Universidade de Lisboa Faculdade de Ciências Secção Autónoma de História e Filosofia da Ciência LUCA PACIOLI AND HIS 1500 BOOK DE VIRIBUS QUANTITATIS TIAGO WOLFRAM NUNES DOS SANTOS HIRTH Dissertação Mestrado em História e Filosofia das Ciências 2015 Universidade de Lisboa Faculdade de Ciências Secção Autónmoa de História e Filosofia das Ciências LUCA PACIOLI AND HIS 1500 BOOK DE VIRIBUS QUANTITATIS TIAGO WOLFRAM NUNES DOS SANTOS HIRTH Dissertação orientada pelo Prof. Jorge Nuno Oliveira Monteiro da Silva Mestrado em História e Filosofia das Ciências 2015 Abstract As the field grows, History of Science has become wider-ranging than a purely progress-oriented view of the history of Science. The History of Mathematics, even though more resilient, has shown to follow the same development. The present dissertation tries to contribute to the general study by shedding some light on a book which has been belittled, misinterpreted or ignored altogether, De Viribus Quantitatis, one of the major historical recreational mathematics books, and its author Luca Pacioli. This text aims to provide a modern updated survey of the content of this book for related studies, as well as a résumé of its contents. Keywords: De Viribus Quantitatis, Luca Pacioli, Recreational Mathematics, Popular Ciênce, History of Mathematics. Resumo Com o crescimento do ramo de História das Ciências este tem vindo a desenvolver um olhar mais abrangente que a clássica visão dedicada ao progresso das ideias científicas. A História da Matemática, embora mais resiliente, também tem vindo a mostrar interesse em expandir os seus horizontes. No presente texto tentamos contribuir para o estudo geral destas disciplinas estudando um tratado que pouca atenção tem tido até ao momento, sendo até mesmo mal interpretado. Trata-se De Viribus Quantitatis, sendo este um dos maiores compêndio de matemática recreativa no seu contexto histórico. O seu autor, Luca Pacioli, sendo uma personalidade de grande interesse e mais conhecido por outras obras suas. Nestas páginas tentamos fornecer uma versão atualisada da documentação relativa ao tratado tal como um resumo dos seus conteúdos. Palavras-chave: De Viribus Quantitatis, Luca Pacioli, Matemática Recreativa, Ciência Popular, História da Matemática. Index Introduction 8 Pacioli 10 A Historiographical sketch of Pacioli’s Biography 15 THE DE VIRIBUS QUANTITATIS 17 Description 17 Historiography DVQ 19 Structure and notation used 21 I. On the Powers of Quantity 23 Algebraic Tricks 23 Numerical Games 41 II. On the virtue and strength of Geometry 58 Geometric Constructions 59 Geometric Marvels 81 III. Other Documents 100 Natura Magistra 101 DE PROBLEMATIBUS ET ENIGMATA 121 CONCLUDING REMARKS 126 BIBLIOGRAPHY 135 8 Introduction De Viribus Quantitatis1 is a unique treatise. It is one of the first (if not the first) to gather multiple mathematical recreations, magical effects and “scientific” experiments, explaining and exposing these within the spirit of its age. It is the work of a mathematician and educator, but is very different to other textbooks written in its time, due to its dedication to recreations. These recreations can be found earlier in correspondence, literature or textbooks, mostly though individually or as interludes, and the sheer size of Pacioli’s work places it in the spotlight. Like most other works of Pacioli the content is mostly mathematical at heart, but also includes other natural sciences and even, at the very end, some literary entertainment. Its discourse is guided by praise of the underlying “power” of mathematics, in its algebraic and geometric form. Various elements such motivation, disclosure, communication and education of science figure in it. The book is certainly a milestone in what today one might call “popular science”. But not only the mathematic inclined individual or the recreational mathematics enthusiast will find interest in DVQ. Many well-known effects appear on paper for the first time. This makes the book popular among magicians. The book is even named a classic of Italian prestigitation by some and even in Portugal many practitioners of the art of illusion will have heard of it. The book holds many illusionist secrets and tricks of the trade teaching many different aptitudes. It also describes and explains some seemingly-miraculous effects. The recreations present in the book take little from the scientific and historic value of its content. It is very likely that a good deal of the book’s sections were used for motivation during classes or education of a general public, while others seem present for the pure pleasure of their effect like a few pranks in the latter part of the book. The author of the DVQ, Luca Pacioli, is a historical figure, known best for his de Summa arithmetica, geometria, proportioni et proportionalita2, a landmark historic textbook on Algebra. With it, Pacioli provides a very general and embracing content for the student of mathematics. The proportione: opera a tutti glingegni perspicaci e curiosi necessaria ove ciascun studioso di philosophia: prospectiva pictura, sculptura, architectura, musica e altre mathematice: suavissima sottile e admirabile doctrina consequeira: e delectarassi cõo varie questione de secretissima scientia3, is another commonly known work of his, unlike the Summa it discusses Geometry. The first treatise of this work focuses on the golden ratio giving the work its name. Although Pacioli’s works are mostly collections and lack major scientific improvement, he is by no means unoriginal, adding to most of his materials and generalizing them. Pacioli is also of great interest for his impact on scientific education and the transmission of science. Pacioli, however, is not too well known as yet, and much work is needed to get a comprehensive image of him. Jayawardene, in a review of the historiography until 19944, gives an eight point list of progress to be made to gain a greater 1 De Viribus Quantitatis shall be referred simply as MS for the Manuscript or DVQ. 2 This work has and will be referred to in short as Summa. 3 This work has and will be referred to in short as Divina. 4 Jaywardene, S.A.“Towards a Biography of Luca Pacioli”, in Luca Pacioli e la matemática del rinascimento. 9 comprehension on Pacioli. Number 2 of that list are the publication of Trattato d’Aritmética and De Viribus Quantitatis. So far studies regarding DVQ have mostly only been accessible to the Italian reading public. For such an ambitious task some editing will be needed and possibly a modernization of the notation. Any additional viewpoint would also be helpful, the book having been tackled so far mostly by mathematicians and illusionists. This dissertation envisions to aid in getting a better understanding of DVQ, by offering connections to other fields of study. Also, it tries to add to bring it into English. Further it is hoped that it may add to the field of History of Science and bring some new perspective to the topic. The content of this dissertation thus provides some analysis of DVQ’s contents in its more recreational mathematics aspects, modernizes notation and terminology, and addresses some of the issues surrounding the text. For a general understanding the DVQ a short annotated biographical chronology regarding Pacioli is given. Some of the major events in his life are listed together with some of his work. This is aided by a small historiographical sketch at the end, presenting some of the major names in the study of Pacioli. Taking as understood the introductory sketch of Pacioli the proper core of this study, the DVQ, is then tackled. To begin with some general remarks and description of the book are provided. A historiography of the book is given, followed by the description and some general structural remarks on the DVQ and notation used. Given this preliminary contextualization, the DVQ is analyzed section by section in three parts echoing those of the DVQ. Finally some concluding remarks are made regarding the book, its contents and several smaller aspects not included in the sectional analyses. 10 Pacioli Either because of the line of work and sometimes less orthodox style and content of his endeavours, or for other reasons, several controversies surround the author of DVQ. The first to is one about his name. Many different renditions have been used, for instance “Patioli”, “Paciuolo”, “Paccioli” or “Paciolus”, last is found in Latin forewords by the author himself. 5 In recent times, however, the Tuscan form “Pacioli” is commonly used, probably popularized by E. Taylor’s Royal Road.6 Within the church order of the Franciscans of the time it was not uncommon to drop the family name, keeping instead only given name and a place of origin. Thus it is no wonder to find the signature of “Lucas de Burgo” or variations of this by the same person. Luca Pacioli was born around 1445. His exact date of birth is unknown and like his name debated. In the Necrologium of the Cloister of Santa Croce of Florence his date of death at 70 is 1517. Lacking more documentation we cannot assume that his age was exact. He was born son of Bartolomeo. He was the nephew of Benedetto and had a brother named Piero, who had two children, Ambrogio and Siniperio. Little more is known about his family, and this knowledge is derived from two testaments, one from 9 of November 1508 and the other from 21 of November 1511. Borgo Of his childhood years little is known. These were spent in care of the Befolci family in Borgo Sansepolcro, now (May 2008 Census) a 16 thousand inhabitant town in the Tiber valley in the Arezzo province in Tuscany, Italy. It is asserted that in these years he had contact with Piero della Francesca (~1410 – 1492), fellow inhabitant of Borgo. Piero is often said to be one of Pacioli’s teachers or tutors.