After Leonardo’s important role in the spread of Hindu-Arabic arithmetic became known in the nineteenth century, scholars began to look for the exact written sources he had consulted in writing Liber abbaci. Trying to identify source materials written more than eight hundred years ago is inevitably problematic, since many of them may have been lost. To be sure, more authorative sources were more likely to be copied, which increased the chance of the work’s survival, but the fact remains that the most historians can do is identify and study sources, or likely sources, among those works that did survive in one form or another.1
What seems certain is that Leonardo consulted many sources to write Liber abbaci, both Latin and Arabic. Occasionally a particular source can be identified with some confidence; for example, his notation for ascending continued fractions came from the Maghreb mathematical school. But for the most part, historians can only speculate on what manuscripts he read.
The earliest extant Arabic work on Hindu arithmetic is the Kitab al-fusul fi’l-hisab al-hindi (Book of chapters on Hindu arithmetic) of Abu’l-Hasan Ahmad ibn Ibrahim al-Uqlidisi, composed in Damascus in 952–53 CE, but that survives only as a manuscript copy written more than two centuries later, in 1186. In fact, Latin manuscripts provide most of the early examples of place-value numerals usage outside India. Thus some of Leonardo’s written sources for Hindu-Arabic arithmetic may have been in Latin, the oldest surviving being a copy of al-Khwārizmī’s Arithmetic. Another available Latin treatise on the system was Liber ysagogorum alchorismi, which may have been written by Adelard of Bath. Unlike the majority of the Italian abbacus books, these earlier works were written by, and to a large extent for, scholars. It is possible that Leonardo read or consulted one, and perhaps several such treatises in preparing his description of Hindu-Arabic arithmetic.
Leonardo most definitely based his treatment of algebra in Liber abbaci on al-Khwārizmī’s Algebra. It may not have been the book from which the young Pisan first learned algebra while he was in North Africa, however, since that work was not available in the Maghreb, despite its wide circulation in al-Andalus. Instead, his first source may have been Abū Kāmil’s Kitāb fīl-jabr wa’l muqābala (Book on algebra). Nevertheless, it is clear that when Leonardo subsequently wrote the more advanced, algebra sections in Liber abbaci, he relied heavily on al-Khwārizmī’s masterpiece, almost certainly a Latin translation to which he had access in Italy.
The Algebra was translated into Latin by Robert of Chester in 1145, by Gherado of Cremona (arguably the greatest translator of the twelfth century, who lived from 1114 to 1187) around 1150, and by Guglielmo de Lunis around 1250. Gherardo’s translation is generally regarded as the best and was the most widely used. He titled it Liber maumeti filii moysi alchoarismi de algebra et almuchabala.2 When the present-day scholar Nobuo Miura compared passages in both Liber abbaci and Gherardo of Cremona’s Latin translation of Algebra, she found that many of the ninety problems in Leonardo’s chapter on algebra came directly from al-Khwārizmī’s text, demonstrating that Leonardo made use of that particular translation.3
One of the difficulties facing the medieval historian is illustrated by the confusion in the literature about al-Khwārizmī’s full name. Most present-day sources give it as Abū 
bdallāh Mu
ammad ibn Mūsā al-Khwārizmī, which can be translated as “Father of ‘Abdallāh, Mohammed, son of Moses, native of the town of al-Khwārizmī”.22 The form parallel to Leonardo Pisano (Leonardo of Pisa) would therefore be Muammad al-Khwārizmī (Muhammad of Khwārizmī), and the one parallel to Leonardo filius Bonacci Pisano (Leonardo, son of Bonacci, of Pisa) would be Muammad ibn Mūsā al-Khwārizmī (Muhammad, son of Moses, of Khwārizmī). This last is the form most present-day scholars use.
Naming conventions are not the only challenge facing the archivist. There are also references in the literature, both ancient and modern, to Abū Ja’far Muammad ibn Mūsā al-Khwārizmī. This could have resulted from an erroneous transcription by a careless inattentive scribe, or perhaps Muammad al-Khwārizmī had two children, one called Abdallāh, the other Ja’far. Among the sources who cite Abdallāh rather than Ja’far as the mentioned son is Frederic Rosen, who in 1831 published an English-language translation of al-Khwārizmī’s Algebra.4 In his preface, Rosen wrote: “ABU ABDALLAH MOHAMMED BEN MUSA, of Khowarezm, who it appears, from his preface, wrote this Treatise at the command of the Caliph AL MAMUN, was for a long time considered as the original inventor of Algebra.” That would seem to settle the matter. Rosen explained how the confusion arose. On page xi of his preface, he wrote, of the author of the famous algebra text: “He lived and wrote under the caliphat of AL MAMUN, and must therefore be distinguished from ABU JAFAR MOHAMMED BEN MUSA [whose father, Rosen colorfully tells us, was a bandit], likewise a mathematician and astronomer, who flourished under the Caliph AL MOTADED (who reigned A.H. 279–289, A.D. 892–902).”
Clearly, then, the two names referred to two different people. The other mathematician, Abu Ja’far Muhammad ibn Musa al-Khwārizmī, was one of three brothers, the “Sons of Musa” (Banū Mūsā), the others being named Ahmad and al-Hasan. But with both “Muammad al-Khwārizmī”s being mathematicians and astronomers, historians have had to exercise caution when citing the literature—particularly since the “father of” part (Abu ‘Abdallah or Abu Ja’far) is not found in most manuscripts.
Another tantalizing puzzle arises from Rosen’s remark that al-Khwārizmī “was for a long time considered as the original inventor of Algebra.” Rosen’s words seem to imply definitive knowledge that the famous Arab author was not the inventor of algebra, and that is indeed the case. On page vii of the preface, Rosen wrote: “From the manner in which our author [al-Khwārizmī], in his preface, speaks of the task he had undertaken, we cannot infer that he claimed to be the inventor. He says that the Caliph AL MAMUN encouraged him to write a popular work on Algebra: an expression which would seem to imply that other treatises were then already extant.”
In fact, algebra (as al-Khwārizmī described it in his book) was being transmitted orally and being used by people in their jobs before he or anyone else started to write it down. Several authors wrote books on algebra during the ninth century besides al-Khwārizmī, all having the virtually identical title Kitāb al-ğabr wa-l-muqābala. Among them were Abū Hanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muammad al-‘Adlī, Abū Yūsuf al-Mişşīşī, ‘Abd al-Hamīd ibn Turk, Sind ibn ‘Alī, Sahl ibn Bisšr, and Šarafaddīn al-Tūsī.
Al-Khwārizmī’s remark, as reported by Rosen in his preface, also states that al-Khwārizmī wrote his algebra book as “a popular work”, aimed at a much wider audience than just his fellow scholars. It is full of examples and applications to a wide range of numerical problems dealing with trade, surveying, and the highly complex issues of Islamic legal inheritance. Such a strong focus on applications was typical of Arabic algebra at the time.
The extent to which al-Khwārizmī and Leonardo filled their books with practical examples is not the only similarity between the two authors; another is the frustrating paucity of information about each. As Rosen wrote of al-Khwārizmī, “Besides the few facts which have already been mentioned in the course of this preface, little or nothing is known of our Author’s life.”5
Still another similarity between them is the uncertainty that has surrounded both their names. Although the confusion about al-Khwārizmī’s full name has finally been resolved—though the incorrect version continues to appear—for Leonardo, the intended meaning of that appended name “Bigollo” (see page 13) remains something of a mystery.
There is some disagreement as to al-Khwārizmī’s mathematical abilities. Did he have creative mathematical talent, or did he merely assemble and transcribe the works of others? Contemporary authorities disagree, saying variously:
[He was] the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time.
[Al-Khwārizmī] may not have been very original.
It is impossible to overstress the originality of the conception and style of al-Khwarīzmī’s algebra.
Al-Khwarīzmī’s scientific achievements were at best mediocre.6
None of these commentators argue that al-Khwārizmī’s two mathematics books were not hugely important. The disagreement is over his abilities as an original mathematician.7 In any event, regardless of how good al-Khwārizmī was at producing original mathematics, regardless of which of al-Khwārizmī’s books Leonardo consulted and to what extent, regardless of which works by others Leonardo consulted, and regardless of which other scholars Leonardo talked to—all factors of which we have little or no knowledge—what is beyond doubt is that the famous Pisan was a clear beneficiary of the work of al-Khwārizmī.
In his introduction to Algebra, al-Khwārizmī stated that the purpose of the book was to explain “what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.” He divided the text into three sections: the first part devoted to algebra, giving various rules together with thirty-nine worked problems, all abstract;8 then a short section on the Rule of Three (see page 74) and mensuration, in which two mensuration problems are solved with algebra; finally, a long section on inheritance problems solved by algebra.
The book begins with an observation about numbers that seems trivial to modern readers but was profound in al-Khwārizmī’s time:
When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand;… so forth to the utmost limit of numeration.
Understanding what al-Khwārizmī meant requires an appreciation that in his day numbers were regarded as different from quantities of length, a distinction still made in the seventeenth century when Newton invented calculus. The great Arabic mathematician was actually making an uncannily accurate prediction about the degree to which numbers would come to dominate mathematics.
The two words al-jabr and al-muqabala in al-Khwārizmī’s title refer to two steps in the simplification of equations. Al-jabr means “restoration” or “completion”, that is, removing negative terms, by transposing them to the other side of the equation to make them positive. For example, using one of al-Khwārizmī’s own examples, but expressing it with modern symbolic notation), al-jabr transforms
x2 = 40x – 4x2
into
5x2 = 40x.
Al-muqabala means “balancing” and is the process of eliminating identical quantities from the two sides of the equation. For example (again in modern notation), one application of al-muqabala reduces
50 + 3x + x2 = 29 + 10x
to
21 + 3x + x2 = 10x
and a second application reduces that to
21 + x2 = 7x.
These are the methods we use today to simplify and hence solve equations, which explains why a meaningful, modern English translation for al-Khwārizmī’s Arabic book title Hisâb al-Jabr wa’l-Muqâbala would be, simply, “Calculation with algebra”.
Today we interpret completion and restoration very differently from medieval mathematicians. The Arabs did not acknowledge negative numbers. For instance, they viewed “ten and a thing” (10 + x) as a composite expression that entailed two types of number (“simple numbers” and “roots”), but they did not see “ten less a thing” (10 – x) as composite. Rather, they thought of it as a single quantity, a “diminished” 10, or a 10 with a “defect” of x. The 10 retained its identity, even though x had been taken away from it. Thus, in a rhetorical equation like “ten less a thing equals five things” the “ten less a thing” was viewed as a deficient “ten” which needed to be restored, and the Arabic mathematicians would write “So restore the ten by the thing and add it to the five things” to get the equation “ten equals six things.” For confrontation, in an equation like “ten and two things equals six things”, they would “confront” the two things with the six things, which entailed taking their difference, to get the equation “ten equals four things”.9
Al-Khwārizmī, like Leonardo after him, developed his algebra in rhetorical fashion, using words, and would not have understood the symbolic derivations above. Arab mathematicians called the unknown quantity the “thing” (shay) or “root” (jidhr). The word jidhr means “the origin” or “the base”, also “the root of a tree”, and that may be the origin of our present-day expression “root of an equation”. (Our word “root” is a translation of the Latin word radix, but its connection to the Arabic is disputed.)
In addition to his two books on mathematics, al-Khwārizmī wrote a revised and completed version of Ptolemy’s Geography, consisting of a general introduction followed by a list of 2,402 coordinates of cities and other geographic features. He gave his book the title Kitāb şūrat al-Ar
(Book on the appearance of the Earth or The image of the Earth) and finished it in 833.23 The complete title of the Latin edition translates as “Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja’far Muhammad ibn Musa al-Khwārizmī, according to the geographical treatise written by Ptolemy the Claudian.” Once again, that incorrect name Abu Ja’far appears. Perhaps the copyist mistook him for Muhammad, one of the Musa brothers. This may in fact be the source of the present-day confusion about the name. The biographer G. J. Toomer probably consulted that Latin text to write the description of al-Khwārizmī for the Dictionary of Scientific Biography (New York, 1970–90), since the entry lists him as Abu Ja’far Muhammad ibn Musa al-Khwārizmī, and presumably it is from there that the error propagated through the literature.
An original work written in Latin, Leonardo’s Liber abbaci was clearly based in part on the earlier writings of al-Khwārizmī and other Arabic mathematicians. Other than his known use of Gherardo’s Latin translation of al-Khwārizmī’s Algebra, however, it is not clear whether Leonardo used Arabic manuscripts or Latin translations, or whether he read them in Bugia, elsewhere in North Africa, or in Italy after his return to Pisa. At that time, many Arabic texts had found their way to Europe, particularly Spain, where Latin translations were made—not just translations of original works by Arab mathematicians but also Arabic translations from the ancient Greek, including Euclid’s Elements and Ptolemy’s Almagest.
Much of the translation work was carried out in the area around the cathedral in the Spanish city of Toledo. Though all European scholars of the time knew Latin, few had mastered Arabic, so the translation was often done in two stages. One scholar—often a Jewish or Muslim scholar living in Spain—would make the translation from the Arabic to some common language, and a second scholar would then translate from that language into Latin. In the same way, many ancient Greek texts, from Aristotle to Euclid, were also translated into Latin.
In addition to Gherado of Cremona, who translated al-Khwārizmī’s Algebra, a colleague called “magister Iohanne”, or “magister Iohannes Hispalensis,” translated the Liber alchoarismi de pratica arismetice, the most complete exposition of Arabic arithmetic and algebra of the twelfth century. It is likely, though not certain, that the same Iohannes wrote Liber mahamalet, an original book on commercial arithmetic based on Arabic material.
Many of the Latin manuscripts produced in Toledo found their way to Italy. So even if Leonardo had no access to a particular Arabic text while on his travels through North Africa, he could have consulted it closer to home. Scholars today seem generally agreed that in writing De practica geometrie he made direct use of both Euclid’s Elements and Plato of Tivoli’s Liber embadorum (1145), which is based on the second book of al-Khwārizmī’s Algebra. (Plato is known only through his writings, at least some of which were produced in Barcelona between 1132 and 1146.)
Aside from al-Khwārizmī’s two books, there is less agreement about Leonardo’s other sources for Liber abbaci. One obvious possibility is that he had access to some of the Arabic texts—or Latin translations thereof—written after al-Khwārizmī. In particular, there are parallels between Liber abbaci and works of the Egyptian-born Abū Kāmil Shujā‘ ibn Aslam ibn Muammad ibn Shujā (ca. 850–ca. 930).10 Abū Kāmil’s Algebra appears to have been a major source for Leonardo’s treatment of algebra, not only in Liber abbaci but also in his other books De practica geometrie and Flos, and may have been the source from which he first learned algebra.11 Abū Kāmil’s book has seventy-four worked-out problems, and many of the more complicated ones, with identical solutions, are found in Liber abbaci. What is not clear is whether Leonardo used an Arabic text or a Latin translation.
Abū Kāmil was the first major Arabic algebraist after al-Khwārizmī. By all accounts he was a prolific author. There are references to works with the titles Book of fortune, Book of the key to fortune, Book of the adequate, Book on omens, Book of the kernel, Book of the two errors, and Book on augmentation and diminution. None of these have survived. Works that have include the Book on algebra, the Book of rare things in the art of calculation, and the Book on surveying and geometry.
Although al-Khwārizmī’s book was primarily intended for practitioners, he included some proofs for those interested in the reasons for some results. In his books, Abū Kāmil extended the range of geometric proofs. He was also the first to work freely with irrational coefficients.
Further advances in algebra were made in the Maghreb in the twelfth to fifteenth centuries, by a highly organized teacher-student network linked to mosque and madrassa teaching. The Maghrebs used abbreviations for both unknowns and their powers and for operations, an innovation that inspired parallel advances in Italian algebra, leading ultimately to the development of modern symbolic algebra.
Since Leonardo’s notation for ascending continued fractions comes from the Maghreb mathematical school, he likely had access to some of their writings, either in Arabic or in a Latin translation. It seems clear that he also consulted the Book on ratio and proportion of Ahmad ibn Yusuf ibn ad-Daya (Ametus filius Iosephi) and the Book on geometry by the Banū Mūsā. He also used problems from the Liber mahamalet.
Leonardo may have used other sources, but recent scholarship has ruled out one obvious candidate: he did not have Omar Khayyám’s Algebra at his disposal.12 Omar Khayyám is better known in the West today as a poet than a mathematician, but that reflects more on the values of today’s Western society than on the inherent merits of Khayyám’s work. While his poetry is competent, and liked by many, few would seriously claim it is on a par with the very best. His mathematical work, on the other hand, is first-rate. Born in 1048 in Nichapur, Persia (now Iran), Khayyám died there in 1131. As a young man he studied philosophy; by the time he was twenty-five, he had written books on arithmetic, algebra, and music. In 1070, he moved to Samarkand in Uzbekistan, and there he wrote his great work in algebra, an analysis of polynomial equations, titled Algebra wa al-muqabala (Proofs of algebra problems).
There is, then, some uncertainty regarding Leonardo’s sources for Liber abbaci. Historians have faced another challenge trying to determine the sequence of events that followed the book’s publication. In particular what role did Liber abbaci play in the arithmetic revolution that swept through Europe after Leonardo completed it. The one thing we know for sure is what Leonardo wrote in Liber abbaci itself—and it was considerable.