euclidea. He freely admitted to using the concepts of Euclid, Boethius, Holy Boasco, Fibonacci ...

The geometric part of Pacioli's Summa is discussed in detail in Section 6:

The geometric part of the sum of Pacioli [Venice, 1494] written in Italian is one of the first books printed in mathematics. Pacioli broadly used Euclid's Elements some rewriting. Also referred to Leonardo of Pisa (Fibonacci).

Another interesting aspect of the Summa is the fact that he studied the play of chance. Pacioli studied the problem of points even if the solution he gave is incorrect.
Ludovico Sforza was the second son of Francesco Sforza, who had proclaimed himself Duke of Milan. When Francis died in 1466, the elder brother of Louis, Galeazzo Sforza became Duke of Milan. Later, Galeazzo was murdered in 1476 and her son seven years became Duke of Milan. Ludovico, after some political intrigue, became tutor to the young man in 1480. With a very generous patronage of artists and scholars, Ludovico Sforza set the stage to make his court in Milan the best in Europe. In 1482 Leonardo da Vinci was taken at the service of Ludovico as a painter and engineer of the court. In 1494 Ludwig became Duke of Milan, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics to his court. This call could have been done at the request of Leonardo who had an interest enthusiasm for mathematics .

At Milan Pacioli and Leonardo became fast friends. Art and mathematics were subjects on which they discussed long and hard, learning a lot each other. In recent years Pacioli began work on the second of his two most famous works: The Divine Proportione illustration which were designed by Leonardo, Pacioli's book on which he worked during 1497 was later to form the first of three books in 1509 that public with the title Divina Proportione. This was the first of three great books that made up the treated and studied the Divine Proportion also called the Golden Section, which is the proportion a: b = b (a + b). It contains the theorems of Euclid that have to do with this proportion, and also studies regular and semiregular polygons. Clearly, Leonardo's interest in this proportion to be aesthetically pleasing from the mathematical point of view that art was an important influence on the work. The golden section was also important in architectural design and this topic would have composed the second part of the Treaty Pacioli wrote later. The third book of the treatise is a translation into Italian of works by Piero della Francesca.
became King Louis XII of France in 1498 and, being a descendant of the first Duke of Milan, claimed the duchy. Louis argued against Milan and Venice in 1499 the French army entered Milan. In the following year Ludovico Sforza was captured in an attempt to regain the city. Leonardo and Pacioli fled together in December of 1499 three months after the capture of Milan. They stopped in Mantua, where they were guests of the Marchesa Isabella d'Este, and in March 1500 went to Venice, then returned to Florence, where the two shared a house.
The University of Pisa had been a revolt in 1494 and moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa to Florence in 1500. He stayed there until 1506. Leonardo, although past ten months out of the city working for Cesare Borgia, he also remained in Florence until 1506.

During his years in Florence, Pacioli was also involved in the affairs of the church. He was elected superior of his order in 1506 and entered the monastery of Santa Croce in Florence. When he left Florence, Pacioli went to Venice where they were granted rights to the publication of his works for the next fifteen years.

In 1510 Pacioli returned to Perugia to teach. He taught again in Rome in 1514 but had 70 years and was already close to his life of teaching and research. He returned to Sansepolcro where he died in 1517 without having published the great work De Viribus amanuensis geometric problems of leisure and proverbs. This work makes frequent references to Leonardo da Vinci who worked with him on this PROJECT, and many of the problems of this treaty are also in Leonardo's notes. Also in this work there is no originality, and the same Pacioli describes it as a compendium.

Despite the lack of original works of Pacioli his contribution to mathematics is particularly important because the influence of his books would last a long time. Pacioli in the importance of the work is discussed, in particular its approximate calculation of the value of the square root (using Newton's method), the incorrect analysis some games of chance (similar to those studied by Pascal), problems concerning the theory of numbers, and his collection of many square "magic". In 1550 was published a biography of Piero della Francesca, written by Giorgio Vasari. This biography accused Pacioli Pacioli of plagiarism and claimed that he had stolen the work of della Francesca on perspective, on arithmetic and geometry. This is an unjust accusation, because although there is some truth in saying that Pacioli took major inspiration from the work of others and certainly that of della Francesca in particular, he never attempted to proclaim His work of others but always quoted the sources he used.

Luca Pacioli is worthy of consideration in the history of mathematics, precisely because the route of "mathematization" of science and technology promoted in all the works of the friar from Sansepolcro and throughout his long career.

The cultural value of his work, however, should not be reduced to the listing of knowledge gathered ee subsequently used in the following centuries, but we must bear in mind the project for the dissemination of mathematics that innervates the tireless work of a teacher and , exerted by Pacioli in the course of its existence. The centrality of mathematics to human knowledge, real innovation in education and humanities in university curricula of the fifteenth century, depends, according to Pacioli, the necessity of the use of proportion in every field of human knowledge. In this context, the proportions are not only the universal language of science and technology, but also the basis on which the Creator has shaped the world.

In Pacioli's books can be traced the origin of the idea of \u200b\u200bmaking the queen of the sciences of mathematics and of making it the foundation of human knowledge, as will also be represented graphically in a clear and compelling by Nicolo Tartaglia on the front Nova's scent.

The idea of \u200b\u200bconsidering the proportions as the universal language of mathematics knowledge and image of the world that emerges from the Platonic cosmology revisited alal cvostituiranno light of divine proportion, in fact, two elements that often occur in the works of the main protagonists of the revolution sceientifica.

And we can not fail to mention the new consideration of the mechanical arts, techniques and crafts that emerges from the books. The math on the one hand provides technical rules to the other provides the scientific legitimacy of those rules, thanks to the logical proofs based on classical geometry (Euclid).

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mathematics in the fifteenth century [ideas for an article]

Abacist vs. Algorismist from Gregor Reisch, Margarita Philosophica
Strassbourg, 1504

In the cultural landscape of the fifteenth century it is customary to distinguish two cultural traditions, different in design and use of mathematical sciences applications.

On the one hand knowledge of the learned, cultivated and expressed in Latin or in the University or in the courts. On the other culture in the intermediate layers widespread literacy: artisans, merchants, architects, teachers of abacus, plumbers, cartographers, mechanics, artillery and other technicians, who used vulgar language and produced a substantial practical mathematical treatises.

The two worlds, however, remained essentially separate and not interconnected, even if there were major exchanges in various fields, through translations from one language to another and interesting matches and acquaintances. For example, some famous painters, cultural education within the tradition of practical and technical of shops and schools of abacus, found in humans and products useful and valid mathematical intermediaries. And 'this is the case of the collaboration of Piero della Francesca with the learned Sansepolcro who created the Latin versions of De Prospectiva dumping and Libellus de Quinque Corporibus Regularibus, oltrer with the drafting of a Latin version the works of Archimedes ¹ .

In general, the two worlds were separated, as evidenced by the fact that an artist geniale come Leonardo, si autodefiniva “omo sanza lettere”, ove con “lettere”  s'allude alle lettere greche e latine, riservate alla cultura colta.

Nelle università italiane del XV secolo erano attivi gli insegnamenti per la  realizzazione di quattro figure professionali: il maestro delle arti, il giurista, il medico  e il teologo. Le arti liberali del trivio (grammatica, retorica, dialettica) e del quadrivio  (aritmetica, geometria, astronomia, musica) erano soprattutto funzionli alla  formazione di medici e teologi, che were the most prominent figures, especially after the introduction of the works of Aristotle (Nicomachean Ethics, the Physica, De Anima, De coelo, De generation and Corruption, the Metaphysica) programs the trivium, enough to overshadow the subjects of the crossroads, grown perloppiù in medical schools, in close association with astrology and music. short, a very marginal role. also arose when groups of scholars who put the crisis in the philosophical system of Aristotle (Franciscans Robert Grosseteste and Roger Bacon reassess the role of geometry as a means of knowledge of nature; Bradwardine in Oxford took the proportions to show the contradictions of Aristotelian dynamics, in Paris Buridan and Nicholas of Oresme recuperarno geometry for the explanation of the theory of shock ) comunqque mathematics was in the service of philosophy, prejudice

crossroads of disciplines, however, are minor compared to those of the trivium.

Mathematics in Italian universities (Pisa, Bologna, Padova) was based on the study of texts summarizing and commenting on some Greek, Hellenistic and Arabic, translated into Latin (like institutione De Arithmetica of Boethius, the ball Sacrobosco, the Almagest, especially the first three books of Euclid's Elements) and some contemporary text from Paris and / or Oxford.

The feverish activity of translation from Arabic to Latin, which took place in Toledo, Salamanca and Barcelona, \u200b\u200bbut also in Sicily during the twelfth century had made available a number of works extremely vasta, ma il loro uso nella didattica  universitaria fu decisamente  istretto, per vari motivi ² . E ai classici si sostituirono dei  compendi medievali dotti.  Solo in epoca Umanistica, mediante l'instancabile opera di recupero dei codici greci,  si concentrarono in Italia gran parte dei classici matematici più importanti (di  tradizione greco-ellenistica e arabo-latina) soprattutto nelle biblioteche di Firenze,  Venezia, Roma e Urbino. Iniziarono nel XV secolo le traduzioni latine di Euclide e di  Archimedes, and studies directly from the Greek codices. The most important figure humanism is certainly the mathematician Regiomontanus that in addition to base modern trigonometry, presented a real publishing project for the revival of mathematics and astronomy.

parallel to the mathematics of the learned, in the '400 developed the tradition of abacus schools ³ and schedules, which was inspired more or less directly from the Liber Abaci ( 1202) of Leonardo Pisano, Fibonacci said, a true model of the manual applied mathematics, written in Latin, however, never entered into university curricula . From Liber Abaci merchants had a good tool with all the basics of algebraic to resolve issues concerning the exchange of currencies, barter, interest, discounts, weights and measures, metal alloys. This work constituted the "textbook" for shops abacus in central and northern Italy until the fifteenth century. In these schools (sometimes imposed by the same masters, sometimes by corporations, sometimes public) boys of 10-12 years were trained in algebra, arithmetic and geometry. The Treaty of Abaco by Piero della Francesca is a bit 'abnormal : Although it is written in the vernacular in order to expose "some reasons mercantesca commo baracti, Meite and

companies, of which it is composed of 128 cards, about 48 are devoted to geometry and to 56 ' algebra, showing great skill in drawing and, above all, direct conoscvenza of Euclid's Elements. It is likely di uno dei massimi  livelli della matematica abachistica.  Attraverso questi manuali e queste scuole anche i tecnici raggiunsero livelli notevoli  di abilità matematiche, su questioni ignorate nella matematica delle università.

Scienza e tecnica restavano separate da un fossato linguistico e sociale.

1 - “l'oratore, poeta e retorico, greco e latino (suo assiduo consotio, e similmente conterraneo) maestro Matheo la reccò

in lingua latina, ornatissimamente, de verbo ad verbum, con esquisiti vocabuli” (L. Pacioli, Summa de arithmetica

geometria proportioni et proportionalità, Venetiis, Paganino de' Paganini, 1494)

2 - Alcune motivazioni: la lenta circolazione dei manoscritti; l'incertezza delle versioni, realizzate spesso da persone poco competenti; il tramite arabo, dotato di ben altri strumenti matematici rispetto a quelli della cultura latina medievale; la scarsa importanza del quadrivio.

3 - Il Medioevo aveva ereditato dal mondo antico una tavoletta di legno con i bordi rilevati e otto bacchette with movable balls used in the calculations: the abacus. But treaties abacus does not presuppose the use of this instrument were taught to solve business problems through cacloli carried out on paper.