Make A Tree Swing Swing Straight

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.