Leonardo of Pisa

Italian mathematician

• Born: c. 1170
• Birthplace: Pisa (now in Italy)
• Died: c. 1240
• Place of death: Pisa (now in Italy)

Leonardo provided Western Europe with the earliest and most heralded Latin account of the Hindu-Arabic number system and its computational methods. He contributed substantially to the acceptance of the Arabic algebraic system and created a revolutionary mathematical technique known as the Fibonacci sequence.

Early Life

Leonardo of Pisa was born Leonardo Fibonacci, the surname meaning “son of Bonaccio.” Although very little is known about his life beyond the few facts gleaned from his mathematical writings, his father, Guglielmo, was a successful merchant who was the chief magistrate of the community of Pisan merchants in the North African port of Bugia (now Bejaïa, Algeria). As a young boy, he joined his father there and began the study of mathematics in this culturally diverse environment.

Desiring his son to be a successful merchant or commercial agent, Guglielmo sent Leonardo to study with a Muslim master who introduced him to the intricacies of Arabic mathematics , especially al-Khwārizmī’s Kītab al-jabr wa al-muqābalah (algebra; c. 820) and the practical value of the Hindu-Arabic numeral system represented by the nine Indian figures (1, 11, 111, 4, 5, 6, 7, 8, 9), the fourth through the ninth symbols representing the first letters of the Hindu names for these integers. As he grew older, he traveled around the Mediterranean area, especially Egypt, Syria, Greece, Sicily, and Provence; visited the dominant commercial centers; acquired knowledge of the arithmetical systems used by hundreds of merchants; and mastered the theoretical subtleties of Greek and Arabic mathematics, chiefly those of Plato of Tivoli, Savasorda, Euclid, Archimedes, Hero of Alexandria, and Diophantus. He even resided for a time at the court of Frederick II, the Holy Roman Emperor, where he engaged in scientific speculations with Frederick and his court philosophers, the most notable being Michael Scot, to whom Leonardo dedicated one of his works.

In his later years, Leonardo probably served Pisa administratively as an examiner of municipal accounts. This commercial expertise, however, was always secondary to his lifelong passion for mathematics.

Life’s Work

The rapid improvements that marked the history of Western mathematics in the thirteenth century, particularly in the fields of arithmetic and algebra, were largely a result of the genius of Leonardo, although a second mathematician of originality, Jordanus Nemorarius, made significant contributions, especially to the theory of numbers and to mechanics. Yet Jordanus showed no trace of Arabic influence. Developing the Greco-Roman arithmetical tradition of Nicomachus and Boethius, he habitually used letters for generalizing proofs in arithmetical problems, an awkward method that Leonardo could avoid because of his employment of Arabic numbers.

Leonardo’s pioneering achievements in mathematics began in 1202, when he wrote his first work, Liber abaci (English translation, 2002). Even though the title, which means “book of the abacus,” is a misnomer because Leonardo eschewed Roman numerals and the methods of the abacus, the work became the earliest in the West to extol the superiority of the nine-numeral Arabic system of numbers when used in conjunction with the zero. When the Liber abaci first appeared, Arabic numerals were known to only a few European philosophers through the Latin translations of al-Khwārizmī’s ninth century treatise. Leonardo understood fully the advantages of this system for mathematical operations. He displayed the system brilliantly in this edition and in a second, revised edition of 1228 dedicated to Michael Scot, the emperor’s chief scholar, and provided more rigorous demonstrations than in any previous or contemporary work. Leonardo realized that the great merit of this system was that it contained the symbol for zero and that any number could be represented simply by arranging digits in order, the value of a digit being shown by its distance from zero or from the first digit on the left.

Predominantly theoretical in nature, the Liber abaci was also valuable for its commercial arithmetic, which covered such operations as profit margins, barter, money changing, conversions of weights and measures, partnerships, and interest. After his death, Italian merchants generally adopted Leonardo’s Arabic system of numeration, and his book remained the standard in Europe for more than two centuries.

Besides popularizing a new system of numerals throughout the West, the Liber abaci was revolutionary for two reasons. First, it introduced Arabic algebra to European civilization. Leonardo’s algebra was rhetorical, but it was unique because of its employment of geometrical methods in its descriptions. He dealt primarily with the extraction of square and cube roots, progressions, indeterminate analysis (an equation with two or more unknowns for which the solution must be in rational numbers, whole numbers or common fractions), false assumptions (when a problem is worked out by incorrect data, then corrected by proportion), the rules of three and five (methods of finding proportions), the solution of equations of the third degree, and other algebraic and geometrical operations.

Of even greater importance was Leonardo’s famous sequence of numbers known before the nineteenth century as the Series of Lamé but now called correctly the Fibonacci sequence. In answer to the problem of how many pairs of rabbits could be produced from a single pair if each pair produced a new pair each month and every new pair became productive from the second month onward (supposing that no pair died), he devised the recurrent, or recursive, series of 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on. In this number sequence, in which the relation between two or more successive terms can be expressed by a formula, each term is equal to the sum of the two preceding ones. In the nineteenth century, the series proved of immense value in the study of divisibility, prime numbers, and Mersenne numbers. In the modern world, the Fibonacci sequence is used in botany for determining the patterns of natural growth.

In addition to the Liber abaci, Leonardo wrote three other significant works. In 1220, his Practica geometriae (practice of geometry) presented theorems based principally on two of Euclid’s works. It applied algebra to the solution of geometrical problems, a radically innovative technique for thirteenth century Europe. In 1225, two smaller works appeared, the Flos (prime) and the Liber quadratorum (The Book of Squares, 1987). More original than the Liber abaci, they were devoted to questions involving quadratic and cubic equations, in addition to several refinements to his earlier algebraic discourses. The Book of Squares may be considered Leonardo’s masterpiece. Although the Liber abaci made his reputation, The Book of Squares made him the most important contributor to number theory until Pierre de Fermat, the celebrated seventeenth century French mathematician who was instrumental in early experimentation aimed at determining the exact length of a quadrant of Earth’s meridian, the scientific basis of the metric system of weights and measures.

Significance

The impact of Leonardo on future generations was enormous. His pioneering achievements helped spread Arabic numeration and Arabic algebra throughout the West. Popular diffusion followed in the form of almanacs, calendars, and literary and poetic productions. Merchants accepted his new system the Italians first and other Europeans by the end of the sixteenth century. Even as early as the second half of the thirteenth century, lectures in the universities incorporated the new numbering system. His use of geometry in algebraic problems and, conversely, his use of algebra in solving geometric problems ushered in a new era in these disciplines. In time, the Fibonacci sequence revolutionized many divergent scientific fields. Last, aside from their scientific merit, his works were of tremendous cultural influence, particularly as they relate to metrology and to the major economic conditions of his time. In short, Leonardo was the greatest Christian mathematician of the Middle Ages. The mathematical renaissance in the West dates from him.

Bibliography

Crombie, A. C. Augustine to Galileo: The History of Science, A.D. 400-1650. 1953. Reprint. Cambridge, Mass.: Harvard University Press, 1979. Includes much discussion of Leonardo’s influence on mathematics and number theory prior to the scientific revolution.

Crombie, A. C. Medieval and Early Modern Science. 2 vols. 1959. Reprint. Cambridge, Mass.: Harvard University Press, 1961. Excellent descriptive bibliographies in both volumes, with coverage of Leonardo’s precursors, his impact on popularizing Arabic numerals, and his contributions to later medieval mathematics.

Fibonacci, Leonardo. Fibonacci’s “Liber abaci”: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. New York: Springer, 2002. Notes on the translation provide information on Leonardo and his work. Bibliography.

Gies, Joseph, and Frances Gies. Leonard of Pisa and the New Mathematics of the Middle Ages. Gainesville, Ga.: New Classics Library, 1969. Contains a summary of Leonardo’s life, a general survey of his works, and a brief overview of the history of numerical notation.

Hardy, G. H., and E. M. Wright. An Introduction to the Theory of Numbers. 1960. Reprint. Oxford, England: Oxford at the Clarendon Press, 1983. This volume provides a detailed account of the Fibonacci numbers and sequences and is meant for the mathematician or serious student of science.

Kibre, Pearl. Studies in Medieval Science: Alchemy, Astrology, Mathematics, and Medicine. London: Hambledon Press, 1984. Leonardo’s standing in the quadrivium (arithmetic, geometry, astronomy, and music) of later thirteenth century universities appears in the first of these republished articles.

Vorobiev, Nicolai N. Fibonacci Numbers. Boston: Birkhäuser Verlag, 2002. A mathematical work examining the Fibonacci numbers.