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Fibonacci's Liber Abaci Fibonacci's Liber Abaci A Translation into Modern English of Leonardo Pisano's Book of Calculation Translated with Notes by Laurence Sigler Fibonacci (Leonardo of Pisa) - Liber abaci 2002 Springer Introduction 4 I. Introduction by Laurence Sigler Liber abaci is one of the most important books on mathematics of the or marble on which were engraved lines. On the lines were manipulated small Middle Ages. Its effect was enormous in disseminating the Hindu number sys­ counters of stone. Another form used dust or powder on the table on which tem and the methods of algebra throughout Europe. This is the first translation marks were made with the finger. During the seventeenth century both Blaise of the Latin manuscript of Liber abaci into a modern language. It is hoped Pascal and Gottfried Leibniz designed mechanical calculating machines. Today that its availability to historians, mathematicians, and the public in general will we have electronic calculators and elaborate computers to assist us with our make a contribution to their knowledge of this part of our cultural heritage. calculations. The inexpensive electronic hand calculator is the abacus of today. Mathematics and science are, after all, as much a part of our culture as litera­ The Hindus and Arabs utilized written numbers with a place system and ture, art, and music. It is as important for a person to know about the classics methods for the basic operations that did not require the abacus. Roman nu­ of mathematics and science as it is to know about the classics of literature and merals and other similar systems of writing numbers did not facilitate calcula­ art. tion. The calculations were done with the abacus and the answers were written Leonardo Pisano, known today to mathematicians and scientists over the down in Roman numerals. The Hindu numerals with the place system are ac­ world by the name Fibonacci, was a citizen of the maritime city-state of Pisa tually used both to make the calculation and to write down the result. These from 1170 until after 1240. This was the time of the Crusades, of strong political are the procedures that children are taught in school when they learn to do addition, multiplication, subtraction, and division with pencil and paper. In Fibonacci conflicts between the Emperor Frederick II of the Holy Roman Empire and the Papacy; it was also the time of the religious fervor of St. Francis of Assisi. the Middle Ages in Europe these new written procedures were called algorithms The Italian maritime states of Pisa, Genoa, Venice and Amalfi were locked in to differentiate them from calculating with the abacus. Leonardo teaches these intense trade rivalry throughout the Mediterranean world, including Byzantium procedures in this book Liber abaci. These written procedures of calculation, (Leonardo and the Muslim countries. algebra, and practical mathematics in general were known in Italy in the Middle Ages as abaca. Leonardo was instructed in mathematics as a youth in Bugia, a trading Liber abaci, or the Book of Calculation, appeared first in 1202, and then enclave established by the city of Pisa and located on the Barbary Coast of again in a second version in 1228. Leonardo's stated intention was to introduce Africa in the Western Muslim Empire. He continued to develop as a mathe­ the Hindu number system and its operations to the Italian people. However, of matician by traveling on business and studying in such places as Egypt, Syria, Pisa) Liber abaci is much more than merely an introduction to the Hindu number Provence, and Byzantium. He developed contacts with scientists throughout system and the algorithms for working with it. Liber abaci is an encyclope­ the Mediterranean world. He became proficient in Euclid's Elements, and the dic work treating much of the known mathematics of the thirteenth century on - Greek mathematical method of definition, theorem, and proof. He learned from Liber arithmetic, algebra, and problem solving. It is, moreover, a theoretical as well the Arabic scientists the Hindu numbers and their place system, and the al­ as practical work; the methods employed in Liber abaci Leonardo firmly estab­ gorithms for the arithmetic operations. He also learned the method of algebra abaci lishes with Euclidean geometric proofs. One must not be misled by the lack of principally found in the work of al-KhwarizmI [K]. Through his study and travel modern mathematical symbolism into thinking that this work is not excellent and learned disputations with world scientists, he became a very superior cre­ or rigorous mathematics. One does not judge the quality of mathematics by ative mathematician. He participated in the academic court of Frederick II who the symbolism with which it is written. Liber abaci was good mathematics sought out and recognized great scholars of the thirteenth century. Leonardo when it was written and it is good mathematics today. Liber abaci is a se­ with his scientific knowledge saw clearly the advantages of the useful mathe­ rious mathematical work written on arithmetic and applied mathematics by a matics known to the Muslim scientists, principally their Hindu numerals and superior creative mathematician. decimal place system, their calculating algorithms, and their algebra. Knowl­ One should here again make the point, that while derived from the word aba­ edge of the Hindu numerals began to reach Europe in the second half of the cus the word abaci refers in the thirteenth century paradoxically to calculation tenth century through the Arabs by way of Spain, however their usage was still without the abacus. Thus Liber abaci should not be translated as The Book not a general practice at Leonardo's time. Leonardo resolved to write his en­ of the Abacus. A maestro d'abbaca was a person who calculated directly with cyclopedic work, Liber abaci, to bring to the Italian people the world's best Hindu numerals without using the abacus, and abaca is the discipline of doing mathematics in a usable form. this. It was Leonardo's purpose to replace Roman numerals with the Hindu Calculation has been an activity of mankind since ancient times. It was numerals not only among scientists, but in commerce and among the common facilitated by various mechanical devices that by Greek and Roman times had people. He achieved this goal perhaps more than he ever dreamed. Italian mer­ developed into the abacus. The best known form consists of a wooden frame chants carried the new mathematics and its methods wherever they went in the strung with wires on which are mounted beads for counters. The efficiency of Mediterranean world. The new mathematics also spread into Germany where this abacus is attested to by its survival and use in some parts of the world even it was propagated by the cossists (a corruption of the Italian cosa, or thing, the today. There were also early forms of the abacus consisting of tables of wood unknown of algebra). 1. Introduction 5 6 1. Introduction For three centuries or so a curriculum based upon Leonardo's Liber abaci place counts only for itself, but the figure in the second place to the left counts as was taught in Tuscany in schools of abaco normally attended by boys intending so many tens. In sequence the third place from the right is valued in hundreds, to be merchants or by others desiring to learn mathematics. Other instructors the fourth in thousands, and so forth. The zero or zephir as Leonardo calls it and some very good mathematicians also wrote books of abaco for use in the counts for nothing and serves as a place holder. Large numbers are organized school. These books vary from primitive rule manuals up to mathematics books by triples to facilitate reading. Accustomed as we are to the use of our decimal of quality, but none was so comprehensive, theoretical, and excellent as the system and our algorithms for addition and the other operations, it is easy to Liber abaci of Leonardo Pisano. overlook that for Europe in the thirteenth century this book brought a new and Leonardo Pisano wrote other books on mathematics: Liber quadratorum revolutionary way to do arithmetic. (1225), Practica geometriae (1223), Flos and Epistola ad Magistrum Theodo­ Leonardo supplements the written numbers with a system of remembering rum (1225). It is his Liber quadratorum, or The Book of Squares lSi]' that numbers by using various finger positions in the hands. When Leonardo says offers best testimony to his power as a mathematician. This work can be said that a number is kept in the hand, he means it literally. This mediaeval memory to stand between the work of Diophantus and the work of Pierre Fermat in the system of keeping numbers in hand was widely used, but has fallen out of theory of numbers. It demonstrates Leonardo's power as a creative mathemati­ use today. This holding of numbers in the hands allowed one to perform the computational algorithms more efficiently and with less writing. Today we Fibonacci cian. Liber abaci is an impressive work on arithmetic, algebra, and applied math­ make small notations with pencil or pen of numbers to be carried or borrowed ematics based upon the theoretical foundation of Euclid's mathematics. Gen­ or we simply remember them as we perform the calculations. Addition and eral methods are established by using the geometric algebra found principally in multiplication of small numbers are presented with tables to be memorized by (Leonardo Book II of the Elements. Leonardo turns to Book X for a foundation of a theory the learner, just as children do today. of quadratic irrational numbers. Throughout Liber abaci proofs are given for In chapter 2 an algorithm for multiplication is given beginning with num­ old methods, methods acquired from the Arabic world, and for methods that are bers of two places by numbers of two places, and numbers of one place by those Leonardo's original contributions.
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