**Tags**

Our tenth A&S Research Paper comes to us from Lady Rosina von Schaffhausen, of the Shire of Quintavia. She introduces us to a fascinating figure from the 13^{th} century – the mathematician Leonardo of Pisa, known most familiarly to us as Fibonacci. (Prospective future contributors, please check out our original Call for Papers.)

**Fibonacci – A Master By Any Name**

Imagine being an Italian merchant in the early 13^{th} century, traveling around the Mediterranean. You visit fascinating places, eat new and unusual foods, see many exotic sights, and trade many of the goods passing through the region.

However, you have a problem. The basic addition and subtraction you need to do to keep your account books you can handle, using the tools you have available, Roman numerals and an abacus. But doing any sort of multiplication or division is difficult. And you need to multiply, or divide, or sometimes both, to do all sorts of important things. You need them to determine how much cinnamon your pepper is worth, how much of your profits each of your investors should receive, how much your cut is, to calculate currency exchange, and to determine how much interest you have earned on the loan your city forced you to give them to build their navy. The methods you know seem much more difficult to deal with than the Arab merchants’ system.

On your next stop at home, a friend is raving about the new system of Hindu reckoning in a book written by one of your compatriots, and you resolve to find a copy and learn this new system…

**Contents**

Leonardo’s Pisa

Leonardo’s Life

Leonardo’s Work

Teaching Europe Math

Blazing New Trails

Conclusion

Bibliography

As commerce began expanding in the Middle Ages in the Mediterranean, the Italian city-states vied for control of lucrative trade routes. Throwing off the control of the Holy Roman Empire, the northern cities began running their own governments and conducting trade. They built navies and were the first governments to go into debt to finance their expenditures, sometimes by forcing their citizens to loan them money. They established customs houses at home and in a number of ports abroad, as well as forming the first corporations. Many of the cities coined their own money, and the first banks started offering interest on deposits.

While the first universities were being founded at this time, they did not add algebra to their curricula until the middle of the 16^{th} century [3, p. 107]. It was the needs of the merchants that drove the eventual revolution in European mathematics. The need for better and easier bookkeeping, computation, and problem-solving methods brought the Hindu-Arabic number system and Arabic algebra into use throughout Europe. Leonardo of Pisa, today known as Fibonacci, was a pivotal figure in the process of changing over from Roman numerals to the system we use today. The practical example problems he included in his best known work, *Liber Abbaci*, displayed the ability of the Hindu-Arabic number system and algebra to solve many of the pressing problems of the medieval merchants.

Fibonacci is best known today for his famous sequence, 1, 2, 3, 5, 8, 13, 21, … where successive numbers are found by adding the previous two together. However, this sequence is not original to him. Since the concept of plagiarism did not exist in the form it does today, many of Fibonacci’s example problems, including this sequence, can be found in identical or similar versions in earlier texts from the Islamic world, India, and even China. The main purpose of Fibonacci’s writing was to educate merchants and surveyors on techniques that were largely unknown or forgotten in Europe. Thus he was influential in bringing in our modern number system, more so than other scholars of his day. In addition, he wrote some impressive original mathematics, which was unheard of in Western Europe at the time. Most of all, he left legacies of education and research in mathematics that lasted long after his death. While today we remember one sequence, in the middle ages and Renaissance, many math texts acknowledged a large debt to one Leonardo of Pisa.

Leonardo of Pisa, known today as Fibonacci, was born in Pisa around 1170. Around this time Pisa was a republic, one of many Italian city-states vying for control of trade in the Mediterranean. At sea, Pisa contended with the other maritime republics of Genoa, Amalfi, and Venice. On land, their main rivals were Florence and Lucca. Pisa is situated on the Arno River on the western coast of the Italian Peninsula, just south of the top of the “boot” of Italy. From Roman times until the 15^{th} century, when the river silted up and Pisa lost its port, Pisa was a prime commercial center in Tuscany.

While other cities mostly concentrated on going east toward Constantinople and the Holy Land for their trade, at first Pisa mostly went west. The city established trading ports, conquered towns, and took over islands in places such as Corsica, Sardinia, and Carthage. During the First Crusade in 1099 Pisa was instrumental in the campaign, and used the opportunity to establish trading centers in the eastern Mediterranean. While other states did the same, using this advantage Pisa became a serious international power. For a time Pisa surpassed Venice as the foremost merchant and military ally of the Byzantine Empire.

At home, Pisa used her spoils to begin building a beautiful city center, including a cathedral and baptistery. As a very young child, Leonardo may have watched the first three floors of the famous bell tower being built and start to lean. Construction on the Leaning Tower, begun in 1173, was stopped in 1178 as the structure, built on unstable ground, began its famous tilt. The tower stayed at that height throughout Leonardo’s lifetime, as building would not resume for nearly 100 years and took nearly 200 to complete. In 1284, the Pisan navy was nearly completely destroyed by Genoa at the Battle Meloria, ending Pisa as a naval power. The city managed to keep up some independent trade until she came under the rule of Florence in 1406. Then the river silted up, ending Pisa’s ability to trade easily.

Fibonacci signs his name in *Liber Abbaci* as Leonardo of Pisa of the Family Bonaci, which might also be translated as Son of Bonaci. However, since his father’s name was Guglielmo, this translation is incorrect. Some scholars think that Bonaci or Bonacio was the name of an ancestor, as a reference to a famous ancestor was a common practice in Italy at this time. The name Fibonacci was first used in the 19^{th} century and has become the most common one used for him today. He signed *Liber Quadratorum* as Leonardo Pisano, or Leonardo of Pisa. However, in his later work *Flos* (1225) and in a legal document (1241) [3, p. 149] his name is listed as Leonardo Pisano Bigollo. Some scholars think Bigollo means traveler, good-for-nothing, or absent-minded, while others think these translations are incorrect. Considering that the legal document is honoring Leonardo, Bigollo could not possibly have been meant as an insult.

Most of what we know about Leonardo’s life comes from the introduction to his book *Liber Abbaci*. When Leonardo was a boy, his father Guglielmo was working in the customs house of the Pisan trading port of Bugia (now Bejaïa) on the Barbary coast of Africa, east of Algiers. Guglielmo had young Leonardo sent over to Africa, most likely just after finishing grammar school, so the boy could learn mathematics. This was a trading center in a Muslim region, so Leonardo learned as much as he could from scholars visiting from many places. Modern scholars believe he learned to read and possibly write Arabic [5, pp. xviii-xx]. He studied a variety of mathematical systems and methods, and decided that the “Indian method”, as he called it, was superior to the rest. Leonardo continued his studies on business trips around the Mediterranean, including Egypt, Syria, Greece, Sicily, and Provence. He later wrote his first book, *Liber Abbaci*, or *Book of Calculation*, in 1202 to spread this method, which is our modern number system [6, p. 15-16].

Leonardo’s first book got him noticed not just in Pisa, but also by the Holy Roman Emperor Frederick II. At this time Pisa was nominally part of the Holy Roman Empire, but had begun governing itself even before a previous emperor permitted the city to function under its own governance. Frederick II was known as the Wonder of the World. At this time, the first universities were still very new, and many scholars worked for wealthy patrons. Frederick encouraged scholarship in his court, and even wrote a treatise himself. Leonardo was introduced to Frederick’s court likely around 1225. Some scholars in the court posed Leonardo some challenge problems, another common practice of the day, used to determine the abilities of scholars. Leonardo was able to solve the challenges posed to him, and his writing on the solutions and related mathematics encompass a large portion of his original mathematics. In the 1220s Leonardo wrote most of his other works, including those on the challenge questions. He wrote a revision of his first book *Liber Abacci*, dedicating it to the scholar who first wrote about him to the court, Michael Scott. In 1241, at the end of his life, Leonardo was presented with an annual stipend from the city of Pisa [3, pp. 148-149] for his contributions to the city.

Leonardo wrote a number of impressive texts in the course of his career. The first and foremost is *Liber Abbaci* (1202; 1228), mentioned above [6], a compilation of arithmetic and the algebra that was known in his day. Next Leonardo wrote *De Practica Geometriae* (1220, meaning Practical Geometry), a book on a variety of geometry problems including practical ones on land area and surveying [5]. *Liber Quadratorum* (1225, meaning Book of Squares), on advanced algebra and number theory, contains some impressive original mathematics [4]. These three are available in English translation. *Flos* (1225, meaning flower) and *Epistola ad Magistrum Theodorum* (date unknown, meaning letter to Master Theodore), both on indeterminate equations like those in *Liber Quadratorum*, have not yet been translated into English. Two of Fibonacci’s works have been lost. One is a tract on Book X of Euclid’s Elements. The other is *Libro di minor guisa* (date unknown, meaning probably The Book of the Lesser Method) on commercial arithmetic. We will go into more details later on some of these books.

Fibonacci’s impressive texts come in two varieties. The first are texts designed to explain mathematics to the common person, as well as to show its usefulness. *De Practica Geometrie*, *Liber Abbaci* and *Libro di minor guisa* fall into this category. *Liber Abaci* is an encyclopedic work which, together with Euclid’s *Elements*, contains most of the mathematics known in the world at that time. The other two texts are shorter and focused more on application. The second group of texts contain Leonardo’s original mathematics. *Flos*, *Liber Quadratorum*, and *Epistola ad Magistrum Theodorum* contain his solutions to problems posed to him by masters in the court of the Holy Roman Emperor Frederick II. In addition, in these books Leonardo expanded on the challenge questions and wrote original proofs of some ancient Greek knowledge as well as general solution methods for the types of problems posed. The tract on Book X of Euclid’s Elements was likely expanded from a chapter in *Liber Abbaci*. Leonardo’s popular works are more extensive than other European texts at this time, and no other European mathematician between the fall of the Roman Empire and the Renaissance has a body of original research.

Leonardo’s longest and most influential work was *Liber Abaci*, first written in 1202 and revised in 1228. The revised version is the one that was recently translated to English, and Leonardo not only made corrections but added some of his original mathematics at the end. In the introduction he stated that his goal was to bring the “Hindu numerals” to the Italian people. He succeeded, at least in planting the seed, since it took was not until after the invention of the printing press for that our modern number system to fully take took hold in Europe. At this time, Leonardo was not the only European familiar with Arab mathematics. Hindu Arabic numerals were known in Europe at least as far back as the 10^{th} century. Gerbert d’Aurilac, later Pope Sylvester II, used them as number symbols but not in calculations. In the 12^{th} century other scholars began translating Arab works into Latin, as well as writing their own texts. However, these translations and texts were aimed at other scholars. Leonardo was the first to deliberately focus on mathematics useful for everyday purposes. Soon after *Liber Abbaci* appeared, other popular arithmetics did also, but few approached the sheer magnitude of Leonardo’s compilation of arithmetic, algebra, geometric proofs, some of his original research, and a wide variety of practical and impractical examples.

While the goal of *Liber Abbac*i is to spread the Hindu Arabic number system, the book contains a wide variety of mathematics. The first chapter explains the basics of the number system. Chapters 2-5 deal with the operations of addition, subtraction, multiplication and division while Chapters 6 and 7 concern operations with fractions. Chapters 8-12 deal with various “word” problems, many of which would naturally arise from business situations of the time. Other problems are abstract, intended to display different solution methods of a problem or to provide further examples on a solution method. Occasionally Leonardo throws in a whimsical problem. One of these is the famous “rabbit problem” from which we get the Fibonacci Sequence. Chapter 13 is on algebra, namely on methods for solving linear equations. Chapter 14 is on extracting roots and arithmetic operations on roots. The 15th chapter deals with geometry and some applications to algebra, namely quadratic equations. The last two chapters are material from Leonardo’s work *Flos* added in the revision.

Most of the mathematics in *Liber Abbaci* can be solved in terms of modern mathematics with elementary school arithmetic, basic first year high school algebra, and geometry. The main exception to this is that most schools no longer teach algorithms for square root extraction, and I have never heard of any that taught cube root extraction. Most of the application problems in the book can be modeled with linear equations, which in modern terms are equations that can be manipulated into the form *Ax+B=C*, where *x* represents the unknown in the problem and *A*, *B*, and *C* are numbers. However, variables as we know them today, as well as almost all modern math symbols, were not used until the late 16^{th}/early 17^{th} centuries. Because of this Leonardo’s solution methods vary from somewhat different to quite different from modern methods. The main methods he used for these problem types are formalized guess-and-check systems called single false position and double false position. In addition, Leonardo uses what he calls “the direct method” which is basically the same as our modern algebraic manipulation, only using words instead of symbols. Most of Leonardo’s paragraphs-long solutions could be condensed to a few lines using modern symbols.

While Leonardo displays a wide range of mathematics and shows a facility for original mathematics in other works, *Liber Abbaci* is a summary of the existing useful arithmetic and algebra of the time. Leonardo had a wide range of sources at his fingertips that he used liberally for this book. Plagiarism was not seen in the same light as today, so many of the example problems can be found elsewhere before *Liber Abbaci* was written. This includes the Fibonacci sequence, which appears in several Indian texts going back to at least 200 BC. In some cases Leonardo acknowledges that he obtained problems elsewhere, but does not always mention names. One example is “A Problem on the Same Thing Proposed to Us by A Master near Constantinople” [6, p. 290]. Other problems come from a variety of texts. Books that Leonardo used included al Khwārizmī’s Algebra text (the oldest known algebra text for which the subject was named) and Euclid’s *Elements*.

Given the massive scope of *Liber Abaci*, there are a number of interesting items within its pages. First, Fibonacci used a fraction system from North Africa where he went to school that seems bizarrely complicated. That is, until one notices that it is incredibly useful for old fashioned monetary and measuring systems that do not work as well with modern decimals. For example, using this fraction system with the current U.S. system of measurement, 1 yard, 2 feet and 3 inches could be expressed as

yards. In some problems with solutions that cannot be expressed as fractions, Leonardo found approximations using this fraction notation that are very much like our modern decimals.

Second, Leonardo partially ignored the existence of negative numbers and considers equations with negative solutions to have no solutions, unless they are part of a merchant account, in which case he considered them debits in the account. He did, however, give directions on how to operate with negative numbers [6, pp. 417-419]. Leonardo never admitted the possibility of square roots of negative numbers for any reason.

Third, *Liber Abbaci* contributed to the financial developments in Italy in multiple ways. Leonardo used a form of present value analysis in some problems, which is the foundational concept of modern financial calculations and decisions [8]. This is the first known use of this method to compare two investments. In addition, the mathematics of minting precious metals into coins in *Liber Abbaci* may have contributed to the more consistent coinage available in subsequent centuries.

Fourth, Leonardo included an early version of the classic “two trains” problems that students love to hate. However, instead of having two trains leave different cities at the same time, he calculated when two ships would meet [6, p. 280]. One scholar examining a 15^{th} century French arithmetic text claims that that text contains possibly the earliest known example of a “two trains” problem [13], but Leonardo’s is 200 years earlier.

While *Liber Abbaci* is an impressive book and has been admired by scholars throughout the centuries, its immense size would be overwhelming to a merchant who just wants to know what his goods are worth in the coin of the city he’s in that week. Most likely for this reason Leonardo wrote another book on the basics of our number system specifically for merchants. This is referred to elsewhere as *di minor guisa* (the minor work), which probably refers to the fact that the book was shorter or contained fewer topics. However, no known copies of this text survive. For a long time scholars wondered why many arithmetic textbooks in the Renaissance had an acknowledgement in them to Leonardo of Pisa, but the books were not similar to any of Leonardo’s surviving texts. Recent scholarship shows a plausible link between the missing *Libro di minor guisa* and the “abbacus” texts used for instruction in Italy in the 14^{th} and 15^{th} centuries [3, Ch. 8]. Recently found manuscripts from the late 13^{th} century and early 14^{th} centuries have some elements in common with later arithmetic texts and some with *Liber Abbaci*, providing the missing link between Leonardo’s work and the tradition of mathematics instruction in later medieval and Renaissance Italy.

The third of Leonardo’s explanatory texts is *De Practica Geometrie*, or Practical Geometry. Practical geometry was the term used at the time for surveying or land measurement. This profession had been very important and well-regarded in Roman times. The art declined during the so called Dark Ages, but began reviving on discovery and transcription of Roman surveyors’ tracts in the ninth and tenth centuries. These works then became common through the 11th century, but were more for teaching than for practical use. In addition to the duties of the Roman surveyors, medieval surveyors also verified weights and measures. [12]

Most other practical geometries of this period contained three sections on measuring heights, areas and volumes, while Hugh of St. Victor [5, p. xxv; 9] replaced the section on volumes with measurement of the heavens. Leonardo included all four of these topics [5, p. xxv], but one seems to have been largely lost between Leonardo’s time and when the existing manuscripts were copied. [5, p. xxv] The other thing that sets Leonardo’s text apart from prior practical geometry texts is his inclusion of the theory underlying the practice. Theoretical validation of surveying procedure became a goal in later texts, sometimes displacing the practice. *De Practica Geometrie* was translated into Italian a couple of times in period, as well as included in various compilations with other geometry texts in Italian.

For his surveying manual, Leonardo used a wide variety of geometry texts from the ancient Greeks and the Arab world. One text he used was Euclid’s *On Division of Figures*, which has since been lost. However, a century ago a scholar took an outline of Euclid’s text that still remained and found that Leonardo’s use of it matched very closely. This scholar then used Leonardo’s text to fill in the missing pieces and come up with a plausible reconstruction of Euclid’s missing text [1]. The scholar who recently translated *De Practica Geometrie* into English opines that it is not only a useful compilation of Greek and Arabic geometry, but is a practical analog of Euclid’s Elements: a stand-alone text containing everything a surveyor would need to solve the mathematics problems inherent in their work.[Back to Top]

In addition to being an excellent explainer of mathematics to the masses, Leonardo of Pisa was also a research mathematician. Between the fall of Rome and the Renaissance, almost all of the writings left by European mathematicians were translations of Greek and Arabic works, expository texts like Leonardo’s *Liber Abbaci*, or textbooks or lists of problems to use in teaching students. Leonardo, however, solved some original mathematics problems and produced new solutions to some previously solved problems. While some of these texts are on other lines of inquiry, some of these were based on the challenge problems he was given in the court of Frederick II. One of these,* Liber Quadratorum*, or *The Book of Squares*, is currently in English translation [4].

Leonardo solved two of the challenge problems and later wrote *Liber Quadratorum* on the solutions and related problems, but never finished or published this book. It contains 24 propositions written in a style similar to Euclid’s Elements, which Leonardo references in the text. The text focuses on whole number and fraction solutions to *indeterminate equations*, or *Diophantine equations*, which are equations with multiple variables, and thus may have multiple solutions. The Greek mathematician Diophantus was the first to study these equations. For example, one of Leonardo’s challenge problems was equivalent to finding x,y,”and” z such that x^{2}+5=y^{2} and y^{2}+5=z^{2}. In the process of finding a solution, Leonardo stated and proved a special case of Lagrange’s identity, although scholars argue over whether his proof is original or based on Arab material. Also, Leonardo wondered what other numbers besides 5 could be used in these equations and the equations would still have solutions. He proved some interesting facts about these numbers, called *congruent* numbers [17], based on the Latin word that Leonardo used for them. The mathematics scholars of the Tuscan school that arose from studying Leonardo’s works were very interested in congruent numbers. Modern mathematicians have not yet determined a general rule for which numbers are congruent numbers.

Another impressive result in *Liber Quadratorum* is that

for any whole numbers *n* and *m* where *n > m*. A conclusion Leonardo reaches in the proof is that no square can be a congruent number. This result is equivalent to the fact that the area of a Pythagorean triangle, a right triangle with whole number side lengths, cannot be a square. The proof about the Pythagorean triangle was one of Fermat’s greatest achievements.

Leonardo’s other higher level works, some of which include some original work, include *Flos* and *Epistola ad Magistrum Theodorum*, and a now lost text on Book X of Euclid’s Elements. *Flos* was included in the revision of *Liber Abbaci* as the last two chapters. The first of the two chapters is also a discussion on Book X of Euclid’s elements, and is basically a discussion on how to deal with numbers that can be expressed using square roots, among other things. Unfortunately we do not know how Leonardo’s lost text would have differed from *Flos*. The second chapter is on solving quadratic equations, which contain a square of the variable in them. Leonardo solved these using completing the square, since methods commonly taught in high school today such as the quadratic formula were not possible without the symbols we now use. He also proved some facts about squares and includes some of his challenge problems. Leonardo wrote *Epistola ad Magistrum Theodorum*, or* Letter to Master Theodore*, to one of the scholars offering challenge problems, with more material on those challenge problems. Leonardo’s works began traditions of scholarship in algebra and in investigating congruent numbers both in Tuscany and in Germany.

Leonardo of Pisa, today called Fibonacci, wrote a number of impressive texts containing the bulk of the practical mathematics known in his day, plus some original mathematics research. His goal in writing his most impressive text was to bring our modern number system to Europe. While other mathematicians were also writing and translating books about this system, the legacy of mathematics scholarship and education that sprang from Leonardo’s works testify to the influence he had in making this happen. His legacy of education continued in the arithmetic and algebra textbooks in Italy and nearby areas until the Renaissance. His legacy of scholarship continued through schools of mathematicians in his native Tuscany and in Germany. Even today mathematicians are studying questions that he pursued in his research. It is no wonder that Leonardo of Pisa is considered the greatest mathematician of the Middle Ages.

1. Archibald, Raymond Clare, *Euclid’s book On divisions of figures…with a restoration based on Woepcke’s text and on the Practica geometriae of Leonardo Pisano*, Cambridge University Press, 1915.

2. Berlinghoff, William P. and Fernando Q. Gouvea, *Math through the Ages: A Gentle History for Teachers and Others*, Oxton House Publishers and the Mathematical Association of America, 2004.

3. Devlin, Keith, *The Man of Numbers: Fibonacci’s Arithmetic Revolution*, Walker & Company, New York, 2011.

4.* Fibonacci, Leonardo, The Book of Squares* / Leonardo Pisano Fibonacci; an annotated translation into modern English by L.E. Sigler, Academic Press, Boston, 1987.

5. Fibonacci, Leonardo, *Fibonacci’s De Practica Geometrie*, ed. and tr. by Barnabas Hughes, Springer Science + Business Media, LLC, 2008.

6. Fibonacci, Leonardo, *Fibonacci’s Liber abaci : a translation into modern English of Leonardo Pisano’s Book of calculation*, tr. by L.E. Sigler, Springer-Verlag, New York 2002.

7. Gies, Frances and Joseph,* Leonard of Pisa and the New Mathematics of the Middle Ages*, Crowell, New York, 1969.

8. Goetzmann, William N., “Fibonacci and the Financial Revolution”, *The Origins of Value: The Financial Innovations That Created Modern Capital Markets*, Goetzmann and Rouwenhorst, eds., Oxford University Press, New York, 2005

9. Hugh of St. Victor, *Practica Geometriae*, tr. by Frederick A. Homann, Marquette University Press, Milwaukee, WI, 1991.

10. Menninger, Karl A., *Number Words and Number Symbols*, tr. by Paul Broneer, MIT Press, Cambridge, MA.

11. Mucillo, Maria, “Fibonacci, Leonardo”, *Medieval Science, Technology, and Medicine: An Encyclopedia*, 2005.

12. Pikulska, Anna, “Agrimensores”, *Medieval Science, Technology, and Medicine: An Encyclopedia*, 2005.

13. Schwartz, Randy K., “‘He Advanced Him 200 Lambs of Gold’: The Pamiers Manuscript,” Convergence (July 2012), DOI:10.4169/loci003888, http://www.maa.org/press/periodicals/convergence.

14. Suzuki, Jeff, *A History of Mathematics*, Prentice Hall, 2002 (ska Master William the Alchemist).

15. Swetz, Frank, *Capitalism and Arithmetic*, Open Court, La Salle, IL, 1987.

16. Swetz, Frank, Ed., *The European Mathematical Awakening: A Journey Through the History of Mathematics from 1000 to 1800*, Dover Publications, Mineola, NY, 2013.

17. *The On-Line Encyclopedia of Integer Sequences*, published electronically at https://oeis.org, 2010, Sequence A003273