bdallāh Mu
ammad ibn Mūsā al-Khwārizmī al-Majūsī. The last part, which translates as “the Mazdean”, indicates that he was a Zoroastrian, but scholars have doubts about the accuracy of that epithet.
1 Leonardo published a completely revised and extended second edition of Liber abbaci in 1228. No copies of the first edition have survived, but three almost complete copies of the 1228 version exist, dating back to the same era, and are kept in libraries in Rome, Florence, and Siena. See chapter 9.
2 Different people use the word “Arab” to mean different things. In this book, I use it in the commonly accepted scholastic sense to mean the peoples whose primary business or cultural language was Arabic—just as we speak of “the Greeks” to refer to the peoples whose primary cultural language was Greek. Used in this sense, the term “Arab” includes peoples from many nationalities, mostly, but not all, Muslim. Similarly, when I write of “Muslim scholars” I am referring to scholars who lived and worked within the Muslim culture, regardless of their race, national origin, or religious beliefs or practices.
4 Several earlier handwritten manuscripts did refer to Leonardo.
5 It was not printed until 1857, when Baron Baldassarre Boncompagni, an Italian bibliophile and medieval mathematical historian, had the manuscript typeset and published it in Rome. Boncompagni’s Liber abbaci comprised the first volume of a two-volume, printed collection of all of Leonardo’s works he compiled under the title Scritti di Leonardo Pisano. The second volume, containing all of Leonardo’s other works, appeared in 1862. A printed English-language translation of Liber abbaci, by the American mathematician Laurence Sigler, was published in 2002. Based on Boncompagni’s edition, it runs to 672 pages and is the only translation of Leonardo’s text into a modern language.
6 Discounting references in some handwritten manuscripts that would not be uncovered until the late twentieth century when scholars began to investigate Leonardo’s heritage.
7 The familiar calculating device comprising beads strung along wires attached to a wooden frame, although popularly called an “abacus”, was not used in medieval Europe, but rather came from China, where it was called a xuanpan. That device is, therefore, sometimes more accurately described as a “Chinese abacus.” The European “abacus” was a board or table (tavola) ruled with a series of parallel lines on which the user slid counters (or “jetons”) to represent numbers. It was used extensively all over Europe in the Middle Ages and could still be found in use in some places as late as the eighteenth century.
8 Historians have not been consistent over this spelling distinction, and it is unfortunate that the English translation of Liber abbaci uses the spelling with one “b”.
9 For doing arithmetic, any other base of roughly the same size would work as well. The satirist, songwriter, and mathematician Tom Lehrer once quipped that base 8 arithmetic is no harder than base 10 “if you’re missing two fingers.”
10 Today, in large part as a result of that one document discovered in Pisa, Roman law provides the foundation of the civil law systems of many countries, including all the countries of continental Europe.
11 The most prominent bankers were those from the northern Italian region of Lombardy, particularly those of its capital, Milan, which is why the major financial streets in both London and San Francisco are called Lombard Street.
12 Today, each bank carries a major road, bustling with traffic—one of those roads being called the Lung’arno Leonardo Fibonacci for part of its length.
13 The very literal translation is by Laurence M. Sigler, 2001. It is based on the second edition of Liber abbaci, published in 1228. Hence Leonardo’s reference to having “corrected” his work.
14 Michael Scott was the philosopher in Frederick II’s court.
15 Leonardo was referring to Frederick.
16 Leonardo was referring to the first edition of Liber abbaci.
17 Leonardo was referring to his book De practica geometrie, which he published in 1220, long after the first edition of Liber abbaci had been completed.
18 While no doubt true compared to earlier Arabic texts, it does not look at all like this to a modern reader.
19 Leonardo used Roman numerals here, perhaps because he knew his readers would not understand Hindu-Arabic numerals until they were well past the introduction.
20 Recall that Gerbert (ca. 980) used the Hindu-Arabic numerals on counters as part of a primitive form of abacus, on which triples of columns were marked with an arc. These were called Pythagorean arcs. When he wrote numbers, Leonardo followed the system of triples, just as we do today when we write numbers like 1,395,281. Leonardo told his readers that even with various enhancements, abacus methods were no match for Hindu-Arabic arithmetic.
21 A more colloquial translation of this last clause would be: “Who up to now have lacked this knowledge.”
22 A few sources give a longer version: Abū bdallāh Muammad ibn Mūsā al-Khwārizmī al-Majūsī. The last part, which translates as “the Mazdean”, indicates that he was a Zoroastrian, but scholars have doubts about the accuracy of that epithet.
23 There is only one surviving copy, which is kept at the University of Strasbourg Library. There is a Latin translation at the Biblioteca Nacional de España in Madrid.
24 One of them, “casting out nines,” was just going out of use in the United Kingdom when I was taught arithmetic there in the 1950s.
25 Teachers today usually introduce addition and subtraction before multiplication.
26 They particularly liked finger reckoning because it allowed them to perform their calculations on the spot, without the need for any apparatus. Given the simplicity of most of their computations, it was an adequate method.
27 The “roll” was a unit of weight, equal to twelve ounces.
28 Recall that 12 denari made 1 solidus.
29 He would remain emperor until his death in 1250.
30 It is possible there were other questions that Leonardo did not feel merited a permanent record. Some scholars have speculated that Frederick organized a mathematical tournament, with Leonardo pitted against other mathematicians, but no evidence exists to support that supposition.
31 A famous problem of this kind in Liber abbaci is about a growing rabbit population. It is discussed in chapter 9 of this book.
32 From a fourteenth-century abbacus book by Piero della Francesca, a leading artist of his time, also known as Piero dell’Abacco. The translation is by Arrighi (1964, p. 78). It is simple to solve using modern symbolic algebra, which reduces the puzzle to the solution of two simultaneous equations, d = f + 40 and 5f = 3d. The answer is 100 fox paces or 60 dog paces.
33 This practice had come to an end by the time printed texts were produced, and Pacioli was the only author of a printed abbacus book who gave such credit.
34 The first is included in codex Palatino 573 of the Biblioteca Nazionale of Florence; the second is part of manuscript L.IV.21 of the Biblioteca Comunale of Siena.
35 The book is often referred to by this shorter title that omits the term di merchaanti.
36 Some of the more significant ones are described in the final section of this chapter.
37 Though people did bind manuscripts into “book form” in Leonardo’s time, they did not resemble modern books, like the one in the statue’s hand.
38 The locations are the Biblioteca Nazionale Centrale di Firenze (BNCF—Florence National Central Library, four copies), the Biblioteca Laurenziana Gadd in Florence, the Biblioteca Riccardiana in Florence, the Biblioteca Ambrosiana in Milan, the Biblioteca Nazionale Centrale in Naples, the Bibliothėque Mazarine in Paris, and the Bibliothėque Nationale de France in Paris (two copies). For more details, see Hughes 2004.