Following Pietro Cossali’s discovery of Leonardo’s work at the end of the eighteenth century, many scholars speculated that the growth of the abbacus books and the abbacus schools—and hence the arithmetic revolution they generated—were direct consequences of the publication of Liber abbaci. In Summa de arithmetica, geometria, proportioni et proportionalità, the book in which Cossali first found reference to Leonardo, the author, Luca Pacioli, had credited the Pisan for instigating the European arithmetic revolution, a view Cossali endorsed in his book. If what they suggested were true, Leonardo’s role in history would be on a par with that of Copernicus or Galileo or Kepler.
Certainly, the contents and structure of Liber abbaci made it an excellent source for the material that drove the revolution. And the timing of events, particularly the growth pattern revealed by Van Egmond’s quarter-century counts of abbacus books, makes Pacioli’s conclusion plausible. Yet despite the fact that medieval authors copied freely from one another all the time, neither Summa nor the majority of the abbacus books contained any passages from Liber abbaci. To credit the revolution to Leonardo, further evidence would be required. In particular, scholars would have to identify the path by which the contents of Liber abbaci were transmitted to the pages of the early abbacus books. Assembling that evidence took historians and archivists many years of painstaking effort, along with some remarkable strokes of luck. The final piece of the puzzle fell into place as recently as 2003.
The easiest case to make is that, whether or not Leonardo was the instigator of the arithmetic revolution, he was at least its inspiration. By the time he completed the second edition of Liber abbaci, Leonardo was famous throughout Italy as a great mathematician and an honored citizen of Pisa. As a result, many people were aware of his book and doubtless wanted to learn of its contents—merchants, scholars, teachers, and parents, as well as the emperor Frederick. In short, the very fact that Liber abbaci had been written by someone of Leonardo’s stature, coupled with a general description of its contents and an appreciation that what he taught was useful in commerce, could well have created a demand for more easily digestible explanations.
When individuals of far lesser mathematical talents than Leonardo sought to meet that demand with books of their own, for the most part they looked elsewhere than in the dense pages of Liber abbaci. In some instances, no doubt, this was because the budding abbacus authors found Liber abbaci far too challenging; in other cases the problem was that they could not read Latin; and some probably sought other sources because they simply could not get hold of one of the few available copies of Leonardo’s manuscript. When authors of abbacus books referred to Leonardo as their inspiration, as many did,33 they may have been simply using his name to add an air of authority to their work, but the very fact that they did so shows that Leonardo was regarded as the primary figure in the tradition.
Even the Danish historian Jens Høyrup—one of the few scholars who have seriously questioned Leonardo’s role as the original source for the abbacus books—finds this scenario plausible. When the famous sixteenth-century mathematician Girolamo Cardano (one of the early pioneers of what we now call probability theory) wrote his great work Ars magna in 1545, he credited Leonardo for introducing modern arithmetic and algebra into Europe. That led Høyrup to speculate, in a 2005 paper: “Instead of being the starting point of abbaco culture Fibonacci may have been an extraordinary representative who, growing, had grown taller and more conspicuous than any other representative—so tall that Cardano saw nobody but him in the landscape who was worth mentioning.”1
By the time Høyrup’s paper had been published, however, a scholar in Italy had already identified the crucial missing text connecting Liber abbaci to the abbacus books—thereby proving that Leonardo was not just the inspiration for the arithmetic revolution but also the instigator in terms of its content and form. To find that missing link, the Italian scholar had to delve deeply into the highly specialized world of archival forensics—the detailed comparison of texts to uncover the historical flow of ideas. Working for many years in the Italian archives, she assembled sufficient evidence not only to identify, by name and author, a key manuscript—long since lost—that led from Liber abbaci to the abbacus books, but also to say with considerable certainty what its contents were. What makes this particular investigation of added interest to the layperson today is that the scholar was uncovering the origins of the modern world we live in. A typical abbacus book looks remarkably like the schoolbooks we all learned from. Replace the old-style language and the local cultural references with those of today, and (apart from the full-color illustrations that today’s publishers are so enamored by) you find yourself looking at a typical modern arithmetic textbook. As Warren Van Egmond wrote in his 1980 catalog of abbacus books that made the scholastic world aware of the abbacus tradition, “However, the tradition does not end [in the sixteenth century] … the abbaci are only the beginning of a tradition of arithmetical writing that extends to the present day. The elementary books that we all used to learn basic arithmetic are the direct descendants of the abbacus books of the fourteenth and fifteenth centuries.”2
One obstacle the mathematical historian faces stems from the way mathematics is recorded and transmitted. To the mathematician, for much of the time, doing mathematics seems to be (some would prefer to say “is”) a process not of invention but of discovery. When the mathematical community views an advance as “routine,” the individual who made it is quickly forgotten. The path was there, waiting to be discovered, and someone had to find it first, but exactly who did so was largely a matter of happenstance and not worthy of further mention. The important thing is the path, which others can now follow. The only exception is when someone establishes a new formula or proves a particular theorem, but only if finding the formula or proving the theorem is deemed to be important and to have required considerable ingenuity.
This is very different from many other areas of human creativity, where we rightly acknowledge the individuals who do things first. In many cases, the creative act is clearly unique. If Shakespeare had not lived, for example, Hamlet would never have been written. In contrast, if Euclid had not proved that there are infinitely many primes, someone else would have. The only uncertainty is how long it would have taken. (As it happened, we do cite Euclid for that theorem, but only because it is an important result and the proof showed great originality and ingenuity—though it’s not entirely clear that Euclid himself did discover the result.)
Hindu-Arabic arithmetic falls into the category of something waiting to be found. The individuals who found it first are, on the whole, not remembered, nor are they acknowledged by others who came later and built on their work. Use of the term “Hindu-Arabic arithmetic” reflects the fact that the Indians developed the system and the Arabs refined it, but with a few exceptions there is little record of exactly when and by whom various advances were made.
Thus, reading an ancient manuscript written by a mathematician is unlikely, on its own, to provide a historian with information about the sources the mathematician consulted in preparing it. For instance, Leonardo clearly obtained much of his material for Liber abbaci from the works of al-Khwārizmī, either directly or indirectly, yet he made no mention of that fact apart from a brief remark in the final chapter. Commentators outside the field of mathematics find this strange, and sometimes seek an explanation. Yet Leonardo was simply following a mathematical practice as old as the subject itself.
If that lack of proper accreditation were not problem enough for the historian, often when a name does become attached to a particular mathematical advance it turns out to be the wrong name. Pythagoras’s theorem is a good illustration; the result was known long before Pythagoras of Samos was born.
One way medievalists trace the development of ideas is by making a detailed comparison of the manuscripts, looking for clues such as passages that one author copied directly from another. Medieval authors frequently copied entire passages from an existing manuscript without crediting the source. If a medievalist finds identical passages in two texts, he or she can conclude that either one author copied from the other or both copied from a third. (This last possibility requires further investigation if the original manuscript is lost, leaving only the two that contain material copied from it.)
For many centuries, copying of books was done almost exclusively by monks in the more scholarly monasteries. It was a slow process. It could take a year or more to make one copy of a book the length of the Bible. From the thirteenth century onward, with the rise of the universities creating a much larger demand for books, bookmaking gradually moved out of the cloisters and into the commercial world. Professional copy companies appeared, usually on the edges of the new universities—an early forerunner of today’s commercial photocopy shops. Some of them grew quite large; one early fifteenth-century copy shop in Florence employed forty-five lay copyists.
Hand copying produced some attractive-looking manuscripts. With considerable time to devote to the task, the scribes, who in Leonardo’s day worked mainly for spiritual reward, developed elaborate calligraphic styles and would often let their creativity run free when it came to adorning and illuminating the page with colorful swirls, drawings, and other embellishments to the text. When a scribe completed a book, he often closed with a personal note. One particularly memorable example—though not in Liber abbaci—reads:
Explicit hoc totum;
Pro Christo da mihi potum,
which can be translated as:
The job is done, I think;
For Christ’s sake give me a drink.3
Unfortunately, the scribes’ creativity frequently extended to the book’s contents as well. They were not averse to making changes, leaving parts out, or copying material from another manuscript, generally without any indication that they had done so. And of course, there was always the possibility they would simply make a copying mistake. All of which makes present-day medieval scholarship fiendishly difficult.
The more influential texts were, of course, copied many times, so errors could—and did—multiply, as copies were made of copies. Faced with a collection of manuscripts all clearly versions of the same original text, the medievalist has the task of putting them in correct sequence. Even if a manuscript gives a date, there can be uncertainty whether it refers to the completion of the original text or when the copy was made. In fact, most of the abbacus texts bear neither the name of the author nor a date, but the authors generally phrased their examples in terms of the local coinage and units of weight—which varied from location to location and over time—so with some painstaking detective work it has been possible to assign a location and date to many of the manuscripts.
Writing history by comparing manuscripts has an obvious limitation: It can be applied only to manuscripts that either have survived or are referred to substantively in a work that has survived. It is always possible that a key manuscript was lost. Sometimes, with a lot of painstaking effort and perhaps a bit of luck, it is possible to infer a particular manuscript’s existence and even determine its author and contents. This is what happened with the birth of the abbacus texts.
If the abbacus manuscripts had contained material taken directly from Liber abbaci, the lineage would have been clear from the outset. To be sure, a few abbacus books were essentially just vernacular extracts from Liber abbaci; but they came later, after many other abbacus texts were in circulation.4 The vast majority of abbacus books, including all the known early ones, had almost nothing in common with Leonardo’s masterpiece. In particular, they typically lacked the organization, precision, and expository standard of Leonardo’s treatise and were much less extensive. Clearly, the abbacus authors must have obtained their material elsewhere. In which case, the first task facing the historian was to decide whether the genre began with material taken from one work or several sources.
In the introduction to his 1980 survey, Van Egmond noted that there was little duplication from one abbacus book to another—the abbacus authors were not simply copying one another. That seems to point to multiple initial sources, but appearances can be deceptive. With roughly four hundred problems in each book, taken altogether the abbacus books provided around 150,000 seemingly different worked examples, but when you ignore the exact words and numbers used and focus on the underlying arithmetic problems, you find a fairly small number. Moreover, the presentations of the problems all followed a standard pattern that varied little over the three-hundred-year extent of the abbacus books. A typical worked problem began with a brief description of a situation, followed by a question, and then the solution. For example:
A solidus of Provins is worth 40 denari of Pisa and a solidus imperiali is worth 32 of Pisa. Tell me how much will I have of these two monies mixed together for 200 lire of Pisa? Do it thus: add together 40 and 32 making 72 [denari] which are 6 solidi and divide 200 lire by 6 which gives 33 lire and 6 solidi and 8 denari, and you will have this much of each of these two monies, that is 33 lire 6 solidi 8 denari for the said 200 lire of Pisa. And it has been done.5
The author would always introduce the question by a phrase such as “Dimmi” (Tell me), “Domandoti” (I ask you), or “Voglio sapere” (I want to know). In many of the earlier manuscripts, the entire problem statement would be preceded by an introductory phrase like “Fammi questa regola” (Do this problem for me) or, if the problem is used to illustrate a rule that has just been introduced, “Voti dare essempro alla detta regola” (I want to give you an example of this rule). The solution to the problem usually began with a phrase such as “Fa cosi” (Do it thus) or “Dei cosi fare” (You must do it this way). The author would often include a restatement of the answer after it had been obtained. The ending too was standard, with a phrase such as “Ed e fatta” (And it has been done) or “Et cosi fa tutte le simigliante ragione” (And do all similar problems in this way).
Thus, when scholars subjected the abbacus books to a closer look, what had initially seemed like evidence of multiple sources turned out to be highly suggestive of at most a small number of original texts, with the differences between the abbacus books being largely in depth, sophistication, and accuracy, and localization to a particular town or region. There were, however, still several possibilities for the original texts.
During the twelfth and thirteenth centuries, a wave of Arabic science flowed into Europe, via Latin translations made in Spain, Italy, and the Crusader states. For instance, in Montpellier, many scribes were translating and copying texts, among them al-Khwārizmī’s Arithmetic. One early book in Latin that some historians have suggested was an abbacus source is De algorismo, written by the English scholar John of Halifax. Born around 1195, in England (presumably in the northern town Halifax), John was educated at the University of Oxford, after which he went to live in Paris, becoming a professor of mathematics at the university. Known also as Johannes de Sacrobosco, he wrote a number of books on mathematics and astronomy. De algorismo was an elementary exposition (in Latin) of Hindu-Arabic arithmetic and included chapters on addition, subtraction, multiplication, division, square roots, and cube roots. Though some abbacus authors may have consulted it, however, there is no firm evidence it was the initial source for the majority of abbacus books.
Since later abbacus books may well have been based on earlier ones, historians trying to identify the origins of the arithmetic revolution focused much of their attention on the earliest manuscripts in the genre. The first vernacular abbacus book may have been a libro di nuovi conti (book of new calculations), written around 1260 in Siena, but no copy has survived.6 The oldest abbacus books still in existence date from around 1290. One of them is Livero del abbecho.7 Its unknown author described it as “lo livero del abbecho secondo la oppenione de maestro Leonardo dela chasa degl’figluogle Bonacie da Pisa” (abbacus book according to the opinion of master Leonardo Fibonacci), perhaps making it one of the earliest to make such use of Leonardo’s name. Another is Columbia algorism,8 which was transcribed by Kurt Vogel in 1977 as Ein italianishes Rechenbuch aud dem 14. Jahrhundert.9 The German phrase “14. Jahrhundert” in Vogel’s title translates as “14th century”, and that underscores the difficulties facing the medieval historian. At the time Vogel published his translation, he believed it was a later manuscript, but subsequent examination of the coins it mentioned showed that it was almost certainly written earlier, most likely around 1290.10 Paolo Gherardi’s book Libro di ragioni (Book of problems), written in Montpellier in 1328, is the oldest known vernacular Italian abbacus book with a chapter on algebra.
In all cases, however, while examination of early abbacus manuscripts suggested some that may have been used as sources for others that were written later, none indicated an original source other than Leonardo, and over the years the majority of scholars came to view the abbacus books and the abbacus schools as part of a tradition initiated by the appearance of Liber abbaci. For instance, in the introduction to his 1980 catalog of abbacus books, Van Egmond declared, “All the manuscripts and books contained in the present catalog can be regarded as members of this tradition [abbacus books] and direct descendants of Leonardo’s book.”11 In his analysis, he wrote: “The printed abbaci are by and large a direct continuation of a tradition that began with the Liber abbaci in 1202. The abbacus books thus represent a continuous and remarkably uniform tradition that stretches from the work of Leonardo Pisano to the end of the sixteenth century, shifting from handwritten to printed books as the primary means of publication changed from manuscript to printing.”12
In a similar vein, Kurt Vogel, in a highly regarded, authoritative article on Leonardo observed:
In surveying Leonardo’s activity, one sees him decisively take the role of a pioneer in the revival of mathematics in the Christian West. Like no one before him he gave fresh consideration to the ancient knowledge and independently furthered it. In arithmetic he showed superior ability in computations. Moreover, he offered material to his readers in a systematic way and ordered his examples from the easier to the more difficult …
With Leonardo a new epoch in Western mathematics began … Leonardo became the teacher of the masters of computation (the maestri d’abbaco) and of the surveyors, as one learns from the Summa of Luca Pacioli, who often refers to Leonardo. These two chief works were copied from the fourteenth to the sixteenth centuries.13
A similar conclusion was also expressed in no uncertain terms by the mathematical historian Ivor Grattan-Guinness: “The immediate origin of this tradition lies unquestionably in the Latin Liber abbaci (‘Book of the Abbacus’) written in 1202 by Leonardo of Pisa—more familiar to modern readers by his nickname, Fibonacci (Boncompagni 1857).”14
Finally, Laurence Sigler, the mathematician who translated Liber abbaci into English, says in his introduction:
For three centuries or so a curriculum based upon Leonardo’s Liber abaci was taught in Tuscany in schools of abaco normally attended by boys intending to be merchants or by others desiring to learn mathematics. Other instructors and some very good mathematicians also wrote books of abaco for use in the school. These books vary from primitive rule manuals up to mathematics books of quality, but none was so comprehensive, theoretical, and excellent as the Liber abaci of Leonardo Pisano.15
These modern scholars were presumably influenced by the few abbacus books that do have sections resembling Liber abbaci; for example, two mid-fifteenth-century Florentine treatises bearing the common title Praticha d’arismetricha, one written around 1450 by an anonymous Florentine master who describes himself as a pupil of Domenico d’Aghostino, the other written about ten years later by Benedetto of Florence.34 Both treatises provided exhaustive summaries of the type of mathematics that was being taught in the abbacus schools, in the same way as Liber abbaci described the type of mathematics being developed in the Arabian countries that Leonardo had visited. Both authors demonstrate their thorough knowledge of Leonardo’s works, which are often cited and from which they quote large excerpts. In the introduction, the anonymous author of the 1450 work places Leonardo first in the list of authors “to be considered” and invokes the Pisan’s authority in many passages of the work. The fifth part of the treatise, “where we will study cases of enjoyment,” is taken entirely out of chapter 12 of Liber abbaci (as the author states). Also the eighth part, “which contains the calculation of roots,” includes vernacular translations of passages in chapter 14 of Liber abbaci. The final section of the treatise, devoted to algebra, presents “cases written by Lionardo Pisano, perfect arithmetician,” and covers problems in the third part of chapter 15 of Liber abbaci. This section opens with the following declaration:
Lionardo Pisano, as he is known for a text in the great volume titled Praticha d’arismetrica, studied in Egypt and there perfected his knowledge by investigating the subject. And in these Tuscan regions, he was the first to bring knowledge and to announce the rules. And this is announced through the words of Master Antonio in the book of his “Fioretti,” where he demonstrates that the intellect of said Lionardo from Pisa is great. Lionardo wrote many books for our science, among which were the ones I know, namely: the Libro di merchaanti detto di minor guisa, the Libro de’ fiori, the Libro de’ numeri quadrati, the Libro sopra il 10° d’Euclide, the Libro di Praticha di geometria, the Libro di Praticha d’arismetricha from which I took what I want to write about presently. And these works are in Santo Spirito and in Santa Maria Novella, and also in the Abbey of Florence, and they are highly regarded by many of our citizens.16
Clearly, with these two manuscripts, we see a link to Liber abbaci, but both were published 250 years after Leonardo’s text, so neither can have played a role in initiating the abbacus movement. Though most scholars believed that all the abbacus books were descendants of Liber abbaci, no one could show how the transmission had taken place for the texts that did not resemble Leonardo’s masterpiece—the vast majority. A crucial first step was missing: Someone—and the simplest explanation was a single individual—must have taken material presented in Liber abbaci in a highly sophisticated way, and reformulated it in the much simpler fashion found in the abbacus books. That key missing text, if it existed, would have been the very first abbacus book, and the true spark that ignited the arithmetic revolution.
A crucial clue came from looking not for similarities between Liber abbaci and the abbacus books but at a major difference. A feature of many abbacus books that is not found in Liber abbaci is the inclusion of a section on geometry, where methods are described for finding lengths or areas or volumes of geometric figures, with examples typically being cast in terms of calculating the height of a tower by triangulation, the area of a field, or the volume of a pond or a barrel. What scholars found particularly curious about this departure from Liber abbaci is that it was the only one. When they started to ask themselves how it could have come about, they found themselves on a path that would eventually lead to the missing link.
Where did the abbacus authors get the material on geometry? Leonardo had touched on some geometric issues in chapter 15 of Liber abbaci in order to use geometric methods to solve problems in arithmetic and algebra, but left a more comprehensive treatment of geometry to another book, De practica geometrie, completed in 1220. Comparing the geometry in abbacus books with that in De practica geometrie revealed the same similarities and differences as those between the arithmetic in the abbacus books and that in Liber abbaci. In other words, the first abbacus book—if there was a single source—was almost certainly a simplified abridgement of not one of Leonardo’s books (Liber abbaci) but two, Liber abbaci and De practica geometrie. The historian Elisabetta Ulivi summarized the evidence in 2002: “[The abbacus books] were written in the vernaculars of the various regions, often in Tuscan vernacular, taking as their models the two important works of Leonardo Pisano, the Liber abaci and the Practica geometrie.”17 If someone wrote an initial, introductory-level summary of Leonardo’s two masterworks that began the abbacus tradition, that person would have to have been not only proficient in Latin but also sufficiently skilled in mathematics to read and understand the contents of two long, dense, and highly sophisticated scholarly texts, select some of their contents, and recast the material in a much simpler fashion.
The most obvious candidate was Leonardo himself, and it was known that he did write a simplified version of Liber abbaci. On a number of occasions, he referred to having written a Liber minoris guise (Book in a smaller manner). One such reference is in Liber abbaci itself,18 another is in his Liber quadratorum, and a third in his Flos. A further reference to the same lost work occurs in the fifteenth-century abbacus book Praticha d’arismetricha,19 whose anonymous author referred to it as Libro di minor guisa o libro di merchaanti (Book in a smaller manner or book for merchants). That description of it as a “book for merchants” is significant, since it suggests that the Libro di minor guisa would most likely have comprised material from the first ten chapters of Liber abbaci together with parts of De practica geometrie. But without a surviving copy of Leonardo’s “book in a smaller manner”, the evidence, while highly suggestive, was not conclusive.
Then, in 2003, the Italian scholar Rafaella Franci published the results of a remarkable study of a manuscript she came across in the Biblioteca Riccardiana in Florence. The manuscript is anonymous and occupies pages 1 to 178 of the library’s codex 2404. The work itself is undated, but dates in some of the problems place its writing at around 1290, and the vernacular language used places it in the Umbria region.20 It may be the earliest vernacular manuscript that has survived to this day. Its author began with the declaration: “This is the book of abacus according to the opinion of master Leonardo of the house of sons of Bonaçie from Pisa.” As always, the author’s reference to Leonardo cannot be taken as evidence that he used any of the Pisan’s writings as sources, though in this case, with the book written not long after Leonardo was alive, the pronouncement should be given some credence.
The book (known as Livero de l’abbecho [Book of abbacus] from the anonymous author’s introduction) is divided into thirty-one short chapters:
1. Three things
2. Things which are sold by the hundreds
3. The pepper rules
4. Rules with no name
5. Merchants’ rules of exchange
6. Rules of barter of money and currency
7. Rules of “marcho Tresçe”
8. How many “cantara” and “charrubbe” and “grane” are in one uncia
9. Buying “bolçone” with currency and in pounds
10. Rules to “consolare” and alloy coins
11. Various rules belonging to “consolare”
12. Rules of interest or usury
13. Rules belonging to the rule of usury
14. Rules of “saldare ragione”
15. Rules of the companies
16. On buying horses
17. On men passing along between one another
18. On men who find purses
19. On men who picked up currency together
20. On rule of “procachio” or travel
21. On men who went earning money at the markets
22. On a cup and its bottom
23. On trees and wooden “vogle”
24. On vases
25. On men who walk together
26. On men who brought pearls for sale in Constantinople
27. On vats and barrels in which the wine comes out of an opening in the bottom
28. On a man who sent his son to Alexandria
29. On a worker who worked on a product
30. On men who walk one after the other
31. On the rules for many tough and light “guise” of many “contintione.”
It is not an original work. Roughly three-quarters of the problems are faithful translations into the vernacular of problems in chapters 8, 9, 10, and 11 of Liber abbaci. Of particular note to a modern reader, on leaf c.107v, Fibonacci’s famous rabbit problem appears, though recast in terms of pigeons. The book was clearly written for a wide audience, since the author began each problem type with simpler problems than those in Liber abbaci. Overall, the treatment is both shorter and less complete than that in Liber abbaci. The author omitted all of the introductory material that occupies the entire first part of Leonardo’s masterpiece, dealing with the Hindu–Arabic representation of numbers, the algorithms for operations, and the methods for calculating with fractions, and proceeded as if these topics were well known to the reader. He began instead by describing the Rule of Three, which he stated thus:
If there is given any calculation, and if that calculation implies three things, we should multiply the thing we want to know by the thing that is not similar and divide by the other.
Franci pointed out two features important to her analysis. First, the author provided four examples to show how the rule works, the first where the known numbers are all integers, the others involving one, two, and three mixed numbers, respectively. Second, the margins contained tables that summarized the operations to be done, but the diagrams were not explained in the text.
Leonardo’s treatment of the Rule of Three in Liber abbaci was very different; he stated it in the most general manner where there are four proportional numbers, three of which are known and one of which needs to be determined; he emphasized the homogeneity of the units of measurement of similar items; and, at the end of the treatise, he provided a rationale for the rule by the theory of proportion. The unknown Umbrian did none of these. Moreover, Leonardo introduced a diagram to proceed more quickly in the calculations, analogous to the tables in the margins of Livero de l’abbecho, but unlike the Umbrian he went on to explain in great detail the way in which it was created and employed. Moreover, the examples in Liber abbaci are different from and more complex than the ones in Livero de l’abbecho.
The topics presented by the Umbrian author in the next three chapters—“things which are sold by the hundreds,” “the pepper rules,” and “the fabrics that are sold by the channa and arm”—are also present in Liber abbaci, but again are discussed in a simpler fashion, using different examples. Also, whereas Leonardo dealt with problems of merchandise pricing, doubtless with an eye on the international wholesale trade, the Umbrian wrote for a readership more interested in local and retail trading. The same is true of the longer, fifth chapter, devoted to the “rules of exchange”; while Liber abbaci had monetary examples from many places, the Umbrian author referred only to the coinage of central and northern Italy, with the exception of one problem referring to coins from Paris.
Up to this point in Franci’s analysis, it would have been possible—though highly unlikely—for the anonymous Umbrian to have taken his material from Liber abbaci. But the most intriguing aspect of his book—the inclusion of three chapters (12, 13, and 14) on calculating interest and depreciation—makes clear that Liber abbaci could not have been the Umbrian’s source. In Liber abbaci, there is hardly any mention of this topic. In fact, in chapter 14 of Livero de l’abbecho, “Rules of ‘saldare ragione’ ”, the author introduced a topic completely absent from Leonardo’s treatise, dealing with questions about mercantile practice, including how to handle payments made on different dates. (The problem he gave makes reference to the years 1288, 1289, and 1290, which is how the work can be dated.) This material was obviously included to make the treatise more useful to merchant readers, but from where did he take it?
Throughout the book, the author gives no indication of any particular mathematical skill, nor of the mathematical knowledge that would be required to make intelligent selections of what to include and what not. He must have simply copied the entire book from another work, with at most minor changes. That other work, Franci suggests, could be none other than Leonardo’s now lost Libro di merchaanti detto di minor guisa (Smaller book for merchants). Part of the reason why such a conclusion can be made with such confidence stems from the nature of mathematics. There is no evidence of anyone living at that time other than Leonardo who had the mathematical ability to write an original work of that kind. With such a demand for arithmetic knowledge, anyone capable of writing a good text would have quickly found an audience and become known.
Taking Franci’s supposition to be correct—and by this stage in her investigation the case she had assembled was as good as one can expect in medieval literary forensics, though more evidence was to come—Livero de l’abbecho provides a close copy of the Libro di minor guisa,35 thereby establishing Leonardo’s lost work as the original source of the hundreds of abbacus books that followed.
Franci’s conclusion has now been supported by examinations of other texts written in the first half of the fourteenth century. Despite the fact that they make no meaningful references to Leonardo, they exhibit striking similarities with the Umbrian’s treatise, especially when it comes to the explanation of the Rule of Three.
Leonardo may have initiated the abbacus book tradition not only by writing his Libro di minor guisa, Franci suggests, but also by lecturing on abbacus methods in Pisa. It is possible, she says, that the first vernacular abbacus book was Libro di nuovi conti (Book of new calculations), written around 1260 in Siena by Ugo Ugurgeri, a Cistercian monk from San Galgano, but no copy has survived.21 Since Siena and Pisa are very close, its author could have had access to a copy of Libro di minor guisa. This supposition gains further support from examinations of the contents of abbacus manuscripts written in Pisa,36 despite the fact that the earliest known date from the late 1300s, over a century after Leonardo’s death.22
For example, in one interesting Pisan text found in manuscript 2186 of the Biblioteca Riccardiana of Florence, the author introduced himself as (c.9v):
I Cristofano di Gherardo di Dino, Pisan citizen of the St. Bastiano chapel in the Chinsica quarter of Pisa, today, May first 1442, with God’s name and His salvation began to write this Book of abbacus.
This is the manner and the way to teach the abbacus in the style of Pisa, that is the beginning, middle and end as we will explain later.
Though Cristofano never mentioned Leonardo by name, the book appears to be a vernacular presentation of some parts of Leonardo’s work. Other manuscripts too indicate that Leonardo’s influence in Pisa was still strong two centuries after his death, with Pisan authors adopting a common way of explaining the abbacus methods. Franci concluded her seminal article: “From our analysis, it appears that, despite the small number of citations of Leonardo Fibonacci in abbacus treatises, the influence of his works—particularly Liber abbaci and Libro di merchaanti—is absolutely clear both in the contents and in the form of the texts, which follow the two typologies that he introduced: one more didactic and the other more scientific.”23
Taken all together, the evidence is overwhelming. Leonardo left two important legacies: One, comprising his scholarly books Liber abbaci, De practica geometrie, and Liber quadratorum, would lead to the development of modern mathematics. The other, his Libro di minor guisa, provided the template for all the abbacus books and the associated growth of practical, commercial arithmetic. Leonardo of Pisa started the modern arithmetic revolution.
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Futher Evidence from the Pisan Manuscripts
Perhaps the most interesting of the Pisan manuscripts that support Franci’s conclusions is the anonymous Tractato dell’arismetica.24 The dates 1270–71 and 1315–16 appear in some calculations, which, together with the Pisan coinage and measures referred to, suggests that it was written there between the end of the thirteenth and the beginning of the fourteenth century. Unlike the Livero de l’abbecho, this later manuscript begins, like Liber abbaci, with an explanation of the representation of numbers in the positional system, use of the hands to do calculations, and a description of algorithms for the four basic arithmetic operations. With those basics out of the way, the topics covered are: Rule of Three Things, ratios of “merito”, 6 “chaselle” of currency exchange, the 13 “chaselle”, ratios of “regare a termine”, discount rules, payments from one town to another, buying from different places, currency exchange, different ways of bartering, rules of the companies, rules of “consolare monete”, “merito” on New Year’s, ratios of the greater thing, how to calculate the root of numbers, and grouping numbers. The presentation of the decimal numeral system and the representation of numbers with the hands are very detailed and accurate. The algorithms taught for the multiplication and division are those presented by Leonardo in Liber abbaci, which the anonymous Pisan author calls moltiplicare in croce and partire a danda, respectively. They are presented through several examples, accompanied by detailed explanations.
The Rule of Three is stated in a sophisticated fashion (c.8r):
The rule for doing all the calculations in which there are three things, either currency or weight or measure, is to multiply the thing we want to know by the one that is not the same and divide by the other. One must also keep in mind that two of those three things have to be similar either by name or by substance or, if they are not, they must be adapted, and if they cannot be adapted, then that calculation cannot be done.
This explanation is similar to the Umbrian author’s, but the later Pisan stresses the homogeneity of the measures (as did Leonardo in Liber abbaci). Overall the Pisan’s treatise is more complete and better organized than that of the earlier Umbrian, but the many commonalities suggest the same source, providing further evidence that both authors took their material from Leonardo’s Libro di minor guisa. A significant difference between the two texts is the inclusion in the Pisan’s manuscript of a chapter on algebra, which Livero de l’abbecho does not cover. The author’s treatment of algebra differs from that presented in Liber abbaci, which leads to the supposition that Leonardo included in Libro di minor guisa a treatment of algebra that was simpler and more similar to that which subsequently appeared in Trattato dell’arismetica. Certainly, all the presentations of algebra that are found in the abbacus books are similar to the one by the anonymous Pisan.