*This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.*

(Previous Section: *Leonardo Pisano, a.k.a Fibonacci*)

Leonardo’s greatest intellectual legacy is his book, *Liber Abaci (The Book of Calculation)*. This book provides almost the only biographical information we have about him. No one knows why Leonardo dedicated the book to Michael Scotus, who was the court astrologer to Holy Roman Emperor Frederick II, nor why he included the autobiographical information (Devlin, *Man* 43). It was uncommon for mathematically-focused texts to include such information. Mathematician/ historian Devlin explains this is because “most mathematicians are interested in mathematical results, not the people who discover them” (*Finding* 68). He says, “Mathematical truth is completely independent of human judgment, immutable, eternal, forms its own abstract world,” so, “to the mathematician, the historical details are of little relevance; it doesn’t matter when someone first proved a theorem; the focus is on the development of the ideas and how one train of thought led to another” (*Finding* 69). The first edition of *Liber Abaci* was a dense work suited more to scholars than the average man. But Leonardo seemed very occupied with producing practical solutions to common problems. Thus, in the preface of the second edition he revealed something of the intellectual heritage which inspired him to write the book in the first place.

The first copy of *Liber Abaci* appeared in Pisa in 1202. “When he finished writing it, he would have taken it to a local monastery to have copies made by the monks.” This is such a laborious method of publication that it could have taken a year or more to copy a manuscript as long as *Liber Abaci* (over 400 pages). After receiving peer commentary and suggestions for improvement, “he made changes and added to its contents, culminating in the second edition published in 1228” (Devlin, *Finding* 85-86).

In modern publications, the most common spelling of the book’s title uses just one letter “b” in the word “Abaci,” a declension of the word ‘abacus’ in medieval Italy. This spelling is nonsensical, though, says Devlin, because “Leonardo was in fact showing how to do arithmetic without the need for any such device as an abacus.” Leonardo himself used the spelling “Abbaci” to mean ‘calculation.’ The words are similar in spelling, but very different in meaning. The first known written use of the word abbacus with this spelling was, in fact, in the prologue of his book, an intentional spelling change to differentiate Leonardo’s new method. It must have caught on, for in the years after Leonardo, the word abbaco was widely used in medieval Italy to describe the practice of calculating with the Hindu-Arabic number system (Devlin, *Finding* 11).

Since medieval authors rarely gave their manuscripts a title, neither did Leonardo; the name for his book comes from his opening statement: “Here begins the Book of Calculation; Composed by Leonardo Pisano, Family Bonacci; In the Year 1202.”

In later writings, he also referred to the work as Liber Numerorum, and in the dedicatory letter for his book Flos he referred to it as his Liber Maior de Numero (Devlin, *Man* 12).

Many consider Leonardo’s book the greatest arithmetic text of the Middle Ages, for he was the first mathematician to demonstrate the superiority of the Hindu- Arabic numeral system versus the Roman system exemplified by Boethius; he did this by providing numerous examples of how to solve problems related to every major contemporary field of business. It is true that Leonardo’s *Liber Abaci* was not the first book written in Italy to introduce and explain the Hindu-Arabic numerals, but no work previously produced was comparable in value, either in content or in quality of the exposition (McClenon).

Leonardo consulted many sources to write *Liber Abaci*, primarily Arabic texts or Latin translations thereof. He undoubtedly included information gleaned from his many discussions with the Arab mathematicians he encountered on his travels. He provided rigorous proofs to justify the methods, in the fashion of the ancient Greeks. For example, Abu Kamil’s book has seventy-four worked-out problems, and many of the more complicated ones, with identical solutions, are found in Liber Abaci (Devlin, *Man* 47, 57, 61). Leonardo explains the mathematical reasoning for each problem extensively, providing numerous examples and variations which were most valuable to the merchants and laymen learning the new calculating system (Devlin, *Finding* 80).

*Liber Abaci* is a general book of mathematics, but it differs from most because the author’s main purpose was to encourage everyone (especially merchants) to abandon Roman numerals and use the superior Indian system of numbers. Leonardo was an advocate for systemic change. Knowing the superiority of the new system for business, he devoted several chapters of his book to showing calculations of profit, interest, and currency conversions. Some scholars think the book was “too advanced for the mercantile class, and too novel for the conservative university circles” which were resistant to adopting Arabic numerals (Smith and Karpinski; Radford). Furthermore, it was so comprehensive, it has been called “encyclopaedic” (“Biography”). Moreover, at the time he published his book, only a few people knew the Hindu-Arabic numerals we use today: privileged European intellectuals who cared to study the translation of the works of Al- Khwˆarizmˆı and Abu Kamil. Nevertheless, because Leonardo was so convinced that the Hindu-Arabic numerals and the place-value principle were far superior to all other methods, he devoted the entire first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications (Livio 92).

The book is relevant for mathematicians today because of “the mathematical insight and originality of the author, which constantly awaken our admiration, and also on account of the concrete problems” (McClenon). It is also of interest to historians, sociologists and economists, because it provides much information about the society in which he lived. For example, through his book, we learn that Pisan ships transported “pepper, a very important item of merchandise, and that the Pisan colony in Constantinople traded extensively with Egypt. Further evidence is also gleaned about the relative values of money coined in the mints of different cities, and about the problem of alloying of coins to be minted” (Horadam).

In Pisa, Leonardo not only became an author of books, but he was likely also a maestri d’ abaco or “teacher of business arithmetic.” As such, he became so famous that the Roman emperor asked the mathematics expositor to give a public demonstration of his ability during one of the emperor’s visits to Pisa (Livio 95).

Holy Roman Emperor Frederick II (1196-1250) was called Stuper Mundi (“Wonder of the World”) by his contemporaries because he was a highly educated and inquisitive man who “encouraged learning and scholarship of every kind;” he even conducted scientific experiments and wrote books of his own. Having a special interest in mathematics and impressed after reading *Liber Abaci*, the emperor invited Leonardo to his palazzo in Pisa (Horadam).

One of the emperor’s court mathematicians, Johannes (John) of Palermo, proposed three mathematical problems for Fibonacci to solve. Leonardo had been provided the problems in advance but the ingenious way he solved and explained the solutions was astounding (Horadam).

The Medieval Italian Cultural Association claims Emperor Federico II was so impressed that he granted Leonardo an annuity which enabled him to devote himself to his studies (“Arabic Numerals”). If this is true, that could explain how Leonardo was able financially to devote more time to writing a revised version of *Liber Abaci*. Indeed, soon after meeting the emperor, the mathematician dedicated his next major work to the emperor, perhaps in appreciation for sponsorship; he also published the solutions he had presented to the imperial court: two in Flos (a copy of which he sent to Frederick) and one in Liber Quadratorum (Devlin, *Man* 90-92).

1202 (1228): *Liber Abaci (The Book of Calculation)*

1220: *Practica Geometriae (The Practice of Geometry)*, a mixture of pure mathematics, theorems, proofs, and practical applications of geometry, such as using similar triangles to calculate the height of tall objects

Before 1225: “Epistola” and “Magistrum Theodorum” (A Letter to Master Theodore), a letter to Frederick II’s philosopher Theodorus Physicus, solving three problems in mathematics

1225: *Liber Quadratorum (The Book of Square Numbers)*, a highly mathematical number theory book dealing with solutions to Diophantine equations

1225: *Flos (The Flower)*, solutions to problems in algebra n.d. (no date known): *Di Liber Minor Guisa (A Smaller Manner)*, a book on commercial arithmetic (No copies exist today.)

n.d.: *Commentary on Book X of Euclid’s Elements* (No copies exist today.)

(Horadam; “Education”)

**Practica Geometriae (1220)**

The *Practica Geometriae (The Practice of Geometry)* is a substantial work on geometric practice (surveying, area and volume formulas for plane figures and bodies); it also contains a wide variety of interesting theorems which represent “a considerable advance over the Geometry of Boethius and Gerbert (Pope Sylvester II)” (Horadam). Leonardo well understood Euclidean geometry and the mathematical methods demonstrated in *Practica Geometriae* reproduce the brilliant techniques found in the works of others, particularly Abu Kamil’s On the Pentagon and the Decagon (Livio 96). Though it “shows no such originality as to enable us to rank Leonardo among the great geometers of history, it is excellently written, and the rigor and elegance of the proofs are deserving of high praise” (McClenon). Furthermore, *Flos, Magistrum Theodorum*, and the *Liber Quadratorum* are so original and instructive, they show well the remarkable genius of this brilliant mathematician.

**Liber Quadratorum (The Book of Squares) (1225)**

Leonardo describes his presentation to Emperor Frederick II at the Pisan court in the dedication to this book, *Liber Quadratorum*, dated 1225 (McClenon). In it, he demonstrates a virtuosic command of number theory. The book, among other things, examines methods to find Pythogorean triples (O’Connor and Robertson). Even more impressive are his presentations of the properties of the squares and tasks that lead to quadratic equations; it has been called the most important work of number theory between Diophantus and Fermat (Gies; “Biography”).

**Liber Minoris Guise (n.d.)**

The *Liber Minoris Guise (Book in a Smaller Manner)* is a manuscript on commercial arithmetic referred to with description (rather than by title) by the Pisan several times as he was comparing it to his more extensive Liber Abaci. Additional proof this missing work once existed is a reference to it by an abbacus author who refers to Leonardo’s *Libro di Minor Guise o Libro di Merchanti (Book in a Smaller Manner or Book for Merchants)* (Devlin, *Finding* 28).

Because Leonardo lived two hundred years before mechanized printing was widely available in Europe, his books were handwritten and the only way to have a copy was to have a scribe handwrite another copy. Of his books, copies remain of *Liber Abaci* (1228), *Practica Geometriae, Flos*, and *Liber Quadratorum*. Regrettably, another of his manuscripts now completely lost is his commentary on *Book X of Euclid’s Elements* (O’Connor and Robertson). No copies of his first *Liber Abaci* exist, and of his 1228 revision, only fourteen copies have been found. Seven are complete or nearly so while seven are fragments, consisting of between one-and-a-half and three of the book’s fifteen chapters (Devlin, *Finding* 82).

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