Luca Pacioli: letters from Venice

Introduction 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.